mixSSG (version 2.1.1)

stoch: 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)

Value

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

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.

Author

Mahdi Teimouri

Details

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

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)