Help for package PoweR

Version:

1.1.4

Date:

2025-07-14

Title:

Computation of Power and Level Tables for Hypothesis Tests

Author:

Pierre Lafaye De Micheaux [aut, cre], Viet Anh Tran [aut], Alain Desgagne [aut], Frederic Ouimet [aut], Steven G. Johnson [aut]

Maintainer:

Pierre Lafaye De Micheaux <lafaye@unsw.edu.au>

Depends:

R (≥ 4.4.0), parallel, Rcpp

Description:

Computes power and level tables for goodness-of-fit tests for the normal, Laplace, and uniform distributions. Generates output in 'LaTeX' format to facilitate reporting and reproducibility. Explanatory graphs help visualize the statistical power of test statistics under various alternatives. For more details, see Lafaye De Micheaux and Tran (2016) <doi:10.18637/jss.v069.i03>.

License:

GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]

Packaged:

2025-07-19 23:15:56 UTC; lafaye

LinkingTo:

Rcpp, RcppArmadillo

NeedsCompilation:

yes

Repository:

CRAN

Date/Publication:

2025-07-22 10:20:13 UTC

Distributions in the PoweR package

Description

Random variate generation for many standard probability distributions are available in the PoweR package.

Details

The functions for the random variate generation are named in the form lawxxxx.

For the Laplace distribution see law0001.Laplace.

For the Normal distribution see law0002.Normal.

For the Cauchy distribution see law0003.Cauchy.

For the Logistic distribution see law0004.Logistic.

For the Gamma distribution see law0005.Gamma.

For the Beta distribution see law0006.Beta.

For the Uniform distribution see law0007.Uniform.

For the Student distribution see law0008.Student.

For the Chi-Squared distribution see law0009.Chisquared.

For the Log Normal distribution see law0010.LogNormal.

For the Weibull distribution see law0011.Weibull.

For the Shifted Exponential distribution see law0012.ShiftedExp.

For the Power Uniform distribution see law0013.PowerUnif.

For the Average Uniform distribution see law0014.AverageUnif.

For the UUniform distribution see law0015.UUnif.

For the VUniform distribution see law0016.VUnif.

For the Johnson SU distribution see law0017.JohnsonSU.

For the Tukey distribution see law0018.Tukey.

For the Location Contaminated distribution see law0019.LocationCont.

For the Johnson SB distribution see law0020.JohnsonSB.

For the Skew Normal distribution see law0021.SkewNormal.

For the Scale Contaminated distribution see law0022.ScaleCont.

For the Generalized Pareto distribution see law0023.GeneralizedPareto.

For the Generalized Error distribution see law0024.GeneralizedError.

For the Stable distribution see law0025.Stable.

For the Gumbel distribution see law0026.Gumbel.

For the Frechet distribution see law0027.Frechet.

For the Generalized Extreme Value distribution see law0028.GeneralizedExtValue.

For the Generalized Arcsine distribution see law0029.GeneralizedArcsine.

For the Folded Normal distribution see law0030.FoldedNormal.

For the Mixture Normal distribution see law0031.MixtureNormal.

For the Truncated Normal distribution see law0032.TruncatedNormal.

For the Normal with outliers distribution see law0033.Nout.

For the Generalized Exponential Power distribution see law0034.GeneralizedExpPower.

For the Exponential distribution see law0035.Exponential.

For the Asymmetric Laplace distribution see law0036.AsymmetricLaplace.

For the Normal-inverse Gaussian distribution see law0037.NormalInvGaussian.

For the Asymmetric Power Distribution see law0038.AsymmetricPowerDistribution.

For the modified Asymmetric Power Distribution see law0039.modifiedAsymmetricPowerDistribution.

For the Log-Pareto-tail-normal distribution see law0040.Log-Pareto-tail-normal.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Pierre Lafaye de Micheaux, Viet Anh Tran (2016). PoweR: A Reproducible Research Tool to Ease Monte Carlo Power Simulation Studies for Goodness-of-fit Tests in R. Journal of Statistical Software, 69(3), 1–42. doi:10.18637/jss.v069.i03

Goodness-of-fit tests for the Laplace distribution

Description

List of goodness-of-fit tests for the Laplace distribution.

Details

The statistic tests for the Laplace distribution are named in the form statxxxx.

For the Glen-Leemis-Barr test see stat0038.Glen.

For the 1st Rayner-Best statistic test see stat0039.Rayner1.

For the 2nd Rayner-Best statistic test see stat0040.Rayner2.

For the Anderson-Darling statistic see stat0042.AndersonDarling.

For the Cramer-von Mises statistic see stat0043.CramervonMises.

For the Watson statistic see stat0044.Watson.

For the Kolmogorov-Smirnov statistic see stat0045.KolmogorovSmirnov.

For the Kuiper statistic see stat0046.Kuiper.

For the 1st Meintanis statistic with moment estimators see stat0047.Meintanis1MO.

For the 1st Meintanis statistic with maximum likelihood estimators see stat0048.Meintanis1ML.

For the 2nd Meintanis statistic with moment estimators see stat0049.Meintanis2MO.

For the 2nd Meintanis statistic with maximum likelihood estimators see stat0050.Meintanis2ML.

For the 1st Choi-Kim statistic see stat0051.ChoiKim1.

For the 2nd Choi-Kim statistic see stat0052.ChoiKim2.

For the 3rd Choi-Kim statistic see stat0053.ChoiKim3.

For the Desgagne-Micheaux-Leblanc statistic see stat0054.DesgagneMicheauxLeblanc-Gn.

For the 1st Rayner-Best statistic see stat0055.RaynerBest1.

For the 2nd Rayner-Best statistic see stat0056.RaynerBest2.

For the Langholz-Kronmal statistic see stat0057.LangholzKronmal.

For the Kundu statistic see stat0058.Kundu.

For the Gulati statistic see stat0059.Gulati.

For the Gel statistic see stat0060.Gel.

For the 1st Gonzalez-Estrada and Villasenor test see stat0091.Gonzales1.

For the 2nd Gonzalez-Estrada and Villasenor test see stat0092.Gonzales2.

For the 1st Hogg test see stat0093.Hogg1.

For the 2nd Hogg test see stat0094.Hogg2.

For the 3rd Hogg test see stat0095.Hogg3.

For the 4th Hogg test see stat0096.Hogg4.

For the Rizzo and Haman test see stat0097.Rizzo.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Goodness-of-fit tests for normality.

Description

List of goodness-of-fit tests for normality.

Details

The statistic tests for normality are named in the form statxxxx.

For the Lilliefors statistic see stat0001.Lilliefors.

For the Anderson-Darling statistic see stat0002.AndersonDarling.

For the 1st Zhang-Wu statistic see stat0003.ZhangWu1.

For the 2nd Zhang-Wu statistic see stat0004.ZhangWu2.

For the Glen-Leemis-Barr statistic see stat0005.GlenLeemisBarr.

For the D'Agostino-Pearson statistic see stat0006.DAgostinoPearson.

For the Jarque-Bera statistic see stat0007.JarqueBera.

For the Doornik-Hansen statistic see stat0008.DoornikHansen.

For the Gel-Gastwirth statistic see stat0009.GelGastwirth.

For the 1st Hosking statistic see stat0010.Hosking1.

For the 2nd Hosking statistic see stat0011.Hosking2.

For the 3rd Hosking statistic see stat0012.Hosking3.

For the 4th Hosking statistic see stat0013.Hosking4.

For the 1st Bontemps-Meddahi statistic see stat0014.BontempsMeddahi1.

For the 2nd Bontemps-Meddahi statistic see stat0015.BontempsMeddahi2.

For the Brys-Hubert-Struyf statistic see stat0016.BrysHubertStruyf.

For the Bonett-Seier statistic see stat0017.BonettSeier.

For the Brys-Hubert-Struyf & Bonett-Seier statistic see stat0018.BrysHubertStruyf-BonettSeier.

For the 1st Cabana-Cabana statistic see stat0019.CabanaCabana1.

For the 2nd Cabana-Cabana statistic see stat0020.CabanaCabana2.

For the Shapiro-Wilk statistic see stat0021.ShapiroWilk.

For the Shapiro-Francia statistic see stat0022.ShapiroFrancia.

For the Shapiro-Wilk statistic modified by Rahman-Govindarajulu see stat0023.ShapiroWilk-RG.

For the D'Agostino statistic see stat0024.DAgostino.

For the Filliben statistic see stat0025.Filliben.

For the Chen-Shapiro statistic see stat0026.ChenShapiro.

For the 1st Zhang statistic see stat0027.ZhangQ.

For the 2nd Zhang statistic see stat0034.ZhangQstar.

For the 3rd Zhang statistic see stat0028.ZhangQQstar.

For the Barrio-Cuesta-Matran-Rodriguez statistic see stat0029.BarrioCuestaMatranRodriguez.

For the Coin statistic see stat0030.Coin.

For the Epps-Pulley statistic see stat0031.EppsPulley.

For the Martinez-Iglewicz statistic see stat0032.MartinezIglewicz.

For the Gel-Miao-Gastwirth statistic see stat0033.GelMiaoGastwirth.

For the Desgagne-LafayeDeMicheaux-Leblanc statistic see stat0035.DesgagneLafayeDeMicheauxLeblanc-Rn.

For the new CS Desgagne-LafayeDeMicheaux statistic see stat0036.DesgagneLafayeDeMicheaux-XAPD.

For the new CS Desgagne-LafayeDeMicheaux statistic see stat0037.DesgagneLafayeDeMicheaux-ZEPD.

For the Spiegelhalter statistic see stat0041.Spiegelhalter.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Pierre Lafaye de Micheaux, Viet Anh Tran (2016). PoweR: A Reproducible Research Tool to Ease Monte Carlo Power Simulation Studies for Studies for Goodness-of-fit Tests in R. Journal of Statistical Software, 69(3), 1–42. doi:10.18637/jss.v069.i03

Goodness-of-fit tests for uniformity

Description

List of goodness-of-fit tests for uniformity.

Details

The statistic tests for uniformity are named in the form statxxxx.

For the Kolmogorov statistic see stat0063.Kolmogorov.

For the Cramer-von Mises statistic see stat0064.CramervonMises.

For the Anderson-Darling statistic see stat0065.AndersonDarling.

For the Durbin statistic see stat0066.Durbin.

For the Kuiper statistic see stat0067.Kuiper.

For the 1st Hegazy-Green statistic see stat0068.HegazyGreen1.

For the 2nd Hegazy-Green statistic see stat0069.HegazyGreen2.

For the Greenwood statistic see stat0070.Greenwood.

For the Quesenberry-Miller statistic see stat0071.QuesenberryMiller.

