mdstab.elliptical

Description

computes the probability density function of a d-dimensional elliptically contoured stable distribution at a given point in R^{d}, see Teimouri et al. (2018).

Usage

mdstab.elliptical(x, alpha, Sigma, Mu)

Arguments

Value

a numeric value

Note

mdstab.elliptical() computes the probability density function of an d-dimensional elliptically contoured stable distribution using either asymptotic series or Monte Carlo approximation.

Author(s)

Mahdi Teimouri, Adel Mohammadpour, and Saralees Nadarajah

References

Teimouri, M., Rezakhah, S., and Mohammadpour, A. (2018). Parameter estimation using the EM algorithm for symmetric stable random variables and sub-Gaussian random vectors, Journal of Statistical Theory and Applications, 17(3),1-20.

Examples

# In the following example, we compute the pdf of a two-dimensional elliptically contoured
# stable distribution with parameters alpha=1.3, Sigma=matrix(c(1,.5,.5,1),2,2), and mu=(0,0)^T.
library("stabledist")
mdstab.elliptical(c(5,5),1.2,matrix(c(1,0.5,0.5,1),2,2),c(0,0))

mfitstab.elliptical

Description

estimates the parameters of a d-dimensional elliptically contoured stable distribution, see Teimouri et al. (2018).

Usage

mfitstab.elliptical(yy, alpha0, Sigma0, Mu0)

Arguments

Value

Author(s)

Mahdi Teimouri, Adel Mohammadpour, and Saralees Nadarajah

References

Teimouri, M., Rezakhah, S., and Mohammadpour, A. (2018). Parameter estimation using the EM algorithm for symmetric stable random variables and sub-Gaussian random vectors, Journal of Statistical Theory and Applications, 17(3),1-20,

Examples

# Here we follow for applying the EM algorithm to Z=(x1, x2)^T using the
# initial values alpha0=1, Sigma0=matrix(c(0.75,0.25,0.25,0.75),2,2), and
# Mu0=(0.5,0.5)^T.
library("stabledist")
x1<-urstab(100,1.2,0,1,2,0)
x2<-urstab(100,1.2,0,0.5,2,0)
z<-cbind(x1,x2)
mfitstab.elliptical(z,1,matrix(c(0.75,0.25,0.25,0.75),2,2),c(0.5,0.5))

mfitstab.ustat

Description

estimates the parameters of a strictly bivariate stable distribution using approaches proposed by Mohammadi et al. (2015)<doi.org/10.1007/s00184-014-0515-7> and Teimouri et al. (2017)<doi.org/10.1155/2017/3483827>. The estimated parameters are tail index and discretized spectral measure on m equidistant points located on unit sphere in R^2.

Usage

mfitstab.ustat(u,m,method=method)

Arguments

Value

Author(s)

Mahdi Teimouri, Adel Mohammadpour, and Saralees Nadarajah

References

Mohammadi, M., Mohammadpour, A., and Ogata, H. (2015). On estimating the tail index and the spectral measure of multivariate alpha-stable distributions, Metrika, 78(5), 549-561.

Nolan. J. P. (2013). Multivariate elliptically contoured stable distributions: theory and estimation, Computational Statistics, 28(5), 2067-2089.

Teimouri, M., Rezakhah, S., and Mohammadpour, A. (2017). U-Statistic for multivariate stable distributions, Journal of Probability and Statistics, https://doi.org/10.1155/2017/3483827.

Examples

# Here, for example, we are interested to estimate the parameters of a bivariate
# stable distribution. For this, two sets of n=400 iid realizations which are
# assumed to distributed jointly as a strictly bivariate stable distribution with
# tail index alpha=1.2 are simulated. Considering m=4, masses of the discrete spectral
# measure are addressed by s_j=(cos(2*pi(j-1)/m), sin (2*pi(j-1)/m)); for j=1,...,4.
library("nnls")
x1<-urstab(400,1.2,-0.50,1,0,0)
x2<-urstab(400,1.2,0.50,0.5,0,0)
z<-cbind(x1,x2)
mfitstab.ustat(z,4,1)

mrstab

Description

generates iid realizations from bivariate stable vectors using the methodology proposed by Modarres and Nolan (1994).

