Improved Ordering Results for Fail-Safe Systems with Exponential Components

Let X_{2: n} and Y_{2: m} be the second order statistics from n independent exponential variables with hazards λ₁, …, λ_n, and an independent exponential sample of size m with hazard change to λ, respectively. When m ? n, we obtain necessary and sufficient conditions for comparing X_{2: n} and Y_{2: m} in mean residual life, dispersive, hazard rate, and likelihood ratio orderings based on some inequalities between λ_i’s and λ. The established results show how one can compare an (n ? 1)-out-of-n system consisting of heterogeneous components with exponential lifetimes with any (m ? 1)-out-of-m system consisting of homogeneous components with exponential lifetimes.     

Ordres stochastiques et efficacité asymptotique des L-estimateurs

The efficiency of an estimator depends heavily on the tails of the distribution of the observations. Several partial orders have been defined to compare probability distributions according to their tails. In this paper we show that the asymptotic relative efficiency of two L-estimators with monotone weight functions is isotonic with respect to the partial orders defined by van Zwet (1964) and Lawrence (1975). We also give results concerning trimmed means.     

The one-way analysis of variance with ordered alternatives: A modification of bartholomew's Ē2 Test

Tests of homogeneity of normal means with the alternative restricted by an ordering on the means are considered. The simply ordered case, μ₁ ≤ μ₂ ≤ ··· ≤ μ_k, and the simple tree ordering, μ₁ ≤ μ_j, for; j= 2, 3,…, k, are emphasized. A modification of the likelihood-ratio test is proposed which is asymptotically equivalent to it but is more robust to violations of the hypothesized orderings. The new test has power at the points satisfying the hypothesized ordering which is similar to that of the likelihood-ratio test provided the degrees of freedom are not too small. The modified test is shown to be unbiased and consistent.     

Tests based on equivariant estimators

Let ?⁽¹⁾ and ?⁽²⁾ be location-equivariant estimators of an unknown location parameter μ. It is shown that the test for H₀: μ ≤ μ₀ versus H_A : μ > μ₀ that rejects H₀ if ?⁽¹⁾ is large is uniformly more powerful than the one that rejects H₀ if ?⁽²⁾ is large if and only if ?⁽²⁾ is “more dispersed” than ?⁽¹⁾. A similar result is obtained for tests on scale using the star-shaped ordering. Examples are given.     

On Uniformly Optimal Networks: A Reversal of Fortune?

In this article, the general problem of comparing the performance of two communication networks is examined. The standard approach, using stochastic ordering as a metric, is reviewed, as are the mixed results on the existence of uniformly optimal networks (UONs) which have emerged from this approach. While UONs have been shown to exist for certain classes of networks, it has also been shown that no UON network exists for other classes. Results to date beg the question: Is the problem of identifying a Uniformly Optimal Network (UON) of a given size dead or alive? We reframe the investigation into UONs in terms of network signatures and the alternative metric of stochastic precedence. While the endeavor has been dead, or at least dormant, for some 20 years, the findings in the present article suggest that the question above is by no means settled. Specifically, we examine a class of networks of a particular size for which it was shown that no individual network was uniformly optimal relative to the standard metric (the uniform ordering of reliability polynomials), and we show, using the aforementioned alternative metric, that this class is totally ordered and that a uniformly optimal network exists after all. Optimality with respect to “performance per unit cost” type metrics is also discussed.     

Reliability and Some Characteristics of Circular Consecutive-k-Within-m-Out-of-n:F System

A system can be classified with respect to the physical arrangement of its components and the functioning principle. A circular consecutive k-within-m-out-of-n:F system consists of n circularly ordered components and fails if and only if there are m consecutive components that include among them at least k failed components. A circular consecutive k-within-m-out-of-n:F system turns into circular consecutive k-out-of-n:F for m = k and k-out-of-n:F system for m = n. In this study, signature-based analysis of circular consecutive k-within-m-out-of-n:F system is performed. A new approximation to this system is provided based on maximum number of failed components and an illustrative example is given for different values of n, m, k to compare the approximate results with simulated and exact results.     

