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.     

STRUCTURES OF SYSTEMS WITH EXPONENTIAL LIFE DISTRIBUTIONS

Abstract. Structures of monotone systems and cold standby systems with     

THE NBELC AND NWELC CLASSES OF LIFE DISTRIBUTIONS

ＴＨＥＮＢＥＬＣＡＮＤＮＷＥＬＣＣＬＡＳＳＥＳＯＦＬＩＦＥＤＩＳＴＲＩＢＵＴＩＯＮＳ￥ＣＡＯＪＩＮＨＵＡ；ＷＡＮＧＹＵＥＤＯＮＧ（ＩｎｓｔｉｔｕｔｅｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ，ＣｈｉｎｅｓｅＡｃａｄｅｍｙｏｆＳｃｉｅｎｃｅ，Ｂｅｉｊｉｎ...     

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.     

Exponential distributions on semigroups

Three fundamental characterizations of the standard exponential distribution on [0, ) are the remaining life, memoryless and constant failure properties. Analogs of these properties are studied for distributions on a class of semigroups in which the semigroup operation replaces addition, a compatible partial order replaces the ordinary order, and a left-invariant measure replaces Lebesgue measure. Partial characterizations of exponential distributions on such semigroups are obtained and the semigroup formulation provides new characterizations of certain aging properties studied in reliability-increasing failure rate, new better than used, and increasing failure rate average.     

Representation of Downton’s bivariate exponential random vector and its applications

Downton’s bivariate exponential distribution is one of the most important bivariate distributions in reliability theory. In this paper a simple representation for Downton’s bivariate exponential random vector is given. As an application of this representation, we consider a reliability model where an item is subject to shocks and obtain an explicit expression for the long-run cost rate.     

Can Data Recognize Its Parent Distribution?

This study is concerned with model selection of lifetime and survival distributions arising in engineering reliability or in the medical sciences. We compare various distributions—including the gamma, Weibull, and lognormal—with a new distribution called geometric extreme exponential. Except for the lognormal distribution, the other three distributions all have the exponential distribution as special cases. A Monte Carlo simulation was performed to determine sample sizes for which survival distributions can distinguish data generated by their own families. Two methods for decision are by maximum likelihood and by Kolmogorov distance. Neither method is uniformly best. The probability of correct selection with more than one alternative shows some surprising results when the choices are close to the exponential distribution.     

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.     

Wrapped weighted exponential distributions

We have developed a new class of circular distributions named wrapped weighted exponential distributions. The estimation of unknown parameters along with some characteristics of these distributions is also investigated. Some theorems that relate the distribution to some other circular distributions are established and we clarify their modeling potential using a classical data set on movements of sea stars.     

On equivalence of weak convergence and moment convergence of life distributions

We investigate the equivalence of weak convergence and moment convergence of life distributions in a family which is larger than the HNBUE family. Also, we point out that within the HNBUE family the exponential distribution can be characterized by one value of its Laplace-Stieltjes transform.     

Approximation with generalized hyperexponential distributions: Weak convergence results

Generalized hyperexponential (GH) distributions are linear combinations of exponential CDFs with mixing parameters (positive and negative) that sum to unity. The denseness of the class GH with respect to the class of all CDFs defined on [0, ) is established by showing that a GH distribution can be found that is as close to a given CDF as desired, with respect to a suitably defined metric. The metric induces the usual topology of weak convergence so that, equivalently, there exists a sequence of GH CDFs that converges weakly to a given CDF. This result is established by using a similar result for weak convergence of Erlang mixtures. Various set inclusion relations are also obtained relating the GH distributions to other commonly used classes of approximating distributions, including generalized Erlang (GE), mixed generalized Erlang (MGE), those with reciprocal polynomial Laplace transforms (K _n ), those with rational Laplace transforms (R _n ), and phase-type (PH) distributions. A brief survey of the history and use of approximating distributions in queueing theory is also included.This research was partially supported by the Office of Naval Research under Contract No. N00014-86-K0029. Much of this work is taken from the first-named author's doctoral dissertation, accepted by the faculty at the University of Virginia.     