Usage

mrstab(n, m, alpha, Gamma, Mu)

Arguments

Value

a vector of n numeric values

Note

mrstab() assumes that masses are located at unit sphere with addresses s_j=(cos(2*pi(j-1)/m), sin(2*pi(j-1)/m)); for j=1,...,4.

Author(s)

Mahdi Teimouri, Adel Mohammadpour, and Saralees Nadarajah

References

Modarres, R. and Nolan, J. P. (1994). A method for simulating stable random vectors, Computational Statistics, 9(1), 11-19.

Examples

# We use the following command to simulate n=200 iid vectors from a two-dimensional stable
# distribution with alpha=1.3, with a vector of 4 masses as gamma=(0.1,0.5,0.5,0.1)^T,
# and mu=(0,0)^T.
library("stabledist")
mrstab(200,4,1.3,c(0.1,0.5,0.5,0.1),c(0,0))

mrstab.elliptical

Description

generates iid realizations from d-dimensional elliptically contoured stable distribution, see Nolan (2013) <doi.org/10.1007/s00180-013-0396-7>.

Usage

mrstab.elliptical(n, alpha, Sigma, Mu)

Arguments

Details

mrstab.elliptical() needs to install the mvtnorm package

Value

an n by d matrix of numeric values

Note

mrstab.elliptical() generates iid realizations from d-dimensional elliptically contoured stable distribution based on definitions given by Nolan (2013) and Samorodnitsky and Taqqu (1994)

Author(s)

Mahdi Teimouri, Adel Mohammadpour, and Saralees Nadarajah

References

Nolan. J. P. (2013). Multivariate elliptically contoured stable distributions: theory and estimation, Computational Statistics, 28(5), 2067-2089.

Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models and Infinite Variance, Chapman and Hall, London.

Examples

# In the following example, we simulate n=200 iid vectors of a two-dimensional elliptically
# contoured stable distribution with parameters alpha=1.3, Sigma=matrix(c(1,.5,.5,1),2,2),
# and mu=(0,0)^T.
library("mvtnorm")
library("stabledist")
mrstab.elliptical(200,1.3,matrix(c(1,.5,.5,1),ncol=2,nrow=2),c(0,0))

udstab

Description

computes the probability density function (pdf) of the univariate stable distribution based on formulas given by Nolan (1997) <doi.org/10.1080/15326349708807450> and asymptotic series, see Teimouri and Amindavar (2008).

Usage

udstab(x, alpha, beta, sigma, mu, param)

Arguments

Value

a numeric value

Note

udstab() computes the pdf of univariate stable distribution using asymptotic series within their convergence regions. For points outside of convergence regions, the pdf is computed using stabledist package based on formulas given by Nolan (1997). So, to compute the pdf using the upstab() we may need to install stabledist package.

Author(s)

Mahdi Teimouri, Adel Mohammadpour, and Saralees Nadarajah

References

Nolan, J. P. (1997). Numerical calculation of stable densities and distribution functions, Communications in statistics-Stochastic models, 13(4), 759-774.

Teimouri, M. and Amindavar, H. (2008). A novel approach to calculate stable densities, Proceedings of the World Congress on Engineering, 1, 710-714.

Examples

# In the following, we compute the pdf of a univariate stable distribution at point 2
# with parameters alpha=1.2, beta=0.9, sigma=1, and mu=0 in S_{0} parameterization.
library("stabledist")
udstab(2,1.2,0.9,1,0,1)

ufitstab.cauchy

Description

estimates the parameters of the Cauchy distribution. Given the initial values of the skewness, scale, and location parameters, it uses the EM algorithm to estimate the parameters of the Cauchy distribution.

