Dynamic mixing probability measures of mixtures

ABSTRACT

Lifetime of heterogeneous population can be modeled as mixture of a family of lifetime distributions according to a mixing probability measure. With the help of dynamic mixing measure, the hazard rate of the mixture can also be expressed as the mixture of the hazard rates of the lifetime distributions. Various local stochastic orderings are defined in this article. Applying these local stochastic orderings, we can explore the behavior of the dynamic mixing measures locally and then compare the hazard rates of two heterogeneous populations in both the local and global ways.     

On the monotonic properties of discrete failure rates

As is well known, the monotonicity of failure rate of a life distribution plays an important role in modeling failure time data. In this paper, we develop techniques for the determination of increasing failure rate (IFR) and decreasing failure rate (DFR) property for a wide class of discrete distributions. Instead of using the failure rate, we make use of the ratio of two consecutive probabilities. The method developed is applied to various well known families of discrete distributions which include the binomial, negative binomial and Poisson distributions as special cases. Finally, a formula is presented to determine explicitly the failure rate of the families considered. This formula is used to determine the failure rate of various classes of discrete distributions. These formulas are explicit but complicated and cannot normally be used to determine the monotonicity of the failure rates.     

DETERMINATION OF CHANGE POINTS OF NON-MONOTONIC FAILURE RATES

This paper was motivated by the problem of the determination of the change points of the failure rate of a mixture of two gamma distributions. For certain values of the parameters the existing methods are not applicable since, in this case, there are two turning points of the failure rate. Thus, we extend the results to models having two or more turning points of the failure rates. The extended procedure helps us to identify failure rates of more complex forms. Finally, the mixture gamma case is completely resolved employing theoretical, graphical and numerical techniques wherever necessary.     

Multivariate discrete reversed hazard rates

In the present paper, we define and study four versions of multivariate discrete reversed hazard rates, namely scalar reversed hazard rate, vector reversed hazard rate, alternative reversed hazard rate, and conditional reversed hazard rate. Various properties of these functions are studied. Interrelationships between these reversed hazard rates are explored. We also present characterization of discrete distributions using these reversed hazard rates.     

On the Characterization of a Bivariate Geometric Distribution

In this article it is shown that a bivariate random variable has a constant failure rate, and mixture geometric marginals, if and only if, it has the loss of memory property and the discrete Freund distribution. This characterization is achieved by extending a key lemma in this area. The mixture geometric can be collapsed to geometric marginals, thus validating the results.     

The generalized inverse Weibull distribution

The inverse Weibull distribution has the ability to model failure rates which are quite common in reliability and biological studies. A three-parameter generalized inverse Weibull distribution with decreasing and unimodal failure rate is introduced and studied. We provide a comprehensive treatment of the mathematical properties of the new distribution including expressions for the moment generating function and the rth generalized moment. The mixture model of two generalized inverse Weibull distributions is investigated. The identifiability property of the mixture model is demonstrated. For the first time, we propose a location-scale regression model based on the log-generalized inverse Weibull distribution for modeling lifetime data. In addition, we develop some diagnostic tools for sensitivity analysis. Two applications of real data are given to illustrate the potentiality of the proposed regression model.     

Tail Behavior of the Failure Rate Functions of Mixtures

The tail behavior of the failure rate of mixtures of lifetime distributions is studied. A typical result is that if the failure rate of the strongest component of the mixture decreases to a limit, then the failure rate of the mixture decreases to the same limit. For a class of distributions containing the gamma distributions this result can be improved in the sense that the behavior of the failure rate of the mixture asymptotically mirrors that of the strongest component in whether it decreases or increases to a limit. This revised version was published online in July 2006 with corrections to the Cover Date.     

Finite mixture of Burr type XII distribution and its reciprocal: properties and applications

In this paper, the finite mixture of Burr type XII distribution with its reciprocal, is proposed as a failure model. The failure rate (FR) of the new model covers several types of failure rates. It is shown that depending on the parameter values, the model is capable of covering different combinations of failure rates. A study of the behavior of the FR curve of the model is made.     

Ordering Properties of Convolutions of Exponential Random Variables

Convolutions of independent random variables are usually compared. In this paper, after a synthetic comparison with respect to hazard rate ordering between sums of independent exponential random variables, we focus on the special case where one sum is identically distributed. So, for a given sum of n independent exponential random variables, we deduce the "best" Erlang-n bounds, with respect to each of the usual orderings: mean ordering, stochastic ordering, hazard rate ordering and likelihood ratio ordering.     

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.     

