Multiple-outlier Models Research Articles

Orderings of fail-safe systems with components having Marshall-Olkin-Harris lifetime distributions

In this paper, we discuss stochastic comparisons of two fail-safe systems in two different directions. First, we present sufficient conditions for the usual stochastic order between two fail-safe systems have different lifetime distributions but with the same Archimedean copula for their joint distribution. Next, hazard rate orderings of two fail-safe systems with independent components following multiple-outlier models are established. Several illustrative examples are also presented.

Ordering results of the smallest order statistics from independent heterogeneous new modified generalized linear failure rate random variables

This article deals with stochastic comparisons of series systems comprising a new modified generalized linear failure rate family of distributions. The results are established in three directions. First, we establish based on vector majorization the usual stochastic order and the hazard rate order of series systems. Next, multiple-outlier models are considered. When the number of components is equal, the results are obtained in terms of the likelihood ratio order. For an unequal number of components, we develop comparisons according to the usual stochastic order, and hazard rate order. Then, based on weak majorization and p-larger orders between the vector of parameters, some characterization results for comparing the lifetimes of series systems are also established. Finally, we discuss based on multivariate chain majorization the usual stochastic order of series systems. Numerical examples are considered to illustrate the results.

Orderings of the Smallest Claim Amounts from Exponentiated Location-Scale Models

In actuarial science, it is often of interest to compare stochastically extreme claim amounts from heterogeneous portfolios. In this regard, in the present work, we compare the smallest order statistics arising from two heterogeneous portfolios in the sense of the usual stochastic, hazard rate, reversed hazard rate and likelihood ratio orderings. We also consider the multiple-outlier model and obtain some ordering results. It is assumed that the portfolios belong to the general exponentiated location-scale model. The results obtained here are based on vector majorization of parameters and multivariate chain majorization with heterogeneity in different parameters. For the purpose of illustration, the derived results are applied to some well known distributions. Various examples and counterexamples are also provided. Finally, a simulation study is conducted to validate some of the results established here.

Stochastic comparisons of series and parallel systems with dependent Burr type XII components

This paper deals with some stochastic comparisons of both series and parallel systems comprising dependent Burr Type XII components under the Archimedean copula. Sufficient conditions are also established for star ordering between the lifetimes of series and parallel systems based on multiple-outlier Burr model. The lifetime of a series system with dependent heterogeneous Burr components is proved to be smaller than that of a system with dependent homogeneous components in the sense of convex transform order.

ORDERING RESULTS ON EXTREMES OF EXPONENTIATED LOCATION-SCALE MODELS

AbstractIn this paper, we consider exponentiated location-scale model and obtain several ordering results between extreme order statistics in various senses. Under majorization type partial order-based conditions, the comparisons are established according to the usual stochastic order, hazard rate order and reversed hazard rate order. Multiple-outlier models are considered. When the number of components are equal, the results are obtained based on the ageing faster order in terms of the hazard rate and likelihood ratio orders. For unequal number of components, we develop comparisons according to the usual stochastic order, hazard rate order, and likelihood ratio order. Numerical examples are considered to illustrate the results.

Some Ordering Results for the Marshall and Olkin’s Family of Distributions

In this paper, we consider Marshall and Olkin’s family of distributions. The parent (baseline) distribution is taken to be a scaled family of distributions. Two models: (i) modified proportional hazard rate scale and (ii) modified proportional reversed hazard rate scale, are considered. Some stochastic comparison results in terms of the usual stochastic, hazard rate and reversed hazard rate orders are studied to compare order statistics formed from two sets of independent observations following these models. Most of the sufficient conditions are obtained depending on various majorization-type partial orderings. Further, the setup with multiple-outlier model is taken. Various stochastic orders between the smallest and largest order statistics are developed. Several numerical examples are provided to illustrate the effectiveness of the established theoretical results.

COMPARISONS OF SAMPLE RANGES ARISING FROM MULTIPLE-OUTLIER MODELS: IN MEMORY OF MOSHE SHAKED

In this paper, we discuss the ordering properties of sample ranges arising from multiple-outlier exponential and proportional hazard rate (PHR) models. The purpose of this paper is twofold. First, sufficient conditions on the parameter vectors are provided for the reversed hazard rate order and the usual stochastic order between the sample ranges arising from multiple-outlier exponential models with common sample size. Next, stochastic comparisons are separately carried out for sample ranges arising from multiple-outlier exponential and PHR models with different sample sizes as well as different hazard rates. Some numerical examples are also presented to illustrate the results established here.

Likelihood ratio order of parallel systems under multiple-outlier models

ABSTRACTThis paper studies the likelihood ratio ordering of parallel systems under multiple-outlier models. We introduce a partial order, the so-called θ-order, and show that the θ-order between the parameter vectors of the parallel systems implies the likelihood ratio order between the systems.

