ForestFit (version 2.4.3)

fitgsm: Estimating parameters of the gamma shape mixture model

Description

Estimates parameters of the gamma shape mixture (GSM) model whose probability density function gets the form as follows. $$f(x,{\Theta}) = \sum_{j=1}^{K}\omega_j \frac{\beta^j}{\Gamma(j)} x^{j-1} \exp\bigl( -\beta x\bigr),$$ where \(\Theta=(\omega_1,\dots,\omega_K, \beta)^T\) is the parameter vector and known constant \(K\) is the number of components. The vector of mixing parameters is given by \(\omega=(\omega_1,\dots,\omega_K)^T\) where \(\omega_j\)s sum to one, i.e., \(\sum_{j=1}^{K}\omega_j=1\). Here \(\beta\) is the rate parameter that is equal for all components.

Usage

fitgsm(data,K)

Value

A list of objects in three parts as

1. The EM estimator of the rate parameter.

2. The EM estimator of the mixing parameters.

3. A sequence of goodness-of-fit measures consist of Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), Hannan-Quinn information criterion (HQIC), Anderson-Darling (AD), Cramer-von Mises (CVM), Kolmogorov-Smirnov (KS), and log-likelihood (log-likelihood) statistics.

Arguments

data

Vector of observations.

K

Number of components.

Author

Mahdi Teimouri

Details

Supposing that the number of components, i.e., \(K\) is known, the parameters are estimated through the EM algorithm developed by the maintainer.

References

A. P. Dempster, N. M. Laird, and D. B. Rubin, 1977. Maximum likelihood from incomplete data via the EM algorithm, Journal of the Royal Statistical Society Series B, 39, 1-38.

S. Venturini, F. Dominici, and G. Parmigiani, 2008. Gamma shape mixtures for heavy-tailed distributions, The Annals of Applied Statistics, 2(2), 756–776.

Examples

n<-100
omega<-c(0.05, 0.1, 0.15, 0.2, 0.25, 0.25)
beta<-2
data<-rgsm(n,omega,beta)
K<-length(omega)
fitgsm(data,K)