Some distributional results through past entropy

Measure of uncertainty in past lifetime distribution plays an important role in the context of Information Theory, Forensic Science and other related fields. In this paper we provide characterizations of quite a few continuous and discrete distributions based on certain functional relationships among past entropy, reversed hazard rate and expected inactivity time. Based on past entropy, a conditional measure of uncertainty has been defined, which has helped in defining a new stochastic order and an ageing class. The properties of the stochastic order and those of the ageing class are also studied here.     

Further results for parallel systems with two heterogeneous exponential components

In this paper, we investigate ordering properties of lifetimes of parallel systems with two independent heterogeneous exponential components with respect to likelihood ratio and hazard rate orders. Two sufficient conditions are provided for likelihood ratio and hazard rate orders to hold between the lifetimes of two parallel systems, respectively. Moreover, we extend the results from exponential case to the proportional hazard rate models. The results established here strength some of the results known in the literature. Finally, some numerical examples are given to illustrate the theoretical results derived here 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 dynamic system for Gompertz model

Two-parameter Gompertz distribution has been introduced as a lifetime model for reliability inference recently. In this paper, the Gompertz distribution is proposed for the baseline lifetimes of components in a composite system. In this composite system, failure of a component induces increased load on the surviving components and thus increases component hazard rate via a power-trend process. Point estimates of the composite system parameters are obtained by the method of maximum likelihood. Interval estimates of the baseline survival function are obtained by using the maximum-likelihood estimator via a bootstrap percentile method. Two parametric bootstrap procedures are proposed to test whether the hazard rate function changes with the number of failed components. Intensive simulations are carried out to evaluate the performance of the proposed estimation procedure.     

Stochastic properties of conditionally independent mixture models

This paper deals with aging properties and stochastic comparisons of multivariate mixture models, having conditionally independent marginal distributions.     

On the failure rate estimation of the inverse gaussian distribution

New estimators of the inverse Gaussian failure rate are proposed based on the maximum likelihood predictive densities derived by Yang (1999). These estimators are compared, via Monte Carlo simulation, with the usual maximum likelihood estimators of the failure rate and found to be superior in terms of bias and mean squared error. Sensitivity of the estimators against the departure from the inverse Gaussian distribution is studied.     

Dynamic proportional hazard rate and reversed hazard rate models

Cox (1972) proportional hazard (PH) model has been used to model failure time data in Reliability and Survival Analysis. Recently, proportional reversed hazard model has been analyzed in the literature. Sometimes, the hazard rate (or the reversed hazard rate) may not be proportional over the whole time interval, but may be proportional differently in different intervals. In order to take care of this kind of problems, in this paper, we introduce the dynamic proportional hazard rate model, and the dynamic proportional reversed hazard rate model, and study their properties for different aging classes. The closure of the models under different stochastic orders has also been studied. Examples are presented to illustrate different properties of the models.     

Reliability Estimation Based on System Data with an Unknown Load Share Rule

We consider a multicomponent load-sharing system in which the failure rate of a given component depends on the set of working components at any given time. Such systems can arise in software reliability models and in multivariate failure-time models in biostatistics, for example. A load-share rule dictates how stress or load is redistributed to the surviving components after a component fails within the system. In this paper, we assume the load share rule is unknown and derive methods for statistical inference on load-share parameters based on maximum likelihood. Components with (individual) constant failure rates are observed in two environments: (1) the system load is distributed evenly among the working components, and (2) we assume only the load for each working component increases when other components in the system fail. Tests for these special load-share models are investigated.     

A new four-parameter discrete distribution with bathtub and unimodal failure rate

In this paper, a discrete counterpart of the general class of continuous beta-G distributions is introduced. A discrete analog of the beta generalized exponential distribution of Barreto-Souza et al. [2], as an important special case of the proposed class, is studied. This new distribution contains some previously known discrete distributions as well as two new models. The hazard rate function of the new model can be increasing, decreasing, bathtub-shaped and upside-down bathtub. Some distributional and moment properties of the new distribution as well as its order statistics are discussed. Estimation of the parameters is illustrated using the maximum likelihood method and, finally, the model with a real data set is examined.     

On Mixture Failure Rates Ordering

Mixtures of increasing failure rate distributions (IFR) can decrease at least in some intervals of time. Usually, this property can be observed asymptotically as t → ∞. This is due to the fact that the mixture failure rate is “bent down” compared with the corresponding unconditional expectation of the baseline failure rate, which was proved previously for some specific cases. We generalize this result and discuss the “weakest populations are dying first” property, which leads to the change in the failure rate shape. We also consider the problem of mixture failure rate ordering for the ordered mixing distributions. Two types of stochastic ordering are analyzed: ordering in the likelihood ratio sense and ordering in variances when the means are equal.