Usage

ufitstab.cauchy(y, beta0, sigma0, mu0, param)

Arguments

Details

Generally the EM algorithm seeks for the ML estimations when the log-likelihood function is not tractable mathematically. This is done by considering an extra missing (or latent) variable when the conditional expectation of the complete data log-likelihood given observed data and a guess of unknown parameter(s) is maximized. So, first we look for a stochastic representation. The representation given by the following proposition is valid for Cauchy distribution. Suppose Y~S_0(1,beta,sigma,mu) and T~S_{1}(1,1,1,0) (here S_0 and S_1 refer to parameterizations S_0 and S_1, respectively). Then Y=sigma*(1-|beta|)*N/Z+sigma*beta*T+mu where N~Z~N(0,1). The random variables N, Z, and T are mutually independent.

Value

Note

The set of data considered here is large recorded intensities (in Richter scale) of the earthquake at seismometer locations in western North America between 1940 and 1980, see Davidian and Giltinan (1995). Among the features, we focus on the 182 distances from the seismological measuring station to the epicenter of the earthquake (in km) as the variable of interest. This set of data can be found in package nlme. We note that ufitstab.cauchy() is robust with respect to the initial values.

Author(s)

Mahdi Teimouri, Adel Mohammadpour, and Saralees Nadarajah

References

Davidian, M. and Giltinan, D.M. (1995). Nonlinear Mixed Effects Models for Repeated Measurement Data, Chapman and Hall.

Examples

# In the following example, using the initial values beta_0=0.5, sigma_0=5, and mu_0=10,
# we apply the EM algorithm to estimate the parameters of Cauchy distribution fitted to
# the earthquake data given by the vector y.
y<-c(7.5,  8.8,   8.9,   9.4,   9.7,   9.7,   10.5,  10.5,  12.0,  12.2,  12.8,  14.6,
     14.9,  17.6,  23.9,  25.0,  2.9,   3.2 ,  7.6,   17.0,  8.0,   10.0,  10.0,  8.0,
     19.0,  21.0,  13.0,  22.0,  29.0,  31.0,  5.8,   12.0,  12.1,  20.5,  20.5,  25.3,
     35.9,  36.1,  36.3,  38.5,  41.4,  43.6,  44.4,  46.1,  47.1,  47.7,  49.2,  53.1,
     4.0,   10.1,  11.1,  17.7,  22.5,  26.5,  29.0,  30.9,  37.8,  48.3,  62.0,  50.0,
     16.0,  62.0,  1.2,   1.6,   9.1,   3.7,   5.3,   7.4,   17.9,  19.2,  23.4,  30.0,
     38.9,  10.8,  15.7,  16.7,  20.8,  28.5,  33.1,  40.3,  8.0,   32.0,  30.0,  31.0,
     16.1,  63.6,  6.6,   9.3,   13.0,  17.3,  105.0, 112.0, 123.0, 5.0,   23.5,  26.0,
     0.5,   0.6,   1.3,   1.4,   2.6,   3.8,   4.0,   5.1,   6.2,   6.8,   7.5,   7.6,
     8.4,   8.5,   8.5,   10.6,  12.6,  12.7,  12.9,  14.0,  15.0,  16.0,  17.7,  18.0,
     22.0,  22.0,  23.0,  23.2,  29.0,  32.0,  32.7,  36.0,  43.5,  49.0,  60.0,  64.0,
     105.0, 122.0, 141.0, 200.0, 45.0,  130.0, 147.0, 187.0, 197.0, 203.0, 211.0, 17.0,
     19.6,  20.2,  21.1,  88.0,  91.0,  12.0,  148.0, 42.0,  85.0,  21.9,  24.2,  66.0,
     87.0,  23.4,  24.6,  25.7,  28.6,  37.4,  46.7,  56.9,  60.7,  61.4,  62.0,  64.0,
     82.0,  107.0, 109.0, 156.0, 224.0, 293.0, 359.0, 370.0, 25.4,  32.9,  92.2,  45.0,
     145.0, 300.0)
library("stabledist")
ufitstab.cauchy(y,0.5,5,10,0)

ufitstab.cauchy.mix

Description

estimates the parameters of a k-component mixture of Cauchy distributions. Assuming that k is known, given the vector of initial values of entire parameter space, it uses the EM algorithm to estimate the parameters of the k-component mixture of Cauchy distributions.

