Model selection: some generalized distributions

Many models have been used to represent the distributions of random variables in statistics, engineering, business, and the physical and social science. This paper considers two, four-parameter generalized bea distributions that include nearly all the models actually used as special or limiting cases. Properties and the interrelationships among these distributions are considered. Expressions are reported that facilitate parameter estimation and the analysis of associated means, variances, hazard functions and other distributional characteristics.

Estimation procedures corresponding to different data types are considered. Maximum likelihood estimation is used and the value of the likelihood function provides and important criterion for model selection. The relative performance of the various models is compared for several data sets.     

Bhattacharyya and Kshirsagar Bounds in Generalized Gamma Distribution

In this article, we consider the exact computation of the famous halfspace depth (HD) and regression depth (RD) from the view of cutting a convex cone with hyperplanes. Two new algorithms are proposed for computing these two notions of depth. The first one is relatively straightforward but quite inefficient, whereas the second one is much faster. It is noteworthy that both of them can be implemented to spaces with dimension beyond three. Some numerical examples are also provided in what follows to illustrate the performances.     

A note on gamma difference distributions

It is the aim of this note to point out that the double gamma difference distribution recently introduced by [Augustyniak M, and Doray, LG. Inference for a leptokurtic symmetric family of distributions represented by the difference of two gamma variables. J Statist Comput Simul. 2012;82:1621–1634] is well known in financial econometrics: it is the symmetric variance gamma family of distributions. We trace back to the various origins of this distribution. In addition, we consider in some detail the difference of two independent gamma distributed random variables with different shape parameters.     

Exponentiated models preserve stochastic orderings of parallel and series systems

Abstract

In this paper, we establish that the usual stochastic, hazard rate, reversed hazard rate, likelihood ratio, dispersive and star orders are all preserved for parallel systems under exponentiated models for lifetimes of components. We then use the multiple-outlier exponentiated gamma models to illustrate this result. Finally, we consider the dual family with exponentiated survival function and establish similar results for series systems. The results established here extend some well-known results for series and parallel systems arising from different exponentiated distributions such as generalized exponential and exponentiated Weibull, established previously in the literature.     

Series evaluation of Tweedie exponential dispersion model densities

Exponential dispersion models, which are linear exponential families with a dispersion parameter, are the prototype response distributions for generalized linear models. The Tweedie family comprises those exponential dispersion models with power mean-variance relationships. The normal, Poisson, gamma and inverse Gaussian distributions belong to theTweedie family. Apart from these special cases, Tweedie distributions do not have density functions which can be written in closed form. Instead, the densities can be represented as infinite summations derived from series expansions. This article describes how the series expansions can be summed in an numerically efficient fashion. The usefulness of the approach is demonstrated, but full machine accuracy is shown not to be obtainable using the series expansion method for all parameter values. Derivatives of the density with respect to the dispersion parameter are also derived to facilitate maximum likelihood estimation. The methods are demonstrated on two data examples and compared with with Box-Cox transformations and extended quasi-likelihoood.     

Accelerated Degradation Models for Failure Based on Geometric Brownian Motion and Gamma Processes

Based on a generalized cumulative damage approach with a stochastic process describing degradation, new accelerated life test models are presented in which both observed failures and degradation measures can be considered for parametric inference of system lifetime. Incorporating an accelerated test variable, we provide several new accelerated degradation models for failure based on the geometric Brownian motion or gamma process. It is shown that in most cases, our models for failure can be approximated closely by accelerated test versions of Birnbaum–Saunders and inverse Gaussian distributions. Estimation of model parameters and a model selection procedure are discussed, and two illustrative examples using real data for carbon-film resistors and fatigue crack size are presented.     

The Posterior Distribution of the Parameters of Component Lifetimes Based on Autopsy Data in a Shock Model

In this paper we consider a binary, monotone system whose component states are dependent through the possible occurrence of independent common shocks, i.e. shocks that destroy several components at once. The individual failure of a component is also thought of as a shock. Such systems can be used to model common cause failures in reliability analysis. The system may be a technological one, or a human being. It is observed until it fails or dies. At this instant, the set of failed components and the failure time of the system are noted. The failure times of the components are not known. These are the so-called autopsy data of the system. For the case of independent components, i.e. no common shocks, Meilijson (1981), Nowik (1990), Antoine et al . (1993) and GTsemyr (1998) discuss the corresponding identifiability problem, i.e. whether the component life distributions can be determined from the distribution of the observed data. Assuming a model where autopsy data is known to be enough for identifia bility, Meilijson (1994) goes beyond the identifiability question and into maximum likelihood estimation of the parameters of the component lifetime distributions based on empirical autopsy data from a sample of several systems. He also considers life-monitoring of some components and conditional life-monitoring of some other. Here a corresponding Bayesian approach is presented for the shock model. Due to prior information one advantage of this approach is that the identifiability problem represents no obstacle. The motivation for introducing the shock model is that the autopsy model is of special importance when components can not be tested separately because it is difficult to reproduce the conditions prevailing in the functioning system. In Gåsemyr & Natvig (1997) we treat the Bayesian approach to life-monitoring and conditional life- monitoring of components     

具扩散项的带免赔额的Erlang(2)问题的破产概率

本文从具有扩散项的模型出发,最终在个体理赔额服从Erlang(2)分布情形下利用解高阶微分-差分方程和鞅的办法得到了与免赔额d有关的原、再保的相应破产概率ψ(u)以及调节系数的表达公式。所得结果不仅仅直接的推广了文献[7]的相关结论,尤其在再保险的场合下研究该模型的文献还不多见,而且在保险资金可以入市的经济背景下也是具有现实意义的。     

Computations via Auxiliary Random Functions for Survival Models

Abstract.  A new simulation method, auxiliary random functions is introduced. When used within a Gibbs sampler, this method enables a unified treatment of exact, right-censored, left-censored, left-truncated and interval censored data, with and without covariates in survival models. The models and methods are exemplified via illustrative analysis.