A Mixture Model of Proportional Reversed Hazard Rate

This article investigates properties of mixture model of proportional reversed hazard rate. Firstly, the mixing random variable and the overall population variable are proved to be positively likelihood dependent. Secondly, lower bounds for the distribution function as well as the conditional distribution are established in the case that the mixing variable belongs to certain nonparametric classes. Finally, some stochastic orders on the mixing (baseline) variables are proved to be translated to the corresponding overall population variables.     

Ordering extremes of interdependent random variables

For random variables with Archimedean copula or survival copula, we develop the reversed hazard rate order and the hazard rate order on sample extremes in the context of proportional reversed hazard models and proportional hazard models, respectively. The likelihood ratio order on sample maximum is also investigated for the proportional reversed hazard model. Several numerical examples are presented for illustrations as well.     

Some characterization results for parallel systems with two heterogeneous exponential components

In this paper, we study ordering properties of lifetimes of parallel systems with two independent heterogeneous exponential components in terms of the likelihood ratio order (reversed hazard rate order) and the hazard rate order (stochastic order). We establish, among others, that the weakly majorization order between two hazard rate vectors is equivalent to the likelihood ratio order (reversed hazard rate order) between lifetimes of two parallel systems, and that the p-larger order between two hazard rate vectors is equivalent to the hazard rate order (stochastic order) between lifetimes of two parallel systems. Moreover, we extend the results to the proportional hazard rate models. The results derived here strengthen and generalize some of the results known in the literature.     

A unified approach to stochastic comparisons of multivariate mixture models

In this paper, we consider a unified approach to stochastic comparisons of random vectors corresponding to two general multivariate mixture models. These stochastic comparisons are made with respect to multivariate hazard rate, reversed hazard rate and likelihood ratio orders. As an application, results are presented for stochastic comparisons of generalized multivariate frailty models.     

Ascertainment-Adjusted Maximum Likelihood Estimation for the Additive Genetic Gamma Frailty Model

The additive genetic gamma frailty model has been proposed for genetic linkage analysis for complex diseases to account for variable age of onset and possible covariates effects. To avoid ascertainment biases in parameter estimates, retrospective likelihood ratio tests are often used, which may result in loss of efficiency due to conditioning. This paper considers when the sibships are ascertained by having at least two affected sibs with the disease before a given age and provides two approaches for estimating the parameters in the additive gamma frailty model. One approach is based on the likelihood function conditioning on the ascertainment event, the other is based on maximizing a full ascertainment-adjusted likelihood. Explicit forms for these likelihood functions are derived. Simulation studies indicate that when the baseline hazard function can be correctly pre-specified, both approaches give accurate estimates of the model parameters. However, when the baseline hazard function has to be estimated simultaneously, only the ascertainment-adjusted likelihood method gives an unbiased estimate of the parameters. These results imply that the ascertainment-adjusted likelihood ratio test in the context of the additive genetic gamma frailty may be used for genetic linkage analysis.     

Stochastic comparisons of sample spacings in single- and multiple-outlier exponential models

This article studies some ordering results for the sample spacings arising from the single- and multiple-outlier exponential models. In the single-outlier exponential models, it is shown that the weak majorization order between the two hazard rate vectors implies the hazard rate order as well as the dispersive order between the corresponding sample spacings. We also extend this result from the single-outlier model to the multiple-outlier model for the special case of the second sample spacing. Furthermore, we obtain some necessary and sufficient conditions such that, on the one hand, the hazard rate, dispersive and usual stochastic orders, and on the other hand, the likelihood ratio and reversed hazard rate orders of the second sample spacings from two independent heterogeneous exponential random variables are equivalent.     

On the Mixture of Proportional Odds Models

This article studies the mixed proportional odds model. We build TP2 dependence between the overall population variable and the unobservable covariate and present some preservation properties. Relations on aging characteristics, such as odds function and (reversed) hazard rate, are discussed. Stochastic comparisons on overall population variables are conducted as well.     

