A Bayesian predictive approach to determining the number of components in a mixture distribution

This paper describes a Bayesian approach to mixture modelling and a method based on predictive distribution to determine the number of components in the mixtures. The implementation is done through the use of the Gibbs sampler. The method is described through the mixtures of normal and gamma distributions. Analysis is presented in one simulated and one real data example. The Bayesian results are then compared with the likelihood approach for the two examples.     

A Note on the Noncentral Beta Distribution Function

The noncentral beta and the related noncentral F distributions have received much attention during the last decade, as is evident from the works of Norton, Lenth, Frick, Lee, Posten, Chattamvelli, and Chattamvelli and Shanmugam. This article reviews the existing algorithms for computing the cumulative distribution function (cdf) of a noncentral beta random variable, and proposes a simple algorithm, based on a sharp error bound, for computing the cdf. A variation of the noncentral beta random variable when the noncentrality is associated only with the denominator χ² and its computational details are also discussed.     

Testing the Fit of Gamma Distributions Using the Empirical Moment Generating Function

ABSTRACT

This article presents goodness-of-fit tests for two and three-parameter gamma distributions that are based on minimum quadratic forms of standardized logarithmic differences of values of the moment generating function and its empirical counterpart. The test statistics can be computed without reliance to special functions and have asymptotic chi-squared distributions. Monte Carlo simulations are used to compare the proposed test for the two-parameter gamma distribution with goodness-of-fit tests employing empirical distribution function or spacing statistics. Two data sets are used to illustrate the various tests.     

Sufficient Conditions for the Existence of a Solution for the Log-Likelihood Equations in the Three-Parameter Gamma Distribution

The existence of maximum likelihood estimators for the three-parameter gamma distribution is still an open problem. It demands the solution of the system of the log-likelihood equations which can be solved only with the use of numerical methods. The aim of this article is to provide sufficient conditions for the existence of a solution for the system of the log-likelihood equations. The conditions are expressed in terms of the geometric, arithmetic, and harmonic mean. Also, the importance of the sign of the third central moment for the existence of a solution is revealed.     

Mixture Models and the Krätzel Integral Transform

Continuous mixture Weibull models arise in many areas of sciences such as reliability studies, communications theory, etc. Due to its wide applicability, we introduce a class of continuous mixture Weibull models which is a combination of Weibull and generalized gamma distributions. Some characteristics of the distribution are obtained. It is seen that Krätzel integral enters into the model naturally, and then the model can be called as a Krätzel density. Applications of the density function related to fading channels and ultrasonic backscatter signals modeling are discussed. A real data analysis is given to illustrate the use of this distribution.     

The skew generalized t distribution as the scale mixture of a skew exponential power distribution and its applications in robust estimation

In this paper, we consider the family of skew generalized t (SGT) distributions originally introduced by Theodossiou [P. Theodossiou, Financial data and the skewed generalized t distribution, Manage. Sci. Part 1 44 (12) ( 1998), pp. 1650–1661] as a skew extension of the generalized t (GT) distribution. The SGT distribution family warrants special attention, because it encompasses distributions having both heavy tails and skewness, and many of the widely used distributions such as Student's t, normal, Hansen's skew t, exponential power, and skew exponential power (SEP) distributions are included as limiting or special cases in the SGT family. We show that the SGT distribution can be obtained as the scale mixture of the SEP and generalized gamma distributions. We investigate several properties of the SGT distribution and consider the maximum likelihood estimation of the location, scale, and skewness parameters under the assumption that the shape parameters are known. We show that if the shape parameters are estimated along with the location, scale, and skewness parameters, the influence function for the maximum likelihood estimators becomes unbounded. We obtain the necessary conditions to ensure the uniqueness of the maximum likelihood estimators for the location, scale, and skewness parameters, with known shape parameters. We provide a simple iterative re-weighting algorithm to compute the maximum likelihood estimates for the location, scale, and skewness parameters and show that this simple algorithm can be identified as an EM-type algorithm. We finally present two applications of the SGT distributions in robust estimation.     

Bivariate generalized gamma distributions of Kibble's type

In this paper, a new type of bivariate generalized gamma (BGG) distribution derived from the bivariate gamma distribution of Kibble [Two-variate gamma-type distribution. Sankh?a 1941;5:137–150] by means of a power transformation is presented. The explicit expressions of statistical properties of the BGG distribution are presented. The estimation of marginal and dependence parameters using the method of moments and the method of inference functions for margins are discussed, and their performance through a Monte Carlo simulation study is assessed. Finally, an example is given to illustrate the applicability of the distributions introduced here.     

Shared frailty models with baseline generalized Pareto distribution

Abstract

In this article, we have considered three different shared frailty models under the assumption of generalized Pareto Distribution as baseline distribution. Frailty models have been used in the survival analysis to account for the unobserved heterogeneity in an individual risks to disease and death. These three frailty models are with gamma frailty, inverse Gaussian frailty and positive stable frailty. Then we introduce the Bayesian estimation procedure using Markov chain Monte Carlo (MCMC) technique to estimate the parameters. We applied these three models to a kidney infection data and find the best fitted model for kidney infection data. We present a simulation study to compare true value of the parameters with the estimated values. Model comparison is made using Bayesian model selection criterion and a well-fitted model is suggested for the kidney infection data.     

Alternative parameter estimators based upon grouped data

The problem of estimating population parameters based upon grouped data is considered and several alternative estimation schemes such as the method of scoring, least lines, least squares, minimum chi square, and a method of approximating method of moments and maximum likelihood estimators are considered. These estimators are compared with maximum likelihood and method of moments estimators based upon individual observations using a Monte Carlo study where the parent population is characterized by a gamma distribution. An application of these techniques to fitting a gamma distribution to 1970-74 census income data is considered.