For the Read-Cressie statistic see stat0072.ReadCressie.

For the Moran statistic see stat0073.Moran.

For the 1st Cressie statistic see stat0074.Cressie1.

For the 2nd Cressie statistic see stat0075.Cressie2.

For the Vasicek statistic see stat0076.Vasicek.

For the Swartz statistic see stat0077.Swartz.

For the Morales statistic see stat0078.Morales.

For the Pardo statistic see stat0079.Pardo.

For the Marhuenda statistic see stat0080.Marhuenda.

For the 1st Zhang statistic see stat0081.Zhang1.

For the 2nd Zhang statistic see stat0082.Zhang2.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Empirical distribution function of p-values.

Description

This function computes, at given points, the value of the empirical distribution function of a sample of p-values.

Usage

calcFx(pval.mat, x = c(seq(0.001, 0.009, by = 0.001), seq(0.01, 0.985,by = 0.005),
       seq(0.99, 0.999, by = 0.001)))

Arguments

pval.mat

matrix whose each column contains a vector of p-values for a given test statistic. The column names of this matrix should be set to the names of the various test statistics considered, whereas the rownames should all be set to the name of the distribution under which the p-values have been computed. This matrix can be obtained using function many.pval.

x

vector of points at which to evaluate the empirical distribution function.

Details

See equation (2) in Lafaye de Micheaux and Tran (2014).

Value

An object of class Fx is returned, which contains a list whose components are:

Fx.mat

matrix whose ith column contains the values of the empirical distribution function (evaluated at the points in vector x) of the p-values of the ith test statistic.

x

same vector x as input.

law

name of the distribution under which the p-values have been computed. Should correspond to the row names of pval.mat.

statnames

names of the test statistics. Should correspond to the column names of pval.mat.

N

number of p-values used.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

stind <- c(43, 44, 42)   # Indices of test statistics.
alter <-list(stat43 = 3, stat44 = 3, stat42 = 3) # Type for each test.
# Several p-values computed under the null.
pnull <- many.pval(stat.indices = stind, law.index = 1,
                        n = 100, M = 10, N = 10, alter = alter,
                        null.dist = 1,
                        method = "direct")$pvals
xnull <- calcFx(pnull)

Check proper behaviour of a random generator

Description

It is desirable to check if a newly added random generator coded in C behaves correctly. To perform this operation, one can superimpose the theoretical density on a histogram of the generated values.

Usage

checklaw(law.index, sample.size = 50000, law.pars = NULL, density =
NULL, trunc = c(-Inf, Inf), center = FALSE, scale = FALSE)

Arguments

law.index

index of the desired law, as given by getindex.

sample.size

number of observations to generate.

law.pars

vector of parameters for the law. The length of this parameter should not exceed 4. If not provided, the default values are used by means of getindex function.

density

a function of two arguments x and pars. Can also be a function of the arguments x and pars[1], ..., pars[k]. See the two examples below.

trunc

vector of left and right truncation thresholds for the generated sample values. Only those values in between will be kept to build the histogram. This can be useful for a distribution with extreme values.

center

Logical. Should we center the data.

scale

Should we scale the data.

Value

Returns invisibly the data generated and make a plot showing histogram and density superimposed.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

dlaplace1 <- function(x, mu, b) {dexp(abs(x - mu), 1 / b) / 2}
checklaw(1, density = dlaplace1)
dlaplace2 <- function(x, pars) {dexp(abs(x - pars[1]), 1 / pars[2]) / 2}
checklaw(1, density = dlaplace2)


checklaw(law.index = 2, sample.size = 50000, law.pars = c(2, 3), density
= dnorm)

## We use the 'trunc' argument to display the density in a region where
## no extreme values are present.
checklaw(27, density = dlaw27, trunc = c(-Inf,10))

# This one (Tukey) does not have a closed form expression for
# the density. But we can use the stats::density() function as
# follows.
res <- checklaw(18)
lines(density(res$sample), col = "blue")

Computation of the quantile values only for one test statistic.

Description

Functions for the computation of the quantile values only for one test statistic at a time and also one n value.

Usage

compquant(n,law.index,stat.index,probs=NULL,M=10^5,law.pars=NULL,
          stat.pars=NULL,model=NULL,Rlaw=NULL,Rstat=NULL,
center=FALSE, scale=FALSE)

Arguments

n

number of observations for each sample to be generated; length(n)=1. This can also be set to 0 if you want to use your own function using the 'Rstat' argument (see below).

law.index

law index as given by getindex; length(law.index)=1.

stat.index

stat index as given by getindex; length(stat.index)=1.

probs

If not NULL, should be a vector of levels from which to compute the quantile values. If NULL, the levels 0.025,0.05,0.1,0.9,0.95,0.975 will be used.

M

Number of Monte Carlo repetitions to use.

law.pars

NULL or a vector of length at most 4 containing 4 possible parameters to generate random values from distribution law(\code{law.pars}[j],j<=4). If NULL, the default parameter values for the law specified by law.index will be used.

stat.pars

A vector of parameters. If NULL, the default parameter values for the statistic specified by this stat.index will be used.

model

NOT YET IMPLEMENTED. If NULL, no model is used. If an integer i>0, the model coded in the C function modelei is used. Else this should be an R function that takes three arguments: eps (vector of \epsilon values), thetavec (vector of \theta values) and xvec (vector or matrix of x values). This function should take a vector of errors, generate observations from a model (with parameters thetavec and values xvec) based on these errors, then compute and return the residuals from the model. See file modele1.R in directory inst/doc/ for an example in multiple linear regression.

Rlaw

The user can provide its own (random generating) R function using this parameter. In this case, 'law.index' should be set to 0.

Rstat

If 'stat.index' is set to 0, an R function that outputs a list with components 'statistic' (value of the test statistic), 'pvalue' (pvalue of the test; if not computable should be set to 0), 'decision' (1 if we reject the null, 0 otherwise), 'alter' (see above), 'stat.pars' (see above), 'pvalcomp' (1L if the pvalue can be computed, 0L otherwise), 'nbparstat' (length of stat.pars).

center

Logical. Should we center the data generated

scale

Logical. Should we center the data generated

Value

A list with M statistic values and also some quantiles (with levels 0.025,0.05,0.1,0.9,0.95,0.975), as well as the name of the law and the name of the test statistic used (just to be sure!).

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

compquant(n=50,law.index=2,stat.index=10,M=10^3)$quant
compquant(n=50,law.index=0,stat.index=10,M=10^3,Rlaw=rnorm)$quant

Create a list giving the type of test statistics.

Description

Create a list giving the type of each test statistic for a given vector of indices of these test statistics.

Usage

create.alter(stat.indices = c(42, 51, 61), values.alter = NULL)

Arguments

stat.indices

vector of indices of test statistics, as given by function getindex.

values.alter

vector of the type of each test statistic in stat.indices. If NULL, the default value will be returned.

Details

See Section 3.3 in Lafaye de Micheaux, P. and Tran, V. A. (2014).

Value

A named list. Each component of the list has the name of the corresponding index in stat.indices (e.g. statxxx) and has the value (in {0,1,2,3,4}) of the type of test (see Details above).

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

create.alter()

Density function

Description

Evaluate the density function at a vector points.

Details

Use the function by typing:

dlawj(x,par1,par2,etc.)

where j is the index of the law and par1, par2, etc. are the parameters of law j.

The indicator function takes a vector x of length n as first argument and two real values a<b. It returns a vector of length n which contains only 0s and 1s (1 if the corresponding value in x is strictly between a and b).

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Generate random samples from a law added in the package.

Description

Generate random samples from a law added in the package as a C function.

Usage

gensample(law.index,n,law.pars = NULL,check = TRUE, center=FALSE, scale=FALSE)

Arguments

law.index

law index as given by function getindex.

n

number of observations to generate.

law.pars

vector of parameters for the law. The length of this parameter should not exceed 4.

check

logical. If TRUE, we check if law.index belongs to the list of laws. If FALSE, we pass on this verification, this will reduce the simulation time.

center

Logical. Should we center the data generated

scale

Logical. Should we center the data generated

Value

A list containing the random sample and the vector of parameters used for the chosen law.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

# This is good to check if the generator of the given law has been well coded.

res <- gensample(2,10000,law.pars=c(-5,2),check=TRUE)
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

# See function checklaw() in this package.
hist(gensample(2,10000,law.pars=c(0,1),check=TRUE)$sample,prob=TRUE,breaks=100,main="Density
histogram of the N(0,1) distribution")
curve(dnorm(x),add=TRUE,col="blue")

Get indices of laws and statistics functions.

Description

Print two correspondence tables between indices and random generators functions or test statistics functions programmed in C in this package. The first table gives indices/laws and the second one gives indices/statistics. These indices can be used in the functions powcomp.easy, powcomp.fast, compquant, gensample, statcompute, checklaw.

Usage

getindex(law.indices = NULL, stat.indices = NULL)

Arguments

law.indices

if not NULL, select only the laws corresponding to this vector of indices.

stat.indices

if not NULL, select only the stats corresponding to this vector of indices.

Value

A list with two matrices. The first one gives the correspondence between the indices and the laws (with also the number of parameters for each law as well as the default values). The second one gives the correspondence between the indices and the test statistics. Note that you can use the law.indices or stat.indices parameters of this function to obtain only some part of these tables of correspondence.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

getindex(1,c(4,3))

Retrieve the default number of parameters of some laws.

Description

Retrieve the default number of parameters of the distributions in the package.

Usage

getnbparlaws(law.indices = NULL)

Arguments

law.indices

vector of the indices of the distributions from which to retrieve the default number of parameters. If NULL, all the distributions will be considered.

Value

The default number of parameters for the laws specified in law.indices.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

## Default numbers of parameters for all the distributions in the package:
getnbparlaws()
## The Gaussian distribution has two parameters:
getnbparlaws(2)

Get numbers of parameters of test statistics.

Description

Return the default numbers of parameters of the test statistics in the package.

Usage

getnbparstats(stat.indices = NULL)

Arguments

stat.indices

if not NULL, select only the statistics corresponding to this vector of indices.

Value

A vector giving the numbers of parameters of test statistics corresponding to the vector of indices.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

getnbparstats(c(42:53))

p-value plot, p-value discrepancy plot and size-power curves.

Description

This function draws a p-value plot, a p-value discrepancy plot or a size-power curves plot.