Stochastic comparisons and properties of conditional generalized order statistics

Generalized order statistics introduced by Kamps [1995. A concept of generalized order statistics. J. Statist. Plann. Inference 48, 1–23] provides a unified approach to a variety of models concerning ordered random variables. This paper carries out stochastic comparisons of conditional generalized order statistics in terms of the likelihood ratio order and the hazard rate order from two samples, and establishes some stochastic monotonicity properties. The main results strengthen and generalize the corresponding results established recently in the literature. Finally, some applications of the main results are given as well.     

Reversed hazard rate order of equilibrium distributions and a related aging notion

This paper deals with preservation of the reversed hazard rate order between equilibrium random variables under formations of some reliability structures. We further investigate a new aging notion based upon the reversed hazard rate order between a random life and its equilibrium version. A nonparametric method is developed to test the exponentiality against such a strict aging property, some numerical results are presented as well. This research was supported by the Program for New Century Excellent Talent in University, China.     

On Redundancy Allocation to Series and Parallel Systems of Two Components

Abstract

For two components and one standby redundancy, we develop a characterization on the hazard rate order and the reversed hazard rate order of the redundant system lifetime in the context of mutually independent components lifetimes. Also, the likelihood ratio order is derived on the lifetime of the series system with two components lifetimes and two matched active redundancies lifetimes both following the proportional hazard model.     

Modeling Multilevel Survival Data Using Frailty Models

Often the dependence in multivariate survival data is modeled through an individual level effect called the frailty. Due to its mathematical simplicity, the gamma distribution is often used as the frailty distribution for hazard modeling. However, it is well known that the gamma frailty distribution has many drawbacks. For example, it weakens the effect of covariates. In addition, in the presence of a multilevel model, overall frailty comes from several levels. To overcome such drawbacks, more heavy-tailed distributions are needed to model the frailty distribution in order to incorporate extra variability. In this article, we develop a class of log-skew-t distributions for the frailty. This class includes the log-normal distribution along with many other heavy tailed distributions, e.g., log-Cauchy, log normal, and log-t as special cases.

Conditional on the frailty, the survival times are assumed to be independent with proportional hazard structure. The modeling process is then completed by assuming multilevel frailty-effects. Instead of tuning a strict parameterization of the baseline hazard function, we consider the partial likelihood approach and thus leave the baseline function unspecified. By eliminating the hazard, the pre-specification and computation are simplified considerably.     

Penalized Empirical Likelihood via Bridge Estimator in Cox's Proportional Hazard Model

We propose the penalized empirical likelihood method via bridge estimator in Cox's proportional hazard model for parameter estimation and variable selection. Under reasonable conditions, we show that penalized empirical likelihood in Cox's proportional hazard model has oracle property. A penalized empirical likelihood ratio for the vector of regression coefficients is defined and its limiting distribution is a chi-square distributions. The advantage of penalized empirical likelihood as a nonparametric likelihood approach is illustrated in testing hypothesis and constructing confidence sets. The method is illustrated by extensive simulation studies and a real example.     

Comparisons between parallel systems with exponentiated generalized gamma components

Consider two parallel systems with their independent components’ lifetimes following heterogeneous exponentiated generalized gamma distributions, where the heterogeneity is in both shape and scale parameters. We then obtain the usual stochastic (reversed hazard rate) order between the lifetimes of two systems by using the weak submajorization order between the vectors of shape parameters and the p-larger (weak supermajorization) order between the vectors of scale parameters, under some restrictions on the involved parameters. Further, by reducing the heterogeneity of parameters in each system, the usual stochastic (reversed hazard rate) order mentioned above is strengthened to the hazard rate (likelihood ratio) order. Finally, two characterization results concerning the comparisons of two parallel systems, one with independent heterogeneous generalized exponential components and another with independent homogeneous generalized exponential components, are derived. These characterization results enable us to find some lower and upper bounds for the hazard rate and reversed hazard rate functions of a parallel system consisting of independent heterogeneous generalized exponential components. The results established here generalize some of the known results in the literature, concerning the comparisons of parallel systems under generalized exponential and exponentiated Weibull models.     