Usage

ufitstab.cauchy.mix(y, k, omega0, beta0, sigma0, mu0)

Arguments

Details

Generally the EM algorithm seeks for the ML estimations when the log-likelihood function is not tractable mathematically. This is done by considering an extra missing (or latent) variable when the conditional expectation of the complete data log-likelihood given observed data and a guess of unknown parameter(s) is maximized. So, first we look for a stochastic representation. The representation given by the following proposition is valid for Cauchy distribution. Suppose Y~ S_{0}(1,beta,sigma,mu) and T~S_{1}(1,1,1,0) (here S_0 and S_1 refer to parameterizations S_0 and S_1, respectively). Then Y=sigma*(1-|beta|)*N/Z+sigma*beta*T+mu where N~Z~N(0,1). The random variables N, Z, and P are mutually independent.

Value

Note

We use the survival times in days of 72 guinea pigs infected with different doses of tubercle bacilli, see Bjerkedal (1960). We note that the EM algorithm is robust with respect to the initial values.

Author(s)

Mahdi Teimouri, Adel Mohammadpour, and Saralees Nadarajah

References

Bjerkedal, T. (1960) Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli, American Journal of Epidemiology, 72, 130-148.

Examples

# In the following, we give an example that fits a two-component mixture of Cauchy distributions
# to the survival times (in days) of 72 guinea pigs through the EM algorithm. For this, the initial
# values are: omega_0=(0.65,0.35), sigma_0=(20,50), beta_0=(0.20,0.05), and mu_0=(95,210).
library("stabledist")
y<-c(10,33,44,56,59,72,74,77,92,93,96,100,100,102,105,107,107,108,108,108,
     109,112,121,122,122,124,130,134,136,139,144,146,153,159,160,163,163,
     168,171,172,176,113,115,116,120,183,195,196,197,202,213,215,216,222,
     230,231,240,245,251,253,254,255,278,293,327,342,347,361,402,432,458,
     555)
ufitstab.cauchy.mix(y,2,c(0.65,0.35),c(0.20,0.05),c(20,50),c(95,210))

ufitstab.skew

Description

using the EM algorithm, it estimates the parameters of skew stable distribution.

Usage

ufitstab.skew(y, alpha0, beta0, sigma0, mu0, param)

Arguments

Details

For any skew stable distribution we give a new representation by the following. Suppose Y~ S_{0}(alpha, beta, sigma, mu), P~ S_{1}(alpha/2,1,(cos(pi*alpha/4))^(2/alpha),0), and V~ S_{1}(alpha,1,1,0). Then, Y=eta*(2P)^(1/2)*N+theta*V+ mu-lambda, where eta=sigma*(1-|beta|)^(1/alpha), theta=sigma*sign(beta)*|beta|^(1/alpha), lambda=sigma*beta*tan(pi*alpha/2), and N~N(0,1) follows a skew stable distribution. All random variables N, P, and V are mutually independent.

Value

Note

Daily price returns of Abbey National shares between 31/7/91 and 8/10/91 (including n=50 business days). By assuming that p_{t} denotes the price at t-th day, the price return at t-th day is defined as (p_{t-1}-p_{t})/p_{t-1}; for t=2,...,n, see Buckle (1995). We note that the EM algorithm is robust with respect to the initial values.

Author(s)

Mahdi Teimouri, Adel Mohammadpour, and Saralees Nadarajah

References

Buckle, D. J. (1995). Bayesian inference for stable distributions, Journal of the American Statistical Association, 90(430), 605-613.

Examples

# For example, We use the daily price returns of Abbey National shares. Using the initial
# values as alpha_{0}=0.8, beta_{0}=0, sigma_{0}=0.25, and mu_{0}=0.25.
price<-c(296,296,300,302,300,304,303,299,293,294,294,293,295,287,288,297,
         305,307,304,303,304,304,309,309,309,307,306,304,300,296,301,298,
         295,295,293,292,307,297,294,293,306,303,301,303,308,305,302,301,
         297,299)
x<-c()
n<-length(price)
for(i in 2:n){x[i]<-(price[i-1]-price[i])/price[i-1]}
library("stabledist")
ufitstab.skew(x[2:n],0.8,0,0.25,0.25,1)

ufitstab.sym

Description

estimates the parameters of a symmetric stable distribution through the EM algorithm, see Teimouri et al. (2018).