Usage

graph(matrix.pval, xi = c(seq(0.001, 0.009, by = 0.001),
      seq(0.01, 0.985, by = 0.005), seq(0.99, 0.999, by = 0.001)),
      type = c("pvalue.plot", "pvalue.discrepancy", "size.power"),
      center = FALSE, scale = FALSE)

Arguments

matrix.pval

a matrix of p-values as returned by function many.pval.

xi

a vector of values at which to compute the empirical distribution of the p-values.

type

character. Indicate the type of plot desired.

center

Logical. Should we center the data generated

scale

Logical. Should we center the data generated

Details

See Section 2.3 in Lafaye de Micheaux, P. and Tran, V. A. (2014).

Value

No return value. Displays a graph.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

stind <- c(43, 44, 42)   # Indices of test statistics.
alter <-list(stat43 = 3, stat44 = 3, stat42 = 3) # Type for each test.
# Several p-values computed under the null.
# You can increase the values of M and N for better results.
matrix.pval <- many.pval(stat.indices = stind, law.index = 1,
                         n = 100, M = 10, N = 10, alter = alter, null.dist = 1,
                        method = "direct")
graph(matrix.pval)

Help Law

Description

Open directly the documentation for a specified law using its index.

Usage

help.law(law.index)

Arguments

law.index

law index as given by function getindex.

Value

No return value. The function opens the help page for the law corresponding to the given index.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Help Stat

Description

Open directly the documentation for a specified goodness-of-fit using its index.

Usage

help.stat(stat.index)

Arguments

stat.index

statistic index as given by function getindex.

Value

No return value. The function opens the help page for the test corresponding to the given index.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Gives information about a given law.

Description

To obtain the name of a law as well as its default number of parameters and default parameter values.

Usage

law.cstr(law.index, law.pars = NULL)

Arguments

law.index

a single integer value corresponding to the index of a distribution as given by function getindex.

law.pars

vector of the values of the parameters of the law specified in law.index. If NULL, the default values are used.

Details

This function can be useful to construct a title for a graph for example.

Value

name

name of the distribution with its parameters and the values they take.

nbparams

default number of parameters of the law.

law.pars

values of the parameters.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

law.cstr(2)

The Laplace Distribution

Description

Random generation for the Laplace distribution with parameters mu and b.

This generator is called by function gensample to create random variables based on its parameters.

Details

If mu or b are not specified they assume the default values of 0 and 1, respectively.

The Laplace distribution has density:

\frac{1}{2b}\exp \left( -\frac{|x-\mu|}{b} \right)

where \mu is a location parameter and b > 0, which is sometimes referred to as the diversity, is a scale parameter.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(1,10000,law.pars=c(9,2))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Normal Distribution

Description

Random generation for the Normal distribution with parameters mu and sigma.

This generator is called by function gensample to create random variables based on its parameters.

Details

If mu or sigma are not specified they assume the default values of 0 and 1, respectively.

The Normal distribution has density:

(\sqrt{2\pi}\sigma)^{-1}\exp^{-\frac{x^2}{2\sigma^2}}

where \mu is the mean of the distribution and \sigma is the standard deviation.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(2,10000,law.pars=c(9,2))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Cauchy Distribution

Description

Random generation for the Cauchy distribution with parameters location and scale.

This generator is called by function gensample to create random variables based on its parameters.

Details

If location or scale are not specified, they assume the default values of 0 and 1 respectively.

The Cauchy distribution has density:

\frac{1}{\pi s(1+(\frac{x-l}{s})^2)}

where l is the location parameter and s is the scale parameter, for all x.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(3,10000,law.pars=c(9,2))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Logistic Distribution

Description

Random generation for the Logistic distribution with parameters location and scale.

This generator is called by function gensample to create random variables based on its parameters.

Details

If location or scale are omitted, they assume the default values of 0 and 1 respectively.

The Logistic distribution with location = \mu and scale = s has distribution function

\frac{1}{1 + exp^{-\frac{(x-\mu)}{s}}}

and density

\frac{exp^{-\frac{(x-\mu)}{s}}}{s(1+exp^{-\frac{(x-\mu)}{s}})^2}

It is a long-tailed distribution with mean \mu and variance (\pi^2)/3 s^2.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(4,10000,law.pars=c(9,2))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Gamma Distribution

Description

Random generation for the Gamma distribution with parameters shape and rate.

This generator is called by function gensample to create random variables based on its parameters.

Details

If shape or rate are not specified they assume the default values of 2 and 1, respectively.

The Gamma distribution has density:

\frac{1}{b^a\Gamma(a)}x^{a-1}exp^{-x/b}

for x \ge 0, a > 0 and b > 0; where a is the shape parameter and b is the rate parameter.

Here \Gamma(a) is the gamma function implemented by R and defined in its help.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(5,10000,law.pars=c(9,2))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Beta Distribution

Description

Random generation for the Beta distribution with parameters shape1 and shape2.

This generator is called by function gensample to create random variables based on its parameters.

Details

If shape1 or shape2 are not specified they assume the default values of 1 and 1, respectively.

The Beta distribution with parameters shape1 = a and shape2 = b has density:

\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1}

for a > 0, b > 0 and 0 \le x \le 1 where the boundary values at x=0 or x=1 are defined as by continuity (as limits).

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(6,10000,law.pars=c(9,2))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Uniform Distribution

Description

Random generation for the Uniform distribution with parameters min and max.

This generator is called by function gensample to create random variables based on its parameters.

Details

If min or max are not specified they assume the default values of 0 and 1, respectively.

The Uniform distribution has density:

\frac{1}{max - min}

for min \le x \le max.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(7,10000,law.pars=c(2,9))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Student t Distribution

Description

Random generation for the Student t distribution with df degrees of freedom.

This generator is called by function gensample to create random variables based on its parameter.

Details

If df is not specified it assumes the default value of 1.

The t distribution with df = k degrees of freedom has density:

(\sqrt{k\pi})^{-1}\frac{\Gamma\left(\frac{k+1}{2} \right)}{\Gamma\left(\frac{k}{2} \right)}\left(1+\frac{t^2}{k} \right)^{-\frac{k+1}{2}}

for all real x. It has mean 0 (for k > 1) and variance k/(k-2) (for k > 2).

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(8,10000,law.pars=8)
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Chi-Squared Distribution

Description

Random generation for the Chi-squared distribution with df degrees of freedom.

This generator is called by function gensample to create random variables based on its parameter.

Details

If df is not specified it assumes the default value of 1.

The Chi-squared distribution with df = k degrees of freedom has density:

2^{-k/2}\Gamma(k/2)^{-1}x^{k/2-1}e^{-x/2}

for x > 0 and k \ge 1. The mean and variance are n and 2n.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(9,10000,law.pars=8)
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Log Normal Distribution

Description

Random generation for the Log Normal distribution whose logarithm has mean equal to meanlog and standard deviation equal to sdlog.

This generator is called by function gensample to create random variables based on its parameters.

Details

If meanlog or sdlog are not specified they assume the default values of 0 and 1, respectively.

The Log Normal distribution has density:

\frac{1}{x\sigma\sqrt{2\pi}}e^{-\frac{(\ln x-\mu)^2}{2\sigma^2}}

where \mu and \sigma are the mean and standard deviation of the logarithm. The mean is E(X) = exp(\mu + 1/2 \sigma^2) and the variance is Var(X) = exp(2*\mu + \sigma^2)*(exp(\sigma^2) - 1).

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(10,10000,law.pars=c(8,6))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Weibull Distribution

Description

Random generation for the Weibull distribution with parameters shape and scale.

This generator is called by function gensample to create random variables based on its parameters.

Details

If shape or scale are not specified they assume the default values of 1 and 1, respectively.

The Weibull distribution with shape parameter k and scale parameter \lambda has density given by

\frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^k}

for x > 0. The cumulative distribution function is F(x) = 1 - e^(-(x/\lambda)^k) on x > 0, the mean is E(X) = \lambda \Gamma(1 + 1/k), and the Var(X) = \lambda^2 * (\Gamma(1 + 2/k) - (\Gamma(1 + 1/k))^2).

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(11,10000,law.pars=c(8,6))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Shifted Exponential Distribution

Description

Random generation for the Shifted Exponential distribution with parameters l and rate.

This generator is called by function gensample to create random variables based on its parameters.

Details

If l or rate are not specified they assume the default values of 0 and 1, respectively.

The Shifted Exponential distribution has density

b\exp\{-(x-l)b\}

for x \le 1, where rate = b. The mean is E(X) = l + 1/b, and the Var(X) = 1/(b^2).

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(12,10000,law.pars=c(8,6))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Power Uniform Distribution

Description

Random generation for the Power Uniform distribution with parameter power.

This generator is called by function gensample to create random variables based on its parameter.

Details

If power is not specified it assumes the default value of 1.

The Power Uniform distribution has density:

\frac{1}{1+j}x^{-\frac{j}{j+1}}

where power = j.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Quesenberry and Miller (1977), Power studies of some tests for uniformity, Journal of Statistical Computation and Simulation, 5(3), 169–191 (see p. 178)

Examples

res <- gensample(13,10000,law.pars=8)
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Average Uniform Distribution

Description

Random generation for the Average Uniform distribution with parameters size, a and b.

This generator is called by function gensample to create random variables based on its parameter.

Details

If size, a and b are not specified they assume the default values of 2, 0 and 1.

The Average Uniform distribution has density:

\frac{k^k}{(k-1)!}\sum_{j=0}^{\lfloor k\frac{x-a}{b-a} \rfloor}(-1)^j{k \choose j}(\frac{x-a}{b-a}-\frac{j}{k})^{k-1}

where size = k and for a \le x \le b.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Quesenberry and Miller (1977), Power studies of some tests for uniformity, Journal of Statistical Computation and Simulation, 5(3), 169–191 (see p. 179)

Examples

res <- gensample(14,10000,law.pars=c(9,2,3))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The UUniform Distribution

Description

Random generation for the UUniform distribution with parameter power.

This generator is called by function gensample to create random variables based on its parameter.

Details

If power is not specified it assumes the default value of 1.

The UUniform distribution has density:

(2(1+j))^{-1}(x^{-j/(1+j)}+(1-x)^{-j/(1+j)})

where power = j.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Quesenberry and Miller (1977), Power studies of some tests for uniformity, Journal of Statistical Computation and Simulation, 5(3), 169–191 (see p. 179)

Examples

res <- gensample(15,10000,law.pars=9)
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The VUniform Distribution

Description

Random generation for the VUniform distribution with parameter size.

This generator is called by function gensample to create random variables based on its parameter.

Details

If size is not specified it assumes the default value of 1.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Quesenberry and Miller (1977), Power studies of some tests for uniformity, Journal of Statistical Computation and Simulation, 5(3), 169–191 (see p. 179)

Examples

res <- gensample(16,10000,law.pars=8)
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Johnson SU Distribution

Description

Random generation for the Johnson SU distribution with parameters mu, sigma, nu and tau.