Stochastic comparisons in frailty models

The frailty model in survival analysis accounts for unobserved heterogeneity between individuals by assuming that the hazard rate of an individual is the product of an individual specific quantity, called “frailty” and a baseline hazard rate. It is well known that the choice of the frailty distribution strongly affects the nonparametric estimate of the baseline hazard as well as that of the conditional probabilities. This paper reviews the basic concepts of a frailty model, presents various probability inequalities and other monotonicity results which may prove useful in choosing among alternative specifications. More specifically, our main result lies in seeing how well known stochastic orderings between distributions of two frailities translate into orderings between the corresponding survival functions. Some probabilistic aspects and implications of the models resulting from competing choices of the distributions of frailty or the baseline are compared.     

Hazard rate ordering of the second-order statistics from multiple-outlier PHR samples

In this paper, we compare the hazard rate functions of the second-order statistics arising from two sets of independent multiple-outlier proportional hazard rates (PHR) samples. It is proved that the submajorization order between the sample size vectors together with the supermajorization order between the hazard rate vectors imply the hazard rate ordering between the corresponding second-order statistics from multiple-outlier PHR random variables. The results established here provide theoretical guidance both for the winner's price for the bid in the second-price reverse auction in auction theory and fail-safe system design in reliability. Some numerical examples are also provided for illustration.     

Ordering properties of largest order statistics from independent and heterogeneous Dagum populations

Abstract

The Dagum distribution has been extensively used to model income data, and its features have been appreciated in economics and financial studies. In this article, we discuss ordering properties of largest order statistics from independent and heterogeneous Dagum populations. We present some sufficient conditions for stochastic comparisons between largest order statistics in terms of the reversed hazard rate order, the usual stochastic order, the convex order, the likelihood ratio order and the dispersive order. Several numerical examples are presented to illustrate the results established here.     

Shared frailty models based on reversed hazard rate for modified inverse Weibull distribution as baseline distribution

The unknown or unobservable risk factors in the survival analysis cause heterogeneity between individuals. Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times, the shared frailty models were suggested. The most common shared frailty model is a model in which frailty act multiplicatively on the hazard function. In this paper, we introduce the shared gamma frailty model and the inverse Gaussian frailty model with the reversed hazard rate. We introduce the Bayesian estimation procedure using Markov chain Monte Carlo (MCMC) technique to estimate the parameters involved in the model. We present a simulation study to compare the true values of the parameters with the estimated values. We also apply the proposed models to the Australian twin data set and a better model is suggested.     

On the Mixture Proportional Mean Residual Life Model

In this article, we propose a new mixture model induced by the model of proportional mean residual life. Under some appropriate assumptions, it is shown that the mixing and overall variables in the model admit the positive likelihood ratio dependence structure. To see how the overall variable is affected by the stochastic variation of the mixing variable, we study some stochastic comparisons using these variables. Finally, some useful bounds for tail probability of the overall variable for large values of the mixing variable are derived.     

Ordering results on extremes of scaled random variables with dependence and proportional hazards

This study deals with random variables equipped with Archimedean copulas and following scale proportional hazards (SPHs) or revered hazards models. We build the usual stochastic order both between minimums of two SPHs samples with Archimedean survival copulas and between maximums from two scale proportional reversed hazards (PRHs) samples with Archimedean copulas. The hazard rate order between minimums of independent SPHs samples and the reversed hazard rate order between maximums of independent scale PRHs samples are both derived. Also we have a discussion on the dispersive order between minimums from samples with a common Archimedean survival copula. The present results either generalize or improve some related ones in the recent literature.     

Modelling survival data using mixtures of frailties

Frailty models are often used to describe the extra heterogeneity in survival data by introducing an individual random, unobserved effect. The frailty term is usually assumed to act multiplicatively on a baseline hazard function common to all individuals. In order to apply the frailty model, a specific frailty distribution has to be assumed. If at least one of the latent variables is continuous, the frailty must follow a continuous distribution. In this paper, a finite mixture of continuous frailty distributions is used in order to describe situations in which one (or more) of the latent variables separates the population in study into two (or more) subpopulations. Closure properties of the unobserved quantity are given along with the maximum-likelihood estimates under the most common choices of frailty distributions. The model is illustrated on a set of lifetime data.