Usage

ufitstab.sym(yy, alpha0, sigma0, mu0)

Arguments

Value

Author(s)

Mahdi Teimouri, Adel Mohammadpour, and Saralees Nadarajah

References

Teimouri, M., Rezakhah, S., and Mohammadpour, A. (2018). Parameter estimation using the EM algorithm for symmetric stable random variables and sub-Gaussian random vectors, Journal of Statistical Theory and Applications, 17(3),1-20.

Examples

# By the following example, we apply the EM algorithm to n=50 iid realization of symmetric
# stable distribution with parameters alpha=1.2, sigma=1, and mu=1. The initial values
# are alpha_0=1.2, sigma_0=1, and mu_0=1.
library("stabledist")
y<-urstab(50,1.2,0,1,1,0)
ufitstab.sym(y,1.2,1,1)

ufitstab.sym.mix

Description

estimates the parameters of a k-component mixture of symmetric stable distributions, Teimouri et al. (2018) <doi.org/10.1080/03610918.2017.1288244>. Having k and given the vector of initial values of entire parameter space, it uses some type of the EM algorithm (ECME) to estimate the parameters of mixture of symmetric stable distributions.

Usage

ufitstab.sym.mix(yy, k, omega0, alpha0, sigma0, mu0)

Arguments

Value

Note

The set of data, used here, is the velocities of 82 distant galaxies, diverging from our own galaxy, called here galaxy. These data are available at https://people.maths.bris.ac.uk/mapjg/mixdata. We note that ufitstab.sym.mix() is robust with respect to the initial values.

Author(s)

Mahdi Teimouri, Adel Mohammadpour, and Saralees Nadarajah

References

Teimouri, M., Rezakhah, S., and Mohammadpour, A. (2018). EM algorithm for symmetric stable mixture model, Communications in Statistics-Simulation and Computation, 47(2), 582-604.

Examples

# In what follows, we apply the EM algorithm to estimate the parameters of the
# mixture of symmetric stable distributions. For this, the initial values for
# fitting a three-component mixture of symmetric stable distribution to the
# galaxy data are: (0.1,0.35,0.55) for weight vector, (1.2,1.2,1.2) for tail
# index vector, (1,1,1) for scale vector, and (8,20,22) for the location vector.
galaxy<-c(9.172,9.350,9.483,9.558,9.775,10.227,10.406,16.084,16.170,18.419,
          18.552,18.600,18.927,19.052,19.070,19.330,19.343,19.349,19.440,
          19.473,19.529,19.541,19.547,19.663,19.846,19.856,19.863,19.914,
          19.918,19.973,19.989,20.166,20.175,20.179,20.196,20.215,20.221,
          20.415,20.629,20.795,20.821,20.846,20.875,20.986,21.137,21.492,
          21.701,21.814,21.921,21.960,22.185,22.209,22.242,22.249,22.314,
          22.374,22.495,22.746,22.747,22.888,22.914,23.206,23.241,23.263,
          23.484,23.538,23.542,23.666,23.706,23.711,24.129,24.285,24.289,
          24.366,24.717,24.990,25.633,26.960,26.995,32.065,32.789,34.279)
library("stabledist")
ufitstab.sym.mix(galaxy,3,c(0.1,0.35,0.55),c(1.2,1.2,1.2),c(1,1,1),c(8,20,22))

ufitstab.ustat

Description

estimates the tail index and scale parameters of a symmetric and zero-location stable distribution using U-statistic proposed by Fan (2006) <DOI: 10.1080/03610920500439992>.

Usage

ufitstab.ustat(x)

Arguments

Value

Note

The ufitstab.ustat() must be applied to a symmetric and zero-location stable distribution.

Author(s)

Mahdi Teimouri, Adel Mohammadpour, and Saralees Nadarajah

References

Fan, Z. (2006). Parameter estimation of stable distributions, Communications in Statistics-Theory and Methods, 35(2), 245-255.

