stoch {mixSSG}

Estimating the tail index of the skewed sub-Gaussian stable distribution using the stochastic EM algorithm given that other parameters are known.

Description

Suppose {\boldsymbol{Y}}_1,{\boldsymbol{Y}}_2, \cdots,{\boldsymbol{Y}}_n are realizations following d-dimensional skewed sub-Gaussian stable distribution. Herein, we estimate the tail thickness parameter 0<\alpha \leq 2 when \boldsymbol{\mu} (location vector in {{{R}}}^{d}, \boldsymbol{\lambda} (skewness vector in {{{R}}}^{d}), and \Sigma (positive definite symmetric dispersion matrix are assumed to be known.

Usage

stoch(Y, alpha0, Mu0, Sigma0, Lambda0)

Arguments

Y

a vector (or an n\times d matrix) at which the density function is approximated.

alpha0

initial value for the tail thickness parameter.

Mu0

a vector giving the initial value for the location parameter.

Sigma0

a positive definite symmetric matrix specifying the initial value for the dispersion matrix.

Lambda0

a vector giving the initial value for the skewness parameter.

Details

Here, we assume that parameters {\boldsymbol{\mu}}, {\boldsymbol{\lambda}}, and \Sigma are known and only the tail thickness parameter needs to be estimated.

Value

Estimated tail thickness parameter \alpha, of the skewed sub-Gaussian stable distribution.

Author(s)

Mahdi Teimouri

Examples

n <- 100
alpha <- 1.4
Mu <- rep(0, 2)
Sigma <- diag(2)
Lambda <- rep(2, 2)
Y <- rssg(n, alpha, Mu, Sigma, Lambda)
stoch(Y, alpha, Mu, Sigma, Lambda)