This generator is called by function gensample to create random variables based on its parameters.

Details

If mu, sigma, nu and tau are not specified they assume the default values of 0, 1, 0 and 0.5, respectively.

The Johnson SU distribution with parameters mu = \mu, sigma = \sigma, nu = \nu and tau = \tau has density:

\frac{1}{c\sigma\tau}\frac{1}{\sqrt{z^2+1}}\frac{1}{\sqrt{2\pi}}e^{-r^2/2}

where r = -\nu + (1/\tau)sinh^-1(z), z = (x - (\mu + c*\sigma (\sqrt(\omega)) sinh(w)))/(c*\sigma), c = ((w-1)(w cosh(2\omega)+1)/2)^-1/2, w = e^(\tau^2) and \omega = -\nu\tau.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(17,10000,law.pars=c(9,8,6,0.5))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Tukey Distribution

Description

Random generation for the Tukey distribution with parameter lambda.

This generator is called by function gensample to create random variables based on its parameter.

Details

If lambda is not specified it assumes the default value of 1.

The Tukey distribution with lambda = \lambda has E[X] = 0 and Var[X] = 2/(\lambda^2) (1/(2\lambda+1) - \Gamma^2(\lambda+1)/\Gamma(2\lambda+2)).

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(18,10000,law.pars=8)
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Location Contaminated Distribution

Description

Random generation for the Location Contaminated distribution with parameters p and m.

This generator is called by function gensample to create random variables based on its parameters.

Details

If p or m are not specified they assume the default values of 0.5 and 0, respectively.

The Location Contaminated distribution has density:

\frac{1}{\sqrt{2\pi}}\left[pe^{-\frac{(x-m)^2}{2}}+(1-p)e^{-\frac{x^2}{2}}\right]

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(19,10000,law.pars=c(0.8,6))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Johnson SB Distribution

Description

Random generation for the Johnson SB distribution with parameters g and d.

This generator is called by function gensample to create random variables based on its parameters.

Details

If g and d are not specified they assume the default values of 0 and 1, respectively.

The Johnson SB distribution has density:

\frac{d}{\sqrt{2\pi}}\frac{1}{x(1-x)}e^{-\frac{1}{2}\left(g+d\ln\frac{x}{1-x} \right)^2}

where d > 0.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(20,10000,law.pars=c(8,6))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Skew Normal Distribution

Description

Random generation for the Skew Normal distribution with parameters xi, omega^2 and alpha.

This generator is called by function gensample to create random variables based on its parameters.

Details

If xi, omega^2 and alpha are not specified they assume the default values of 0, 1 and 0, respectively.

The Skew Normal distribution with parameters xi = \xi, omega^2 = \omega^2 and alpha = \alpha has density:

\left(\frac{2}{\omega}\right)\phi\left(\frac{x-\xi}{\omega}\right)\Phi\left(\alpha\left(\frac{x-\xi}{\omega}\right)\right)

where \phi(x) is the standard normal probability density function and \Phi(x) is its cumulative distribution function.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(21,10000,law.pars=c(8,6,2))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Scale Contaminated Distribution

Description

Random generation for the Scale Contaminated distribution with parameters p and d.

This generator is called by function gensample to create random variables based on its parameters.

Details

If p or d are not specified they assume the default values of 0.5 and 0, respectively.

The Scale Contaminated distribution has density:

frac{1}{\sqrt{2\pi}}\left[\frac{p}{d}e^{-\frac{x^2}{2d^2}}+(1-p)e^{-\frac{x^2}{2}}\right]

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(22,10000,law.pars=c(0.8,6))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Generalized Pareto Distribution

Description

Random generation for the Generalized Pareto distribution with parameters mu, sigma and xi.

This generator is called by function gensample to create random variables based on its parameters.

Details

If mu, sigma and xi are not specified they assume the default values of 0, 1 and 0, respectively.

The Generalized Pareto distribution with parameters mu = \mu, sigma = \sigma and xi = \xi has density:

\frac{1}{\sigma}\left(1+\frac{\xi(x-\mu)}{\sigma} \right)^{(-\frac{1}{\xi}-1)}

where x \ge \mu if \xi \ge 0 and x \le \mu - \sigma/\xi if \xi < 0.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(23,10000,law.pars=c(8,6,2))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Generalized Error Distribution

Description

Random generation for the Generalized Error distribution with parameters mu, sigma and p.

This generator is called by function gensample to create random variables based on its parameters.

Details

If mu, sigma and p are not specified they assume the default values of 0, 1 and 1, respectively.

The Generalized Error distribution with parameters mu = \mu, sigma = \sigma and p = p has density:

\frac{1}{2p^{1/p}\Gamma(1+1/p)\sigma}\exp\left[-\frac{1}{p\sigma^p}|x-\mu|^p\right]

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(24,10000,law.pars=c(8,6,2))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Stable Distribution

Description

Random generation for the Stable distribution with parameters stability, skewness, scale and location.

This generator is called by function gensample to create random variables based on its parameters.

Details

If stability, skewness, scale and location are not specified they assume the default values of 2, 0, 1 and 0, respectively.

The Stable distribution with parameters stability = \alpha, skewness = \beta, scale = c and location = \mu doesn't have an analytically expressible probability density function, except for some parameter values. The parameters have conditions : 0 < \alpha \le 2, -1 \le \beta \le 1 and c > 0.

The mean of Stable distribution is defined \mu when \alpha > 1, otherwise undefined.

The variance of Stable distribution is defined 2 c^2 when \alpha = 2, otherwise infinite.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(25,10000,law.pars=c(2,1,1,2))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Gumbel Distribution

Description

Random generation for the Gumbel distribution with parameters mu and sigma.

This generator is called by function gensample to create random variables based on its parameters.

Details

If mu or sigma are not specified, they assume the default values of 1.

The Gumbel distribution with parameters mu = \mu and sigma = \sigma has density:

\frac{1}{\sigma}\exp\left\{-\exp\left[-\left(\frac{x-\mu}{\sigma}\right)\right]-\left(\frac{x-\mu}{\sigma}\right)\right\}

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(26,10000,law.pars=c(9,2))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Frechet Distribution

Description

Random generation for the Frechet distribution with parameters mu, sigma and alpha.

This generator is called by function gensample to create random variables based on its parameters.

Details

If mu, sigma and alpha are not specified they assume the default values of 0, 1 and 1, respectively.

The Frechet distribution with parameters mu = \mu, sigma = \sigma and alpha = \alpha has density:

frac{\alpha}{\sigma}\left(\frac{x-\mu}{\sigma}\right)_{+}^{-\alpha-1}\exp\left\{-\left(\frac{x-\mu}{\sigma}\right)^{-\alpha}\right\}

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(27,10000,law.pars=c(8,6,2))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Generalized Extreme Value Distribution

Description

Random generation for the Generalized Extreme Value distribution with parameters mu, sigma and xi.

This generator is called by function gensample to create random variables based on its parameters.

Details

If mu, sigma and xi are not specified they assume the default values of 0, 1 and 1, respectively.

The Generalized Extreme Value distribution with parameters mu = \mu, sigma = \sigma and xi = \xi has density:

[1+z]_{+}^{-\frac{1}{\xi}-1}\exp\left\{-[1+z]_{+}^{-\frac{1}{\xi}}\right\}/\sigma

for \xi > 0 or \xi < 0, where z = \xi (x - \mu)/\sigma. If \xi = 0, PDF is as same as in the Gumbel distribution.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(28,10000,law.pars=c(8,6,2))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Generalized Arcsine Distribution

Description

Random generation for the Generalized Arcsine distribution with parameters alpha.

This generator is called by function gensample to create random variables based on its parameter.

Details

If alpha is not specified it assumes the default value of 0.5.

The Generalized Arcsine distribution with parameter alpha = \alpha has density:

\frac{\sin(\pi\alpha)}{\pi}x^{-\alpha}(1-x)^{\alpha-1}

for 0 < \alpha < 1.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(29,10000,law.pars=0.8)
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Folded Normal Distribution

Description

Random generation for the Folded Normal distribution with parameters mu and sigma.

This generator is called by function gensample to create random variables based on its parameters.

Details

If mu and sigma are not specified they assume the default values of 0 and 1, respectively.

The Folded Normal distribution with parameters mu = \mu and sigma = \sigma has density:

dnorm(x,mu,sigma2)+dnorm(x,-mu,sigma2)

for x \ge 0.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(30,10000,law.pars=c(8,6))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Mixture Normal Distribution

Description

Random generation for the Mixture Normal distribution with parameters p, m and d.

This generator is called by function gensample to create random variables based on its parameters.

Details

If p, m and d are not specified they assume the default values of 0.5, 0 and 1, respectively.

The Mixture Normal distribution has density:

p\frac{1}{d\sqrt{2\pi}}e^{-\frac{(x-m)^2}{2d^2}}+(1-p)\frac{1}{d\sqrt{2\pi}}e^{-\frac{x^2}{2}}

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(31,10000,law.pars=c(0.9,8,6))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Truncated Normal Distribution

Description

Random generation for the Truncated Normal distribution with parameters a and b.

This generator is called by function gensample to create random variables based on its parameters.

Details

If a and b are not specified they assume the default values of 0 and 1, respectively.

The Truncated Normal distribution with parameters mu = \mu and sigma = \sigma has density:

\frac{\exp(-x^2/2)}{\sqrt{2\pi}(\Phi(b)-\Phi(a))}

for a \le x \le b, where \phi(x) is the standard normal probability density function and \Phi(x) is its cumulative distribution function.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(32,10000,law.pars=c(2,3))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Normal with outliers Distribution

Description

Random generation for the Normal with outliers distribution with parameter a which belongs to {1,2,3,4,5}.

This generator is called by function gensample to create random variables based on its parameter.

Details

If a is not specified it assumes the default value of 1.

Five cases of standard normal distributions with outliers, hereon termed Nout1 to Nout5, consisting of observations drawn from a standard normal distribution where some of the values are randomly replaced by extreme observations.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Romao, X., Delgado, R. and Costa, A. (2010), An empirical power comparison of univariate goodness-of-fit tests for normality, Journal of Statistical Computation and Simulation, 80(5), 545–591.

Examples

res <- gensample(33,10000,law.pars=4)
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Generalized Exponential Power Distribution

Description

Random generation for the Generalized Exponential Power distribution with parameters t1, t2 and t3.

This generator is called by function gensample to create random variables based on its parameters.

Details

If t1, t2 and t3 are not specified they assume the default value of 0.5, 0 and 1, respectively.