Examples

# We are estimating the parameters of a symmetric stable distribution. For this, firstly,
# we simulate a sample of n=100 iid realizations from stable distribution in S_1 parameterization
# with parameters alpha=1.2, beta=0, sigma=1, and mu=0.
x<-urstab(100,1.2,0,1,0,1)
ufitstab.ustat(x)

upstab

Description

computes the cumulative distribution function (cdf) of the univariate stable distribution based on formulas given by Nolan (1997) <doi.org/10.1080/15326349708807450> and asymptotic series, see Teimouri and Amindavar (2008).

Usage

upstab(x, alpha, beta, sigma, mu, param)

Arguments

Value

a numeric value

Note

upstab() computes the cdf of univariate stable distribution using asymptotic series within their convergence regions. For points outside of convergence regions, the cdf is computed using stabledist package based on formulas given by Nolan (1997). So, to compute the cdf using the upstab() we may need to install stabledist package.

Author(s)

Mahdi Teimouri, Adel Mohammadpour, and Saralees Nadarajah

References

Nolan, J. P. (1997). Numerical calculation of stable densities and distribution functions, Communications in statistics-Stochastic models, 13(4), 759-774.

Teimouri, M. and Amindavar, H. (2008). A novel approach to calculate stable densities, Proceedings of the World Congress on Engineering, 1, 710-714.

Examples

# In the following, we compute the cdf of a univariate stable distribution at point 2
# with parameters alpha=1.2, beta=0.9, sigma=1, and mu=0 in S_{0} parameterization.
upstab(2,1.2,0.9,1,0,1)

urstab

Description

simulates iid realizations from univariate stable distribution based on formulas given by Chambers et al. (1976) <DOI: 10.1080/01621459.1976.10480344> and Weron (1996) <doi.org/10.1016/0167-7152(95)00113-1>.

Usage

urstab(n,alpha,beta,sigma,mu,param)

Arguments

Value

a vector of n numeric values

Author(s)

Mahdi Teimouri, Adel Mohammadpour, and Saralees Nadarajah

References

Chambers, J. M., Mallows, C. L., and Stuck, B. W. (1976). A method for simulating stable random variables, Journal of the american statistical association, 71(354), 340-344.

Weron, R. (1996). On the Chambers-Mallows-Stuck method for simulating skewed stable random variables, Statistics & probability letters, 28(2), 165-171.

Examples

# By the following example, we simulate n=200 iid realizations from univariate stable
# distribution with parameters alpha=1.2, beta=0.5, sigma=2, and mu=0 in S_0 parameterization.
x <- urstab(200, 1.2, 0.5, 2, 0, 0)

urstab.trunc

Description

using the methodology given by Soltan and Shirvani (2010), Shirvani and Soltani (2013) for simulating iid truncated stable random variable, it simulates truncated stable realizations.

Usage

urstab.trunc(n, alpha, beta, sigma, mu, a, b, param)

Arguments

Value

a vector of n numeric values

Author(s)

Mahdi Teimouri, Adel Mohammadpour, and Saralees Nadarajah

References

Shirvani, A. and Soltani, A. R. (2013). A characterization for truncated Cauchy random variables with nonzero skewness parameter, Computational Statistics, 28(3), 1011-1016.

Soltani, A. R. and Shirvani, A. (2010). Truncated stable random variables: characterization and simulation, Computational Statistics, 25(1), 155-161.

Teimouri, M. and Nadarajah, S. (2013). On simulating truncated stable random variables, Computational Statistics, 28(5), 2367-2377.

Teimouri, M. and Nadarajah, S. (2017). On simulating truncated skewed Cauchy random variables, Communications in Statistics-Simulation and Computation, 46(2), 1318-1321.

Examples

# We simulate n=200 iid realizations from truncated stable distribution with parameters
# alpha=1.3, beta=0.5, sigma=2, and mu=0 which is truncated over (-5,5) in S_0 parameterization.
urstab.trunc(200,1.3,0.5,2,0,-5,5,0)