Comparisons Between Largest Order Statistics from Multiple-outlier Models with Dependence

We study stochastic comparisons between the largest order statistics from samples which may contain outliers. The data in each sample can also be dependent. Under these assumptions we study three cases. In the first one we consider the general case without additional assumptions. In the second we assume that the data come from two different distributions. In the third one we assume that the data come from a proportional hazard rates model. The results obtained here can be applied to compare parallel systems. Some illustrative examples are provided.

Dispersive order of lifetimes of series systems in multiple-outlier Weibull models

This paper considers the multiple-outlier Weibull model, and derives some sufficient conditions for the comparison of lifetimes of series systems with respect to dispersive order. In these models, order statistic is closed under minima, so convex transform, star and Lorenz orders are not investigated because they are scale variant. The results established here strengthen some of the results presented by Fang and Tang (2014).

SOME NEW RESULTS ON THE LARGEST ORDER STATISTICS FROM MULTIPLE-OUTLIER GAMMA MODELS

In this paper, we carry out stochastic comparisons of the largest order statistics arising from multiple-outlier gamma models with different both shape and scale parameters in the sense of various stochastic orderings including the likelihood ratio order, star order and dispersive order. It is proved, among others, that the weak majorization order between the scale parameter vectors along with the majorization order between the shape parameter vectors imply the likelihood ratio order between the largest order statistics. A quite general sufficient condition for the star order is presented. The new results established here strengthen and generalize some of the results known in the literature. Numerical examples and applications are also provided to explicate the theoretical results.

Comparisons between largest order statistics from multiple-outlier models

In this work, we discuss stochastic comparisons between the largest order statistics from multiple-outlier models when the numbers of independent and identically distributed random variables are different. That is, if is the largest order statistic among and is the largest order statistic among , where and , we discuss whether and are ordered in some stochastic sense.

Likelihood ratio order of the second spacing in multiple-outlier exponential models

In this paper, we study the ordering properties of the second sample spacing arising from multiple-outlier exponential models in terms of the likelihood ratio order. Let [] be independent exponential random variables with [] having common hazard rate [] and [] having common hazard rate []. Let and denote the corresponding second sample spacing, respectively. It is proved here that is stochastically greater than in the sense of the likelihood ratio order, under two different kinds of parameter conditions. The results established here strengthen and generalize some of the results known in the literature. Two applications are also presented to illustrate the results.

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.

Comparisons of Largest Order Statistics from Multiple-outlier Gamma Models

In this paper, we discuss stochastic comparisons of largest order statistics from multiple-outlier gamma models in terms of different stochastic orderings including the likelihood ratio order, hazard rate order, star order and dispersive order. It is proved, among others, that the weak majorization order between the two scale parameter vectors implies the likelihood ratio order between the largest order statistics, and that the p-larger order between the two scale parameter vectors implies the hazard rate order between the largest order statistics. We also present a general sufficient condition for the star order. The results established here strengthen and generalize some of the results known in the literature. Some numerical examples are also presented to illustrate the established results.

Comparisons of [formula omitted]-out-of-[formula omitted] systems with heterogenous components

The k-out-of-n structure is a very popular type of redundancy in fault-tolerant systems which has been widely applied in reliability engineering and system security. In this paper, we carry out stochastic comparisons between k-out-of-n systems with independent heterogeneous components with respect to various stochastic orders such as hazard rate order, mean residual life order, and reversed hazard rate order. Moreover, we also establish likelihood ratio ordering results of series and parallel systems in multiple-outlier models.

On sample ranges in multiple-outlier models

In this paper, we investigate the ordering properties of sample ranges arising from multiple-outlier models in terms of the reversed hazard rate order and the usual stochastic order. Under the setup of an exponential model, it is shown that the weak majorization order between the two hazard rate vectors is equivalent to the reversed hazard rate order between exponential sample ranges; the p-larger order between two hazard rate vectors implies the usual stochastic order between exponential sample ranges. Under the setup of a proportional hazard rate (PHR) model, we prove that the majorization order between two parameter vectors implies the usual stochastic order between sample ranges. The results established here strengthen and generalize some of the results known in the literature. Some numerical examples are provided to illustrate the theoretical results.

On the Skewness of Order Statistics in Multiple-Outlier Models

Kochar and Xu (2009) proved that a parallel system with heterogeneous exponential component lifetimes is more skewed (according to the convex transform order) than the system with independent and identically distributed exponential components. In this paper we extend this study to the general k-out-of-n systems for the case when there are only two types of component in the system. An open problem proposed in Pǎltǎnea (2008) is partially solved.

On the Skewness of Order Statistics in Multiple-Outlier Models

Kochar and Xu (2009) proved that a parallel system with heterogeneous exponential component lifetimes is more skewed (according to the convex transform order) than the system with independent and identically distributed exponential components. In this paper we extend this study to the generalk-out-of-nsystems for the case when there are only two types of component in the system. An open problem proposed in Pǎltǎnea (2008) is partially solved.

Likelihood Ratio Order of m-Spacings in Multiple-Outlier Models

This article deals with the multiple-outlier exponential model. The likelihood ratio order between m-spacings of the combined sample is developed, some results extend the conclusions on simple spacings in Wen et al. (2007).