The Generalized Exponential Power distribution has density:

p(x;\gamma,\delta,\alpha,\beta,z_0) \propto e^-{\delta|x|^\gamma} |x|^{-\alpha}(log|x|)^{-\beta}

for x \ge z_0, and the density equals to p(x;\gamma,\delta,\alpha,\beta,z_0) for x < z_0.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Desgagne, A., Lafaye de Micheaux, P. and Leblanc, A. (2013), Test of Normality Against Generalized Exponential Power Alternatives, Communications in Statistics - Theory and Methods, 42(1), 164–190.

Examples

res <- gensample(34,10000,law.pars=c(1,8,4))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Exponential Distribution

Description

Random generation for the Exponential distribution with rate rate (i.e., mean 1/rate).

This generator is called by function gensample to create random variables based on its parameter.

Details

If rate is not specified it assumes the default value of 1.

The Exponential distribution with rate = \lambda has density:

\lambda exp^{-\lambda x}

for x \ge 0.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(35,10000,law.pars=8)
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Asymmetric Laplace Distribution

Description

Random generation for the Asymmetric Laplace distribution with parameters mu, b and k.

This generator is called by function gensample to create random variables based on its parameters.

Details

If mu, b or k are not specified they assume the default values of 0, 1 and 2, respectively.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(36,10000,law.pars=c(9,2,6))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Normal-inverse Gaussian Distribution

Description

Random generation for the Normal-inverse Gaussian distribution with parameters shape, skewness, location and scale.

This generator is called by function gensample to create random variables based on its parameters.

Details

If shape, skewness, location and scale are not specified they assume the default values of 1, 0, 0 and 1, respectively.

The Normal-inverse Gaussian distribution with parameters shape = \alpha, skewness = \beta, location = \mu and scale = \delta has density:

\frac{\alpha\delta K_1(\alpha\sqrt{\delta^2+(x-\mu)^2})}{\pi\sqrt{\delta^2+(x-\mu)^2}}e^{\delta\gamma+\beta(x-\mu)}

where \gamma = \sqrt(\alpha^2 - \beta^2) and K_1 denotes a modified Bessel function of the second kind.

The mean and variance of NIG are defined respectively \mu + \beta \delta / \gamma and \delta \alpha^2 / \gamma^3.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

res <- gensample(37,10000,law.pars=c(3,2,1,0.5))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Asymmetric Power Distribution

Description

Random generation for the Asymmetric Power Distribution with parameters theta, phi, alpha and lambda.

This generator is called by function gensample to create random variables based on its parameters.

Details

If theta, phi, alpha and lambda are not specified they assume the default values of 0, 1, 0.5 and 2, respectively.

The Asymmetric Power Distribution with parameters theta, phi, alpha and lambda has density:

f(u) = \frac{1}{\phi}\frac{\delta^{1/\lambda}_{\alpha,\lambda}}{\Gamma(1+1/\lambda)}\exp\left[-\frac{\delta_{\alpha,\lambda}}{\alpha^{\lambda}}\left|\frac{u-\theta}{\phi}\right|^{\lambda}\right]

u\leq0

and

f(u) = \frac{1}{\phi}\frac{\delta^{1/\lambda}_{\alpha,\lambda}}{\Gamma(1+1/\lambda)}\exp\left[-\frac{\delta_{\alpha,\lambda}}{(1-\alpha)^{\lambda}}\left|\frac{u-\theta}{\phi}\right|^{\lambda}\right]

u\leq0,

where 0<\alpha<1, \lambda>0 and \delta_{\alpha,\lambda}=\frac{2\alpha^{\lambda}(1-\alpha)^{\lambda}}{\alpha^{\lambda}+(1-\alpha)^{\lambda}}.

The mean and variance of APD are defined respectively by

E(U) = \theta+\phi\frac{\Gamma(2/\lambda)}{\Gamma(1/\lambda)} [1-2\alpha]\delta_{\alpha,\lambda}^{-1/\lambda}

and

V(U) = \phi^2 \frac{\Gamma(3/\lambda)\Gamma(1/\lambda)[1-3\alpha+3\alpha^2]-\Gamma(2/\lambda)^2[1-2\alpha]^2}{\Gamma^2(1/\lambda)} \delta_{\alpha,\lambda}^{-2/\lambda}.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Komunjer, I. (2007), Asymmetric Power Distribution: Theory and Applications to Risk Measurement, Journal of Applied Econometrics, 22, 891–921.

Examples

res <- gensample(38,10000,law.pars=c(3,2,0.5,1))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The modified Asymmetric Power Distribution

Description

Random generation for the modified Asymmetric Power Distribution with parameters theta, phi, alpha and lambda.

This generator is called by function gensample to create random variables based on its parameters.

Details

If theta, phi, alpha and lambda are not specified they assume the default values of 0, 1, 0.5 and 2, respectively.

The modified Asymmetric Power Distribution with parameters theta, phi, theta1 and theta2 has density:

f(x\mid\boldsymbol{\theta})=\frac{(\delta_{\boldsymbol{\theta}}/2)^{1/\theta_2}}{\Gamma(1+1/\theta_2)}\exp\left[-\left(\frac{2(\delta_{\boldsymbol{\theta}}/2)^{1/\theta_2}}{1+sign(x)(1-2\theta_1)}|x|\right)^{\theta_2}\right]

where \boldsymbol{\theta}=(\theta_2, \theta_1)^T is the vector of parameters, \theta_2>0, 0<\theta_1<1 and

\delta_{\boldsymbol{\theta}}=\frac{2(\theta_1)^{\theta_2} (1-\theta_1)^{\theta_2}}{(\theta_1)^{\theta_2}+(1-\theta_1)^{\theta_2}}

The mean and variance of APD are defined respectively by

E(U) = \theta + 2 ^ {1 / \theta_2} \phi \Gamma(2 / \theta_2) (1 - 2 \theta_1) \delta ^ {-1 / \theta_2} / \Gamma(1 / \theta_2)

and

V(U) = 2 ^ {2 / \theta_2} \phi ^ 2 \left(\Gamma(3 / \theta_2) \Gamma(1 / \theta_2) (1 - 3 \theta_1 + 3 \theta_1 ^ 2) - \Gamma^2(2 / \theta_2) (1 - 2 \theta_1) ^ 2\right) \delta ^ {-2 / \theta_2} / \Gamma^2(1 / \theta_2).

Author(s)

P. Lafaye de Micheaux

References

Desgagne, A. and Lafaye de Micheaux, P. and Leblanc, A. (2016), Test of normality based on alternate measures of skewness and kurtosis, ,

Examples

res <- gensample(39, 10000, law.pars = c(3, 2, 0.5, 1))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)

The Log-Pareto-tail-normal Distribution

Description

Random generation for the Log-Pareto-tail-normal distribution with parameters alpha, mu and sigma.

This generator is called by function gensample to create random variables based on its parameters.

Details

If alpha, mu and sigma are not specified they assume the default values of 1.959964, 0.0 and 1.0 respectively.

The log-Pareto-tailed normal distribution has a symmetric and continuous density that belongs to the larger family of log-regularly varying distributions (see Desgagne, 2015). This is essentially a normal density with log-Pareto tails. Using this distribution instead of the usual normal ensures whole robustness to outliers in the estimation of location and scale parameters and in the estimation of parameters of a multiple linear regression.

The density of the log-Pareto-tailed normal distribution with parameters alpha, mu and sigma is given by

g(y\mid\alpha,\mu,\sigma)=\left\{ \begin{array}{ccc} \frac{1}{\sigma}\phi\left(\frac{y-\mu}{\sigma}\right) & \textrm{ if } & \mu - \alpha\sigma \le y\le \mu + \alpha\sigma, \\ &\\ \phi(\alpha)\frac{\alpha}{|y-\mu|}\left(\frac{\log \alpha}{\log (|y-\mu|/\sigma)}\right)^\beta & \textrm{ if } & |y-\mu|\ge \alpha\sigma, \end{array} \right.

where \beta = 1+2\,\phi(\alpha)\,\alpha\log(\alpha)(1-q)^{-1} and q=\Phi(\alpha)-\Phi(-\alpha). The functions \phi(\alpha)=\frac{1}{\sqrt{2\pi}}\exp[-\frac{\alpha^2}{2}] and \Phi(\alpha) are respectively the p.d.f. and the c.d.f. of the standard normal distribution. The domains of the variable and the parameters are -\infty<y<\infty, \alpha>1, -\infty<\mu<\infty and \sigma>0.

Note that the normalizing constant K_{(\alpha,\beta)} (see Desgagne, 2015, Definition 3) has been set to 1. The desirable consequence is that the core of the density, between \mu-\alpha\sigma and \mu+\alpha\sigma, becomes exactly the density of the N(\mu,\sigma^2). This mass of the density corresponds to q. It follows that the parameter \beta is no longer free and its value depends on \alpha as given above.

For example, if we set \alpha=1.959964, we obtain \beta=4.083613 and q=0.95 of the mass is comprised between \mu-\alpha\sigma and \mu+\alpha\sigma. Note that if one is more comfortable in choosing the central mass $q$ instead of choosing directly the parameter \alpha, then it suffices to use the equation \alpha=\Phi^{-1}((1+q)/2), with the contrainst q>0.6826895\Leftrightarrow \alpha>1.

The mean and variance of Log-Pareto-tail-normal are not defined.

Author(s)

P. Lafaye de Micheaux

References

Desgagne, Alain. Robustness to outliers in location-scale parameter model using log-regularly varying distributions. Ann. Statist. 43 (2015), no. 4, 1568–1595. doi:10.1214/15-AOS1316. http://projecteuclid.org/euclid.aos/1434546215.

Examples

res <- gensample(40, 10000, law.pars = c(1.959964, 0.0, 1.0))
res$law
res$law.pars

Computation of critical values for several test statistics

Description

Computation of critical values for several test statistics, several n values, and several level values, for a given distribution

Usage

many.crit(law.index,stat.indices,M = 10^3,vectn = c(20,50,100),levels = c(0.05,0.1),
  alter = create.alter(stat.indices),law.pars = NULL,parstats = NULL,model = NULL,
  Rlaw=NULL, Rstats = NULL,center=FALSE, scale=FALSE)

Arguments

law.index

law index as given by function getindex. length(law.index)=1

stat.indices

vector of statistic indices as given by function getindex.

