mixSSG (version 2.1.1)

fitBayes: Estimating parameters of the symmetric \(\alpha\)-stable (S\(\alpha\)S) distribution using Bayesian paradigm.

Description

Let \({{y}}_1,{{y}}_2, \cdots,{{y}}_n\) are \(n\) realizations form S\(\alpha\)S distribution with parameters \(\alpha, \sigma\), and \(\mu\). Herein, we estimate parameters of symmetric univariate stable distribution within a Bayesian framework. We consider a uniform distribution for prior of tail thickness, that is \(\alpha \sim U(0,2)\). The normal and inverse gamma conjugate priors are designated for \(\mu\) and \(\sigma^2\) with density functions given, respectively, by $$ \pi(\mu)=\frac{1}{\sqrt{2\pi}\sigma_{0}}\exp\Bigl\{-\frac{1}{2}\Bigl(\frac{\mu-\mu_0}{\sigma_0}\Bigr)^{2}\Bigr\}, $$ and $$ \pi(\delta)= \delta_{0}^{\gamma_{0}}\delta^{-\gamma_0-1}\exp\Bigl\{-\frac{\delta_0}{\delta}\Bigr\}, $$ where \(\mu_0 \in R\), \(\sigma_0>0\), \(\delta=\sigma^2\), \(\delta_0>0\), and \(\gamma_0>0\).

Usage

fitBayes(y, mu0, sigma0, gamma0, delta0, epsilon)

Value

Estimated tail thickness, location, and scale parameters, number of iterations to attain convergence, the log-likelihood value across iterations, the Bayesian information criterion (BIC), and the Akaike information criterion (AIC).

Arguments

y

vector of realizations that following S\(\alpha\)S distribution.

mu0

the location hyperparameter corresponding to \(\pi(\mu)\).

sigma0

the standard deviation hyperparameter corresponding to \(\pi(\mu)\).

gamma0

the shape hyperparameter corresponding to \(\pi(\delta)\).

delta0

the rate hyperparameter corresponding to \(\pi(\delta)\).

epsilon

a positive small constant playing the role of threshold for stopping sampler.

Author

Mahdi Teimouri

Examples

n <- 100
alpha <- 1.4
mu <- 0
sigma <- 1
y <- rnorm(n)
fitBayes(y, mu0 = 0, sigma0 = 0.2, gamma0 = 10e-5, delta0 = 10e-5, epsilon = 0.005)