THE NBELC AND NWELC CLASSES OF LIFE DISTRIBUTIONS

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SOME MULTIVARIATE DMRL AND NBUE DEFINITIONS BASED ON CONDITIONAL STOCHASTIC ORDER

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Proximity between life distributions and exponential distributions (I)

Many classes of life distributions have been introduced into reliability theory. Because of the importance of exponential distributions in reliability theory, it is interesting to study the difference between life distributions and exponential distributions. In this paper, we study the proximity between the life distribution in various classes and the exponential distribution. We shall give some simple upper bounds.This research was partially supported by the National Natural Science Foundation of China.     

Multivariate new better than used in expectation distributions

Various definitions of multivariate new better than used in expectation (MNBUE) life distributions are introduced and their interrelationships are studied. The definitions are multivariate generalizations of the very useful univariate aging property new better than used in expectations (NBUE), which is weaker than new better than used (NBU) and decreasing mean residual life (DMRL) properties. Various closure properties of the different MNBUE classes are proved. Some examples are given to illustrate the relationship between the proposed MNBUE properties and MNBUE properties presented by Buchanan and Singpurwalla (1977). Examples of well known life distributions, that give rise to the MNBUE distributions are discussed. We also study the dual multivariate new worse than used in expectation (MNWUE) distributions.     

SOME NEW PARTIAL ORDERINGSDESCRIBING AGING PROPERTY

The concept of aging, which is very users in relability describes the adverse orbeneficial effects of age on the lifetime of systems. Several partial orderings of life distributionshave been introduced. In this article,some more partial orderings of life distributions are imtroduced mainly by means of the equilibrium distribution. And conversely, a group of new aging clasps oflife distrbutions can be derived from these orderings.     

Characterizations of geometric distributions based on moments

Restricted to certain classes of discrete life distributions, and based on moments, conditional binomial moments, order statistics, spacing or record values, characterizations of geometric distributions are given.This project is supported by the National Natural Science Foundation of China Grant No. 1880492.     

Proximity between life distributions and exponential distributions (II)

This paper is a continuation of the foregoing one [1]. In this paper, we study the proximity between exponential distributions and life distributions in variousW-type classes, whereW-type classes indicate DFR, DFRA, NWU, NWUE, IMRL and HNWUE.This research was partially supported by the National Natural Science Foundation of China.     

Representing the mean residual life in terms of the failure rate

In survival or reliability studies, the mean residual life or life expectancy is an important characteristic of the model. Whereas the failure rate can be expressed quite simply in terms of the mean residual life and its derivative, the inverse problem—namely that of expressing the mean residual life in terms of the failure rate—typically involves an integral of a complicated expression. In this paper, we obtain simple expressions for the mean residual life in terms of the failure rate for certain classes of distributions which subsume many of the standard cases. Several results in the literature can be obtained using our approach. Additionally, we develop an expansion for the mean residual life in terms of Gaussian probability functions for a broad class of ultimately increasing failure rate distributions. Some examples are provided to illustrate the procedure.     

Asymptotic behaviour of reliability functions

We relate the classes of life distributions based on notions of aging with those that depend on the tail behaviour. This last classification introduces the concepts of outlier-neutral, outlier-prone and outlier-resistant distributions. We prove some results that connect IFR, IFRA, NBU and NBUE definitions with the outlier-resistance concept. Similarly the relationship of DFR, DFRA, NWU and NWUE to outlier-proneness is obtained.     

Reliability Studies of Bivariate Distributions with Pareto Conditionals

In this paper we study Arnold's (1987, Statist. Probab. Lett.5, 263–266) class of bivariate distributions with Pareto conditionals from a reliability point of view. Failure rates and mean residual life function of the marginal distributions and their monotonic properties are studied. The hazard components and their properties are investigated and their relationships with some measures of dependence are established. Finally, the failure rate of the minimum of the two components is examined and its monotonicity is investigated. Some of the results presented here are general and would be useful in studying the dependence structure in other classes of bivariate distributions.     

A new construction for skew multivariate distributions

This paper considers a new approach to develop a very general class of skew multivariate distributions. The approach is based on a linear combination of an elliptically distributed random variable with a linear constraint. Using this approach two different classes of multivariate distributions are constructed based on original distribution. These new classes include different types of skew normal (type A and type B) and other skew elliptical distributions, exist in the literature. We also derive the moment generating function, marginal and conditional density of our proposed classes of distributions. Straightforward explanations are applied to demonstrate the relationships among previous approaches by others with our proposed class of skew distributions.     