M

number of Monte Carlo repetitions to use.

vectn

vector of number of observations for the samples to be generated.

levels

vector of required level values.

alter

named-list with type of test for each statistical test: alter[["statj"]]=0, 1 ,2, 3 or 4; for each j in stat.indices (0: two.sided=bilateral, 1: less=unilateral, 2: greater=unilateral, 3: bilateral test that rejects H0 only for large values of the test statistic, 4: bilateral test that rejects H0 only for small values of the test statistic)

law.pars

NULL or a vector of length at most 4 containing 4 possible parameters to generate random values from distribution law(law.pars[j],j<=4)

parstats

named-list of parameter values for each statistic to simulate. The names of the list should be statj, j taken in stat.indices. If statj=NA, the default parameter values for the test statistic statj will be used.

model

NOT IMPLEMENTED YET. If NULL, no model is used. If an integer i>0, the model coded in the C function modelei is used. Else this should be an R function that takes three arguments: eps (vector of \epsilon values), thetavec (vector of \theta values) and xvec (vector or matrix of x values).This function should take a vector of errors, generate observations from a model (with parameters thetavec and values xvec) based on these errors, then compute and return the residuals from the model. See function modele1.R in directory inst/doc/ for an example in multiple linear regression.

Rlaw

If 'law.index' is set to 0 then 'Rlaw' should be a (random generating) function.

Rstats

A list of same length as stat.indices. If a value of the vector stat.indices is set to 0, the corresponding component of the list Rstats should be an R function that outputs a list with components statistic (value of the test statistic), pvalue (pvalue of the test; if not computable should be set to 0), decision (1 if we reject the null, 0 otherwise), alter (see above), stat.pars (see above), pvalcomp (1L if the pvalue can be computed, 0L otherwise), nbparstat (length of stat.pars). If a value of stat.indices is not 0, then the corresponding component of Rstats should be set to NULL.

center

Logical. Should we center the data generated

scale

Logical. Should we center the data generated

Value

An object of class critvalues, which is a list where each element of the list contains a matrix for the corresponding statistic. This column matrices are: n values, level values, parameters of the test statistic (NA if none), left critical values and right critical values).

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

critval <- many.crit(law.index=2,stat.indices=c(10,15),M=10^3,vectn=c(20,50,100),
                     level=c(0.05,0.1),alter=list(stat10=3,stat15=3),law.pars=NULL,
                     parstats=NULL)
print(critval,digits=3,latex.output=FALSE)

Computes several `p`-values for many test statistics.

Description

This function generates a sample of n observations from the law specified in law.index. It then computes the value of each test statistic specified in stat.indices and use it to obtain the corresponding p-value under the null. The computation of these p-values can be done using a Monte-Carlo simulation.

Usage

many.pval(stat.indices, law.index, n = 100, M = 10^5, N = 100,
          alter = create.alter(stat.indices), law.pars = NULL, parstats = NULL,
          null.dist = 2, null.pars = NULL, method = c("direct", "MC"), Rlaw.index = NULL,
Rnull.dist = NULL, Rstats = NULL, center=FALSE, scale=FALSE)

Arguments

stat.indices

vector of test statistic indices as given by function getindex (some components can be 0 if you want to use your own function for some test statistics; see 'Rstats' argument).

law.index

index of the distribution from which to generate observations used to compute the values of the test statistics specified with stat.indices.

n

integer. Size of the samples from which to compute the value of the test statistics.

M

integer. Number of Monte-Carlo repetitions. Only used when method = 'MC'.

N

integer. Number of p-values to compute for each test statistic.

alter

integer value in {0,1,2,3,4}. Type of test. See function create.alter.

law.pars

vector of the parameter values for the law specified in law.index.

parstats

named-list of vectors of parameters for the test statistics specified in stat.indices. The names of the list should be statxxx where xxx are the indices specified in stat.indices.

null.dist

used only if method = 'MC'. Integer value (as given by function getindex) specifying the distribution under the null hypothesis.

null.pars

vector of parameters for the null distribution

method

character. Either 'direct' to compute the p-value under the null using for example the asymptotic distribution of the test statistic under the null. This is not possible for all test statistics; or 'MC' to use a Monte-Carlo simulation to approximate the distribution of the test statistic under the null (specified by null.dist).

Rlaw.index

If 'law.index' is set to 0 then 'Rlaw.index' should be a (random generating) function.

Rnull.dist

If 'null.dist' is set to 0 then 'Rnull.dist' should be a (random generating) function.

Rstats

center

Logical. Should we center the data generated

scale

Logical. Should we center the data generated

Value

pvals

the N x length(stat.indices) matrix of p-values.

stat.indices

same as input.

n

same as input.

M

same as input.

alter

same as input.

parstats

same as input.

null.dist

same as input.

method

same as input.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

stind <- c(43,44,42)   # Indices of test statistics.
alter <-list(stat43=3,stat44=3,stat42=3) # Type for each test.
# Several p-values computed under the null.
# You can increase the values of M and N for better results.
matrix.pval <- many.pval(stat.indices=stind,law.index=1,
                        n=100,M=10,N=10,alter=alter,null.dist=1,
                        method="direct")

Expectation and variance.

Description

Evaluate the expectation and variance of a law.

Details

Use the function by typing:

momentsj(x,par1,par2,etc.)

where j is the index of the law and par1, par2, etc. are the parameters of law j.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

`p`-value discrepancy plot.

Description

This function produces a p-value discrepancy plot.

Usage

## S3 method for class 'discrepancy'
plot(x,legend.pos=NULL,...)

Arguments

x

Fx object as returned by function calcFx.

legend.pos

If NULL, position of the legend will be computed automatically. Otherwise, it should be either a character vector in "bottomright", "bottom", "bottomleft", "left", "topleft", "top", "topright", "right" and "center". Or a numeric vector of length 2 giving the x-y coordinates of the legend.

...

further arguments passed to the plot or points functions.

Details

See Section 2.3 in Lafaye de Micheaux, P. and Tran, V. A. (2014).

Value

No return value. Displays a graph.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

stind <- c(43,44,42)   # Indices of test statistics.
alter <-list(stat43=3,stat44=3,stat42=3) # Type for each test.
# Several p-values computed under the null.
pnull <- many.pval(stat.indices=stind,law.index=1,
                   n=100,N=10,alter=alter,null.dist=1,
                   method="direct")$pvals
xnull <- calcFx(pnull)
plot.discrepancy(xnull)

`p`-value plot.

Description

This function produces a p-value plot.

Usage

## S3 method for class 'pvalue'
plot(x,legend.pos=NULL,...)

Arguments

x

Fx object as returned by function calcFx.

legend.pos

...

further arguments passed to the plot or points functions.

Details

See Section 2.3 in Lafaye de Micheaux, P. and Tran, V. A. (2014).

Value

No return value. Displays a graph.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

stind <- c(43,44,42)   # Indices of test statistics.
alter <-list(stat43=3,stat44=3,stat42=3) # Type for each test.
# Several p-values computed under the null.
pnull <- many.pval(stat.indices=stind,law.index=1,
                   n=100,N=10,alter=alter,null.dist=1,
                   method="direct")$pvals
xnull <- calcFx(pnull)
plot(xnull)

size-power curves.

Description

This function produces a size-power curves plot.

Usage

## S3 method for class 'sizepower'
plot(x, xnull,legend.pos=NULL,...)

Arguments

x

Fx object as returned by function calcFx.

xnull

Fx object as returned by function calcFx, but computed under the null.

legend.pos

...

further arguments passed to the plot or points functions.

Details

See Section 2.3 in Lafaye de Micheaux, P. and Tran, V. A. (2014).

Value

No return value. Displays a graph.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

## You can increase M for better results:
stind <- c(43,44,42)   # Indices of test statistics.
alter <-list(stat43=3,stat44=3,stat42=3) # Type for each test.
# Several p-values computed under the null.
pnull <- many.pval(stat.indices=stind,law.index=1,
                   n=100,M=100,N=10,alter=alter,null.dist=2,
                   method="MC")$pvals
Fxnull <- calcFx(pnull)
p <- many.pval(stat.indices=stind,law.index=4,n=100,
               M=100,N=10,alter=alter,null.dist=2,
               method="MC")$pvals
Fx <- calcFx(p)
plot.sizepower(Fx,Fxnull)

Computation of power and level tables for hypothesis tests.

Description

Functions for the computation of power and level tables for hypothesis tests, in LaTeX format.

Usage

powcomp.easy(params,M=10^5,model=NULL,Rlaws=NULL,Rstats=NULL,center=FALSE, scale=FALSE)

Arguments

M

number of Monte Carlo repetitions to use.

params

matrix with (at least) 11 named-columns with names (n, law, stat, level, cL, cR, alter, par1, par2, par3, par4). Each row of params gives the necessary parameters for a simulation of powers.

n : sample size;

law : integer giving the index of the law considered;

stat : integer giving the index of the test statistic considered (can be 0 if you want to use your own function for some test statistic; see 'Rstats' argument);

level : double, this is the significance level desired;

cL : left critical value (can be NA);

cR : right critical value (can be NA);

alter : type of test (integer value in {0,1,2,3,4});

parj: values of the parameters of the distribution specified by law (can be NA).

See 'Details section'.

model

NOT YET IMPLEMENTED. If NULL, no model is used. If an integer i>0, the model coded in the C function modelei is used. Else this should be an R function that takes three arguments: eps (vector of \epsilon values), thetavec (vector of \theta values) and xvec (vector or matrix of x values). This function should take a vector of errors, generate observations from a model (with parameters thetavec and values xvec) based on these errors, then compute and return the residuals from the model. See function modele1.R in directory inst/doc/ for an example in multiple linear regression

Rlaws

When some law indices in second column of 'params' are equal to 0, this means that you will be using some R random generators not hardcoded in C in the package. In that case, you should provide the names of the random generation functions in the corresponding components of a list; the other components should be set to NULL.

Rstats

A list. If in a given row of the 'params' matrix, the value of 'stat' is set to 0, the corresponding component of the list 'Rstats' should be an R function that outputs a list with components 'statistic' (value of the test statistic), 'pvalue' (pvalue of the test; if not computable should be set to 0), 'decision' (1 if we reject the null, 0 otherwise), 'alter' (see above), 'stat.pars' (see above), 'pvalcomp' (1L if the pvalue can be computed, 0L otherwise), 'nbparstat' (length of stat.pars). If the value of 'stat' is not 0, then the corresponding component of 'Rstats' should be set to 'NULL'.

center

Logical. Should we center the data generated

scale

Logical. Should we center the data generated

Details

If both cL and cR are NA, no critical values are used and the decision to reject (or not) the hypothesis is taken using the p-value.

If a test statistic depends upon some parameters, these can be added (in a correct order) in the last columns of params. If other test statistics are considered simultaneously (in the same params matrix) and if not all the test statistics have the same number of parameters, NA values should be used to complete empty cells of the matrix.