A Unified Approach to Ordering Comparison of GPS Distributions with their Mixtures

Recently, several authors have been concerned with ordering comparison of known distributions of the family of generalized power series (GPS) distributions with their mixtures in various senses. In this article, we shall employ a unified approach and obtain similar results, more generally, for all members of the class of the GPS distributions. Some of the previous findings of Misra et al. (2003 Misra , N. , Singh , H. , Harner , E. J. ( 2003 ). Stochastic comparisons of Poisson and binomial random variables with their mixtures . Statist. Probab. Lett. 65 : 279 – 290 .[Crossref], [Web of Science ®] , [Google Scholar]), Alamatsaz and Abbasi (2008 Alamatsaz , M. H. , Abbasi , S. ( 2008 ). Ordering comparison of negative binomial random variables with their mixtures . Statist. Probab. Lett. 78 : 2234 – 2239 . [Google Scholar]), and Aghababaei Jazi and Alamatsaz (2010 Aghababaei Jazi , M. , Alamatsaz , M. H. ( 2010 ). Ordering comparison of logarithmic series random variables with their mixtures . Commun. Statist. Theor. Meth. 39 : 3252 – 3263 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) in this connection, then, follow as corollaries. Further, we have derived some more ordering comparison results.     

A note on the mean residual life of a coherent system with a cold standby component

In this paper, we investigate the effect of a cold standby component on the mean residual life (MRL) of a system. When the system fails, a cold standby component is immediately put in operation. We particularly focus on the coherent systems in which, after putting the standby component into operation, the failure of the system is due to the next component failure. For these systems, we define MRL functions and obtain their explicit expressions. Also some stochastic ordering results are provided. Such systems include k-out-of-n systems. Hence, our results extend some results in literature.     

Analytical results for IGFR and DRPFR distributions,with actuarial applications

An increasing generalized failure rate of a lifetime X defines an ageing concept, denoted by IGFR. Another notion, denoted by DRPFR, is defined by the decreasingness of the reversed proportional failure rate. In this article, we provide characterizations for both IGFR and DRPFR absolutely continuous lifetimes, based on monotonicity of quotients of probabilistic functionals and a result by Nanda and Shaked (2001 Nanda, A.K., Shaked, M. (2001). The hazard rate and the reversed hazard rate orders, with applications to order statistics. Ann. Inst. Stat. Math. 53:853–864.[Crossref], [Web of Science ®] , [Google Scholar]). We derive the necessary conditions for the IGFR notion, based on stochastic orderings of truncated distributions, and we prove that the product of DRPFR lifetimes is also DRPFR; that the IGFR property is preserved by composition with certain risk aversion utility functions; and that the order statistics and the records (and the subsequent order statistic (record)) are IGFR under suitable assumptions, with similar results for DRPFR lifetimes. Also, we provide sufficient conditions for the hazard rate ordering of products and random products of IGFR lifetimes, and similar results for the reversed hazard rate order and DRPFR lifetimes, with a complementary result for the mean residual life order of random products of two families of IGFR lifetimes, we derive the upper and lower bounds for the cumulative distribution function of the product of IGFR lifetimes, and we provide the lower bounds for the risk function of an IGFR lifetime based on the distribution moments, and these bounds are extended for the product of IGFR lifetimes. We discuss extensively the applications of the results in insurance portfolios.     

On Dependence Properties of Random Minima and Maxima

Let X _(n) and X ₍₁₎ be the largest and smallest order statistics, respectively, of a random sample of fixed size n. Quite generally, X ₍₁₎ and X _(n) are approximately independent for n sufficiently large. In this article, we study the dependence properties of random extremes in terms of their copula, when the sample size has a left-truncated binomial distribution and show that they tend to be more dependent in this case. We also give closed-form formulas for the measures of association Kendall's τ and Spearman's ρ to measure the amount of dependence between two extremes.     

Concomitants in a Sequence of Independent Nonidentically Distributed Random Vectors

ABSTRACT

Concomitants of order statistics are considered for the situation in which the random vectors (X ₁, Y ₁), (X ₂, Y ₂),…, (X _n , Y _n ) are independent but otherwise arbitrarily distributed. The joint and marginal distributions of the concomitants of order statistics and stochastic comparisons among the concomitants of order statistics are studied in this situation.     

Modification of Pickands' Dependence Function for Ordered Bivariate Extreme Distribution

We study the characteristics of the Pickands' dependence function for bivariate extreme distribution for minima, BEVM, when considering the stochastics ordering of the two variables, X < Y. The existing Pickand's dependence function terminologies and theories are modified to suit the dependence functions of extreme minimum cases. The main result is the introduction of the restricted logistic dependence function, A _RL , and the restricted exponential function, V _RL (x, y).     

On the distribution of some ordered variables

Abstract

Motivated by a recent article published by Adam and Tawn, we characterize the distribution of two random variables X, Y ordered linearly like X < Y. We suppose that the random variables follow a bivariate extreme value distribution.     