Characterizations of the exponential distribution by stochastic ordering properties of the geometric compound

Under the reliability NBU/NWU conditions, the exponential distribution is characterized by stochastic ordering properties which link the geometric compound with minimum order statistics or spacings of order statistics. This somewhat answers a question posed by Kakosyan, Klebanov and Melamed (1984,Characterization of Distributions by the Method of intensively Monotone Operators, Springer, New York). We also show the related results based on the residual life in a renewal process and on record values. Finally, some fundamental properties of the NBUC/NWUC classes of life distributions are investigated.     

On Structural and Asymptotic Properties of Some Classes of Distributions

We consider the two-parametric classes of distributions which comprise exponential dispersion models possessing a simple structure. Some of our models are related to Galton-Watson processes and branching diffusions. We study asymptotic properties of related classes of distributions emphasizing their convergence to Poisson-exponential law. We establish/refute infinite divisibility for certain classes of distributions. The critical value for the deflation of zeros from a geometric law which preserves infinite divisibility is determined. A class of distributions related to zero-modified geometric and logarithmic laws is considered.     

Relationships among classes of spherical matrix distributions

Several classes of spherical matrix distributions have been studied by many authors. In this paper the relationships among the classes are discussed in the following aspects: the characteristic functions in these classes, coordinate transformations, marginal distributions, marginal densities, and sphericity. Some statistics that are invariant in these classes are listed for applications.This paper was printed as Technical Report No. 10, ARO Contract DAAG 29-82-K-0156, Department of Statistics, Stanford University, Stanford, California, April 1984.     

The preservation of classes of discrete distributions under convolution and mixing

In this paper we consider some widely utilized classes of discrete distributions and aim to provide a systematic overview about their preservation under convolution and mixing. Moreover, inclusion properties among these classes are discussed. This paper will serve as a detailed reference for the study and applications of the preservation of the classes of discrete distributions.     

Multivariate Meixner classes of invariant distributions

We define multivariate Meixner classes of invariant distributions of random matrices as those whose generating functions for the associated orthogonal polynomials are of certain special integral or summation forms, generalizing the univariate Meixner classes of distributions which were first characterized by Meixner [21]. Characterization theorems and properties of these multivariate Meixner classes are established. The zonal polynomials, the extended invariant polynomials with matrix arguments, and their related results in multivariate distribution theory are utilized in the discussion.     

Some Classes of Multivariate Life Distributions in Discrete Time

New classes of multivariate survival distribution functions based on monotonic behaviour of a multivariate failure rate are developed in the discrete set up. Relationship among the classes along with multivariate geometric distributions that act as boundaries of the various classes are identified.     

The expected discounted penalty function for two classes of risk processes perturbed by diffusion with multiple thresholds

In this paper, we consider a perturbed risk model with two independent classes of risks under multiple thresholds in which both of the two inter-claim times have phase-type distributions. We obtain the integro-differential equations with boundary conditions for the expected discounted penalty function. Explicit expressions are derived if the two classes claim amount distributions both belong to the rational family.     

The maximum surplus before ruin for two classes of perturbed risk model

In this paper, we consider the distribution of the maximum surplus before ruin in a perturbed risk model with two independent classes of risks, in which both of the two inter-claim times have phase-type distributions. We obtain the integro-differential equations for the distribution of the maximum surplus before ruin. Explicit expressions are derived if the two classes claim amount distributions both belong to the rational family.     

On HNBUE class after specific age

In this article we introduce new classes of life distributions namely harmonic new better (worse) than used in expectation after specific age t₀ HNBUE¹t₀ (HNWUE¹t₀). The closure properties under various reliability operations such as convolution, mixture, mixing and the homogeneous Poisson shock model of these classes are studied. Furthermore, nonparametric test is proposed to test exponentiality vs. the HNBUE¹t₀ class. The critical values and the powers of this test are calculated to assess the performance of the test. It is shown that the proposed test have high efficiencies for LFR and Weibull distributions. Sets of real data are used as examples to elucidate the use of the proposed test for practical problems.