Value

The powers for the different statistics and laws specified in the rows of params, NOT YET provided in the form of a LaTeX table. This version is easier to use (but slower) than the powcomp.fast version. It should be used in the process of investigating the power of test statistics under different alternatives. But when you are ready to produce results for publication in a paper, please use the powcomp.fast version and its print method..

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

# Warning: the order of the parameters of the law (4 maximum) is important!
sim1 <- c(n=100,law=2,stat=10,level=0.05,cL=NA,cR=0.35,alter=3,
          par1= 2.0,par2=NA,par3=NA,par4=NA,parstat1=NA,parstat2=NA)
sim2 <- c(n=100,law=2,stat=17,level=0.10,cL=-0.30,cR=NA,alter=1,
          par1=-1.0,par2=3.0,par3=NA,par4=NA,parstat1=NA,parstat2=NA)
sim3 <- c(n=100,law=2,stat=31,level=0.10,cL=NA,cR=0.50,alter=3,
          par1=-1.0,par2=3.0,par3=NA,par4=NA,parstat1=0.7,parstat2=NA)
sim4 <- c(n=100,law=7,stat=80,level=0.10,cL=NA,cR=9.319,alter=3,
          par1=NA,par2=NA,par3=NA,par4=NA,parstat1=1,parstat2=5)
params <- rbind(sim1,sim2,sim3,sim4)
powcomp.easy(params,M=10^2)
sim5 <- c(n=100,law=0,stat=80,level=0.10,cL=NA,cR=9.319,alter=3,
          par1=NA,par2=NA,par3=NA,par4=NA,parstat1=1,parstat2=5)
params <- rbind(params,sim5)
powcomp.easy(params,M=10^2,Rlaws=list(NULL,NULL,NULL,NULL,rnorm))

Computation of power and level tables for hypothesis tests.

Description

Functions for the computation of power and level tables for hypothesis tests, with possible use of a cluster.

Usage

powcomp.fast(law.indices,stat.indices,vectn = c(20,50,100),M = 10^3,levels = c(0.05,0.1),
         critval = NULL,alter = create.alter(stat.indices),parlaws = NULL, 
         parstats = NULL,nbclus = 1,model = NULL,null.law.index = 2,null.law.pars = NULL,
         Rlaws=NULL, Rstats = NULL, center=FALSE, scale=FALSE, pvalcomp = 1L)

Arguments

law.indices

vector of law indices as given by function getindex.

stat.indices

vector of statistic indices as given by function getindex (some components can be 0 if you want to use your own function for some test statistics; see 'Rstats' argument).

vectn

vector of sample sizes (n) values.

M

number of Monte Carlo repetitions.

levels

vector of significance levels for the test.

critval

if not NULL, a named-list of critical values for each test statistic. The names of the list should be statj, j taken in stat.indices. Note that if a single value of critval$statj is povided, then it is the right critical value. If two values are provided, then these are the left and right critical values, in that order. If NULL, critval is computed using the function many.crit; in that case, be sure to provide the correct value for null.law.index.

alter

named-list of integer values (0: two.sided=bilateral, 1: less=unilateral, 2: greater=unilateral, 3: bilateral test that rejects H0 only for large values of the test statistic, 4: bilateral test that rejects H0 only for small values of the test statistic). The names of the list should be statj, j taken in stat.indices.

parlaws

named-list of parameter values for each law to simulate. The names of the list should be lawj, j taken in law.indices. The length of vector lawj should not be greater than 4 (we supposed than no common distribution has more than 4 parameters!).

parstats

named-list of parameter values for each statistic to simulate. The names of the list should be statj, j taken in stat.indices (in the same order). If NULL, the default parameter values for these statistics will be used.

nbclus

number of slaves to use for the computation on a cluster. This needs parallel or Rmpi package to be installed and functionnal on the system. Also the mpd daemon sould be started.

model

NOT YET IMPLEMENTED. If NULL, no model is used. If an integer i>0, the model coded in the C function modelei is used. Else this shoud be an R function that takes three arguments: eps (vector of \epsilon values), thetavec (vector of \theta values) and xvec (vector or matrix of x values). This function should take a vector of errors, generate observations from a model (with parameters thetavec and values xvec) based on these errors, then compute and return the residuals from the model. See function modele1.R in directory inst/doc/ for an example in multiple linear regression.

null.law.index

index of the law under the null. Only used, by many.crit function, if critval is NULL.

null.law.pars

vector of parameters corresponding to null.law.index.

Rlaws

When some law indices in 'law.indices' are equal to 0, this means that you will be using some R random generators. In that case, you should provide the names of the random generation functions in the corresponding components of 'Rlaws' list, the other components should be set to NULL.

Rstats

center

Logical. Should we center the data generated

scale

Logical. Should we center the data generated

pvalcomp

Integer. 1L to compute p-values, 0L not to compute them.

Details

This version is faster (but maybe less easy to use in the process of investigating the power of test statistics under different alternatives) than the powcomp.easy version.

Value

A list of class power whose components are described below:

M

number of Monte Carlo repetitions.

law.indices

vector of law indices as given by function getindex.

vectn

vector of sample sizes.

stat.indices

vector of test statistic indices as given by function getindex.

decision

a vector of counts (between 0 and M) of the decisions taken for each one of the levels.len * laws.len * vectn.len * stats.len combinations of (level,law,sample size,test statistic), to be understood in the following sense. The decision for the l-th level (in levels), d-th law (in law.indices), n-th sample size (in vectn) and s-th test statistic (in stat.indices) is given by:

decision[s + stats.len*(n-1) + stats.len*vectn.len*(d-1) + stats.len*vectn.len*laws.len*(l-1)]

where stats.len, vectn.len, laws.len and levels.len are respectively the lengths of the vectors stat.indices, vectn, law.indices and levels.

levels

vector of levels for the test.

cL

left critical values used.

cR

right critical values used.

usecrit

a vector of 1s and 0s depending if a critical value has been used or not.

alter

type of each one of the tests in stat.indices used (0: two.sided=bilateral, 1: less=unilateral, 2: greater=unilateral, 3: bilateral test that rejects H0 only for large values of the test statistic, 4: bilateral test that rejects H0 only for small values of the test statistic).

nbparlaws

default number of parameters used for each law in law.indices.

parlaws

default values of the parameters for each law.

nbparstats

default number of parameters for each test statistic in stat.indices.

parstats

default values of the parameters for each test statistic.

nbclus

number of CPUs used for the simulations.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

## Regenerate Table 6 from Puig (2000) (page 424)

law.index <- 1
# Take M = 50000 for accurate results 
M <- 10
vectn <- c(10,15,20,35,50,75,100)
level <- c(0.05)
stat.indices <- c(43,44,42,45,46)
law.indices <- c(2,3,4)
alter <- list(stat43 = 3,stat44 = 3,stat42 = 3,stat45 = 3,stat46 = 3)
critval <- many.crit(law.index,stat.indices,M,vectn,level,alter,
                     law.pars = NULL,parstats = NULL)
table6 <- powcomp.fast(law.indices,stat.indices,vectn,M,level,critval = critval,alter,
                       parlaws = NULL,parstats = NULL,nbclus = 1)
table6

PoweR GUI

Description

Graphical user interface (GUI) for the package.

Usage

power.gui()

Details

This GUI is a 5-tabbed notebook whose goal is to make our package easier to use :

- Tab 1 gensample : generate random samples from a law added in the package;

- Tab 2 statcompute : perform the test for a given index value of test statistic;

- Tab 3 many.crit : computation of critical values for several test statistics;

- Tab 4 powcomp.fast : computation of power and level tables for hypothesis tests;

- Tab 5 Examples : reproduce results from published articles.

Important note concerning 'Iwidgets': for the GUI to work, a third party software has to be installed.

Under Microsoft Windows:

First, install ActiveTcl following indications given here: 'http://www.sciviews.org/_rgui/tcltk/TabbedNotebook.html'

After the installation of ActiveTcl and the modification of the PATH variable, launch from an MsDOS terminal (accessible through typing 'cmd' in the Start Menu) the following command: C:\Tcl\bin\teacup.exe install Iwidgets

You can then check the existence of a directory called 'Iwidgets4.0.2' in 'C:\Tcl\lib\teapot\package\tcl\lib'.

Under Linux:

Install 'iwidgets'.

Value

No return value. Displays a graphical user interface.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Latex table for critical values

Description

Transform the critical values given by function many.crit into a LaTeX code for creating the table of critical values.

Usage

## S3 method for class 'critvalues'
print(x, digits = 3, latex.output =  FALSE, template = 1, ...)

Arguments

x

critical values given by function many.crit.

digits

integer indicating the number of decimal places to be used.

latex.output

logical. If TRUE, we output LaTeX code for the table of critical values. If FALSE, we output this table in the R Console.

template

integer, template to use for the (LaTeX) printing of values. Only template = 1 is defined for the moment.

...

further arguments passed to or from other methods.

Value

No return value. The function prints a formatted representation of critical values, optionally in 'LaTeX' format. The object printed is of class "critvaluesX", where X is the template number.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Puig, P. and Stephens, M. A. (2000), Tests of fit for the Laplace distribution, with applications, Technometrics, 42, 417–424.

Examples

## Regenerate Table 1 from Puig (2000) (page 419)
# Take M = 10000 for accurate results
M <- 10
law.index <- 1
vectn <- c(10,15,20,35,50,75,100,1000)
level <- c(0.50,0.25,0.10,0.05,0.025,0.01) 
table1 <- many.crit(law.index,stat.indices = c(43),M,vectn,level,
                    alter = list(stat43=3),law.pars = NULL,parstat = NULL)
print.critvalues(table1,digits=3,latex.output=TRUE)

Latex table for power simulations

Description

Transform the power values given by function powcomp.fast into a LaTeX code for creating the table of power simulations.

Usage

## S3 method for class 'power'
print(x, digits = 3, latex.output = FALSE, template = 1,
summaries = TRUE, ...)

Arguments

x

power values given by function powcomp.fast.

digits

control the number of decimal places. It can take values from 0 to 3.

latex.output

logical. If TRUE, we output LateX code for the table of power simulations. If FALSE, we output this table in the R Console.

template

integer, template to use for the (LaTeX) printing of values. Only template = 1 is defined for the moment.

summaries

logical, to display the summaries Average power table, Average gap table and Worst gap table.

...

further arguments passed to or from other methods.

Value

No return value. The function prints a formatted representation of power analysis results, optionally in 'LaTeX' format. The printed object is of class "powerX", where X is the template number.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Puig, P. and Stephens, M. A. (2000), Tests of fit for the Laplace distribution, with applications, Technometrics, 42, 417–424.