Chain graph models: topological sorting of meta-arrows and efficient construction of $${\mathcal{B}}$$ -essential graphs

Essential graphs and largest chain graphs are well-established graphical representations of equivalence classes of directed acyclic graphs and chain graphs respectively, especially useful in the context of model selection. Recently, the notion of a labelled block ordering of vertices was introduced as a flexible tool for specifying subfamilies of chain graphs. In particular, both the family of directed acyclic graphs and the family of “unconstrained” chain graphs can be specified in this way, for the appropriate choice of . The family of chain graphs identified by a labelled block ordering of vertices is partitioned into equivalence classes each represented by means of a -essential graph. In this paper, we introduce a topological ordering of meta-arrows and use this concept to devise an efficient procedure for the construction of -essential graphs. In this way we also provide an efficient procedure for the construction of both largest chain graphs and essential graphs. The key feature of the proposed procedure is that every meta-arrow needs to be processed only once.     

On tail behaviour of kth upper order statistics under fixed and random sample sizes via tail equivalence

For a fixed positive integer k, limit laws of linearly normalized kth upper order statistics are well known. In this article, a comprehensive study of tail behaviours of limit laws of normalized kth upper order statistics under fixed and random sample sizes is carried out using tail equivalence which leads to some interesting tail behaviours of the limit laws. These lead to definitive answers about their max domains of attraction. Stochastic ordering properties of the limit laws are also studied. The results obtained are not dependent on linear norming and apply to power norming as well and generalize some results already available in the literature. And the proofs given here are elementary.     

The power functions of the likelihood ratio tests for a simple tree ordering in normal means: unequal weights

Likelihood ratio tests are considered for two testing situations; testing for the homogeneity of k normal means against the alternative restricted by a simple tree ordering trend and testing the null hypothesis that the means satisfy the trend against all alternatives. Exact expressions are given for the power functions for k = 3 and 4 and unequal sample sizes, both for the case of known and unknown population variances, and approximations are discussed for larger k. Also, Bartholomew’s conjectures concerning minimal and maximal powers are investigated for the case of equal and unequal sample sizes. The power formulas are used to compute powers for a numerical example.     

On Estimating The Quantile Function For Distributions Constrained by Star-Shaped Ordering

Consider distributions F and G such that G ^-1 F is star-shaped. In the problem of estimating the quantile functions for lifetime distributions, the estimators developed by Rojo (1998) are compared with the commonly used empirical quantile function. Both the one-sample and the two-sample methods of estimation are considered for a wide class of lifetime distributions. In addition, the behavior of the estimators is examined for star-shaped ordered lifetime distributions of the important class of coherent k- out-of-n reliability systems. Results of a Monte Carlo study are presented which compare the behavior of the new estimators with that of the empirical quantile function interms of bias and mean-squared error. As the behavior of these estimators typically depends on the tail behavior of the underlying distributions, the examples presented here include distributions with short, medium and long tails. A formula for the inverse of the Kaplan-Meier estimator is provided and used to generate the simulations in the case of censored data.     

Ordering Comparison of Logarithmic Series Random Variables with their Mixtures

The problem of comparing some known distributions in various types of stochastic orderings has been of interest to many authors. In particular, several authors have been recently concerned with the comparison of Poisson, binomial, and negative binomial distributions with their respective mixtures. Incidentally, these distributions are among the four well-known distributions of the family of generalized power series distributions (GPSD's). The remaining distribution is the logarithmic series distribution. In this paper, we shall be concerned with comparing this remaining distribution of the class GPSD with its mixture in terms of various types of stochastic orderings such as the simple stochastic, likelihood ratio, uniformly more variable, convex, hazard rate and expectation orderings. Derivation of the results in this case prove to be computationally trickier than the other three. The special case when the means of the two distributions are the same is also discussed. Finally, an illustrative explicit example is provided.     

Reliability analysis of systems with components having two dependent subcomponents

In this article, a system that consists of n independent components each having two dependent subcomponents (A_i, B_i), i = 1, …, n is considered. The system is assumed to compose of components that have two correlated subcomponents (A_i, B_i), and functions iff both systems of subcomponents A₁, A₂, …, A_n and B₁, B₂, …, B_n work under certain structural rules. The expressions for reliability and mean time to failure of such systems are obtained. A sufficient condition to compare two systems of bivariate components in terms of stochastic ordering is also presented.