修理设备可更换且有修理延迟的N部件串联系统分析

假定部件的寿命服从指数分布,修理延迟时间和修理时间均服从任意分布,并且修理设备的寿命服从指数分布,其更换时间服从任意分布的情况下,利用马尔可夫更新过程理论和拉普拉斯变换工具,研究了修理有延迟且修理设备可更换的n部件串联可修系统,求得了系统的可用度和(0,t]时间内的平均故障次数.进一步,在定义修理设备“广义忙期”下,利用全概率分解,提出了一种新的分析技术,讨论了修理设备的可靠性指标,得到修理设备的一些重要可靠性结果.     

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.     

Family of multivariate generalized t distributions

In this paper, we introduce a new family of multivariate distributions as the scale mixture of the multivariate power exponential distribution introduced by Gómez et al. (Comm. Statist. Theory Methods 27(3) (1998) 589) and the inverse generalized gamma distribution. Since the resulting family includes the multivariate t distribution and the multivariate generalization of the univariate GT distribution introduced by McDonald and Newey (Econometric Theory 18 (11) (1988) 4039) we call this family as the “multivariate generalized t-distributions family”, or MGT for short. We show that this family of distributions belongs to the elliptically contoured distributions family, and investigate the properties. We give the stochastic representation of a random variable distributed as a multivariate generalized t distribution. We give the marginal distribution, the conditional distribution and the distribution of the quadratic forms. We also investigate the other properties, such as, asymmetry, kurtosis and the characteristic function.     

Estimation of the ratio of the scale parameters of two exponential distributions with unknown location parameters

We consider the estimation of the ratio of the scale parameters of two independent two-parameter exponential distributions with unknown location parameters. It is shown that the best affine equivariant estimator (BAEE) is inadmissible under any loss function from a large class of bowl-shaped loss functions. Two new classes of improved estimators are obtained. Some values of the risk functions of the BAEE and two improved estimators are evaluated for two particular loss functions. Our results are parallel to those of Zidek (1973, Ann. Statist., 1, 264–278), who derived a class of estimators that dominate the BAEE of the scale parameter of a two-parameter exponential distribution.     

二元Friday-Patil型指数分布的特征及其应用

导出了二元Friday-Patil型指数分布的一个特征,利用该特征获得了二元Friday-Patil型指数分布参数的最大似然估计及矩估计,给出了强度服从二元Friday-Patil型指数分布时系统可靠度的估计.     

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.     

Reliability analysis of 1-for-2 shared protection systems with general repair-time distributions

In this paper, we investigate the reliability of a type of 1-for-2 shared protection systems. The 1-for-2 shared protection system is the most basic fault-tolerant configuration with shared backup units. We assume that there are two working units each serving a single user and one shared protection (spare) unit in the system. We also assume that the times to failure and to repair are subject to exponential and general distributions respectively. Under these assumptions, we derive the Laplace transform of the survival function (the cdf that the system will survive beyond a given time) for each user as well as the user-perceived Mean Time to First Failure (MTTFF) by combining the state transition analysis and the supplementary variable method. We also show the effect of the repair-time distribution, the failure rates and the repair rates of the units through the case study of small-sized two enterprises that share one spare device for backup purpose. The analysis reveals what is important and what should be done in order to improve the user-perceived reliability of shared protection systems.     

Characterizations of continuous distributions through inequalities involving the expected values of selected functions

Nanda (2010) and Bhattacharjee et al. (2013) characterized a few distributions with help of the failure rate, mean residual, log-odds rate and aging intensity functions. In this paper, we generalize their results and characterize some distributions through functions used by them and Glaser’s function. Kundu and Ghosh (2016) obtained similar results using reversed hazard rate, expected inactivity time and reversed aging intensity functions. We also, via w(·)-function defined by Cacoullos and Papathanasiou (1989), characterize exponential and logistic distributions, as well as Type 3 extreme value distribution and obtain bounds for the expected values of selected functions in reliability theory. Moreover, a bound for the varentropy of random variable X is provided.     

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.