Examples

## Regenerate Table 6 from Puig (2000) (page 424)
# Change M = 50000 for more accurate results
M <- 10
law.index <- 1
vectn <- c(10,15,20,35,50,75,100)
level <- c(0.05)
stat.indices <- c(43,44,42,45,46)
law.indices <- c(2,3,4)
alter <- list(stat43 = 3,stat44 = 3,stat42 = 3,stat45 = 3,stat46 = 3)
critval <- many.crit(law.index,stat.indices,M,vectn,level,alter,law.pars = NULL,parstat = NULL)
table6 <- powcomp.fast(law.indices,stat.indices,vectn,M,level,critval = critval,alter,
                       parlaws = NULL,parstats = NULL,nbclus = 1)
print.power(table6,digits=0,latex.output = TRUE)

Monte-Carlo computation of a p-value for one single test statistic.

Description

This function can compute the p-value associated with a test statistic value from a sample of observations.

Usage

pvalueMC(data, stat.index, null.law.index, M = 10^5, alter, null.law.pars = NULL,
         stat.pars = NULL, list.stat = NULL, method = c("Fisher"),
         center = FALSE, scale = FALSE)

Arguments

data

sample of observations.

stat.index

index of a test statistic as given by function getindex.

null.law.index

index of the distribution to be tested (the null hypothesis distribution), as given by function getindex.

M

number of Monte-Carlo repetitions to use.

alter

value (in {0,1,2,3,4}) giving the the type of test (See Section 3.3 in Lafaye de Micheaux, P. and Tran, V. A. (2014)).

null.law.pars

vector of parameters for the law. The length of this parameter should not exceed 4. If not provided, the default values are taken using getindex function.

stat.pars

a vector of parameters. If NULL, the default parameter values for the statistic specified by this statistic wil be used.

list.stat

if not NULL, a vector of test statistic values should be provided. If NULL, these values will be computed.

method

method to use for the computation of the p-value. Only 'Fisher' method is available for the moment.

center

Logical. Should we center the data generated

scale

Logical. Should we center the data generated

Value

The Monte-Carlo p-value of the test.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

x <- rnorm(100)
statcompute(1,x,level = c(0.05),alter = 3)$pvalue
pvalueMC(x,stat.index = 1,null.law.index = 2,M = 10^5,alter = 3)

Gives information about a test statistic.

Description

To obtain the name of a test as well as its default number of parameters and default parameter values.

Usage

stat.cstr(stat.index, stat.pars = NULL, n = 0)

Arguments

stat.index

a single integer value corresponding to the index of a test statistic as given by function getindex.

stat.pars

vector of the values of the parameters of the test specified in stat.index. If NULL, the default values are used.

n

integer giving the sample size (useful since some default values of the parameters might depend on the sample size).

Value

name

name of the test.

nbparams

default number of parameters of the test.

law.pars

values of the parameters

alter

0: two.sided=bilateral, 1: less=unilateral, 2: greater=unilateral, 3: bilateral test that rejects H0 only for large values of the test statistic, 4: bilateral test that rejects H0 only for small values of the test statistic.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

stat.cstr(80)

The combination test for normality of Brys-Hubert-Struyf & Bonett-Seier is used

- to compute its statistic and p-value by calling function statcompute;

- to compute its power by calling function powcomp.fast or powcomp.easy.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

References

Zhang, P (1999), Omnibus test of normality using the Q statistic, Journal of Applied Statistics, 26(4), 519–528.

Description

The Desgagne-LafayeDeMicheaux Z_{EPD} test for normality is used

- to compute its statistic and p-value by calling function statcompute;

- to compute its quantiles by calling function compquant or many.crit;

- to compute its power by calling function powcomp.fast or powcomp.easy.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

P. Lafaye de Micheaux, V. A. Tran

References

Choi, B. and Kim, K. (2006), Testing goodness-of-fit for Laplace distribution based on maximum entropy, Statistics, 40(6), 517–531.

The 2nd Choi-Kim test for the Laplace distribution

Description

The 2nd Choi-Kim test T_{m,n}^{E} for the Laplace distribution is used

- to compute its statistic and p-value by calling function statcompute;

- to compute its quantiles by calling function compquant or many.crit;

- to compute its power by calling function powcomp.fast or powcomp.easy.

Details

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Choi, B. and Kim, K. (2006), Testing goodness-of-fit for Laplace distribution based on maximum entropy, Statistics, 40(6), 517–531.

The 3rd Choi-Kim test for the Laplace distribution

Description

The 3rd Choi-Kim test T_{m,n}^{C} for the Laplace distribution is used

- to compute its quantiles by calling function compquant or many.crit;

- to compute its power by calling function powcomp.fast or powcomp.easy.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Hegazy, Y. A. S. and Green, J. R. (1975), Some new goodness-of-fit tests using order statistics, Applied Statistics, 24, 299–308.

The 2nd Hegazy-Green test for uniformity

Description

The 2nd Hegazy-Green test T_2 for uniformity is used

- to compute its statistic and p-value by calling function statcompute;

- to compute its quantiles by calling function compquant or many.crit;

- to compute its power by calling function powcomp.fast or powcomp.easy.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Hegazy, Y. A. S. and Green, J. R. (1975), Some new goodness-of-fit tests using order statistics, Applied Statistics, 24, 299–308.

The Greenwood test for uniformity

Description

The Greenwood test G(n) for uniformity is used

- to compute its statistic and p-value by calling function statcompute;

- to compute its quantiles by calling function compquant or many.crit;

- to compute its power by calling function powcomp.fast or powcomp.easy.

Details

Note that n is the sample size.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Greenwood, M. (1946), The statistical study of infectious diseases, Journal of Royal Statistical Society Series A, 109, 85–110.

The Quesenberry-Miller test for uniformity

Description

- to compute its quantiles by calling function compquant or many.crit;

- to compute its power by calling function powcomp.fast or powcomp.easy.

Details

If \lambda and m are not specified they assume the default values of 0 and 2, respectively.

There are 3 choices for value of \lambda : \lambda = 0, \lambda = -1, and \lambda != 0, != -1.

Note that n is the sample size.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Morales, D., Pardo, L., Pardo, M. C. and Vajda, I. (2003), Limit laws for disparities of spacings, Journal of Nonparametric Statistics, 15(3), 325–342.

M. A. Marhuenda, Y. Marhuenda, D. Morales, (2005), Uniformity tests under quantile categorization, Kybernetes, 34(6), 888–901.

The Pardo test for uniformity

Description

The Pardo test E_{m,n} for uniformity is used

- to compute its statistic and p-value by calling function statcompute;

- to compute its quantiles by calling function compquant or many.crit;

- to compute its power by calling function powcomp.fast or powcomp.easy.

Details

If m is not specified it assumes the default value of 2. Note that m < (n/2) where n is the sample size.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Pardo, M. C. (2003), A test for uniformity based on informational energy, Statistical Papers, 44, 521–534.

The Marhuenda test for uniformity

Description

The Marhuenda test T_{n,m}^{lambda} for uniformity is used

- to compute its statistic and p-value by calling function statcompute;

- to compute its quantiles by calling function compquant or many.crit;

References

Rizzo, M. L., Haman, J. T. 2016. Expected distances and goodness-of-fit for the asymmetric Laplace distribution. Statist. Probab. Lett., 117, 158-164.

Performs a hypothesis test for the given value of statistic.

Description

Performs the hypothesis test for those added in the package.

Usage

statcompute(stat.index, data, levels = c(0.05,0.1), critvalL = NULL,
            critvalR = NULL, alter = 0, stat.pars = NULL, pvalcomp = 1L,
            check = TRUE)

Arguments

stat.index

one statistic index as given by function getindex.

data

sample from which to compute the statistic.

levels

vector of desired significance levels for the test.

critvalL

NULL or vector of left critival values.

critvalR

NULL or vector of right critival values.

alter

stat.pars

a vector of parameters. If NULL, the default parameter values for this statistic will be used.

pvalcomp

1L to compute the p-value, 0L otherwise.

check

Logical. If FALSE it will execute much faster, but in this case be sure to give a value to the 'stat.pars' argument; if you don't know what value to give, use rep(0.0, getnbparstats(stat.index)) as a default value.

Details

The function statcompute() should not be used in simulations since it is NOT fast. Consider instead using powcomp.easy or powcomp.fast. See also in the Example section below for a fast approach using the .C function (but be warned that giving wrong values of arguments can crash your session!).

Value

A list with components:

statistic

the test statistic value

pvalue

the p-value

decision

the vector of decisions, same length as levels

alter

stat.pars

symbol

how the test is noted

Author(s)

P. Lafaye de Micheaux, V. A. Tran

References

Examples

data <- rnorm(50)
# Shapiro-Wilk test:
statcompute(21, data, levels = c(0.05, 0.1), critvalL = NULL, critvalR = NULL,
            alter = 0, stat.pars = NULL)
# Identical to:
shapiro.test(data)

# The function statcompute() should not be used in simulations since it
#  is NOT fast. Consider instead the call below (but see the Details
#  Section):
.C("stat21", data = data, n = 50L, levels = 0.05, nblevels = 1L, name =
rep(" ", 50), getname = 0L, statistic = 0, pvalcomp = 1L, pvalue = 0, cL = 0.0,
cR = 0.0, usecrit = 0L, alter = 4L, decision = 0L, stat.pars = 0.0, 
nbparstat = 0L)

# Another option is to use the 'pvalcomp' and 'check' arguments as
#  follows which can be much faster (when computing the p-value takes time) 
statcompute(21, data, levels = c(0.05, 0.1), critvalL = NULL, critvalR = NULL,
            alter = 0, stat.pars = NULL, pvalcomp = 0L, check = FALSE)

Computation of test statistic values in pure R.

Description

Alternate way to compute test statistic values (only) in pure R instead of C/C++, for clarity reasons.

Details

Use the function by typing:

statj(x,par1,par2,etc.)

where j is the index of the test and par1, par2, etc. are the parameters of test j, if any.

Author(s)

P. Lafaye de Micheaux, V. A. Tran

Distributions in the PoweR package

Description

Details

Author(s)

References

See Also

Goodness-of-fit tests for the Laplace distribution

Description

Details

Author(s)

References

See Also

Goodness-of-fit tests for normality.

Description

Details

Author(s)

References

See Also

Goodness-of-fit tests for uniformity

Description

Details

Author(s)

References

See Also

Empirical distribution function of p-values.

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Check proper behaviour of a random generator

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Computation of the quantile values only for one test statistic.

Description

Usage

Arguments

Value

Author(s)

References

Examples

Create a list giving the type of test statistics.

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Density function

Description

Details

Author(s)

References

Generate random samples from a law added in the package.

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Get indices of laws and statistics functions.

Description

Usage

Arguments

Value