Usual Stochastic Order Research Articles

Stochastic Comparisons for Finite Mixtures from Location-scale Family of Distributions

In this study, we consider two finite mixture models (FMMs) with location-scale family distributed components, in which ordering results are established in various stochastic senses. For heterogeneity in one parameter, the comparisons are obtained with respect to usual stochastic order, hazard rate order, reversed hazard rate order and likelihood ratio order. Further, for heterogeneity in two parameters, we derive sufficient conditions for the stochastic comparison of FMMs with respect to usual stochastic order and hazard rate order. Various examples and counterexamples are presented to illustrate the proposed results.

Stochastic ordering results on extreme order statistics from dependent samples with Archimedean copula

This paper considers parallel and series systems with heterogeneous components having dependent exponential lifetimes. The underlying dependence is assumed to be Archimedean and the component lifetimes are supposed to be connected according to an Archimedean copula. Sufficient conditions are found to dominate a parallel system with heterogenous exponential components, with respect to the dispersive order, by another parallel system with homogenous exponential components where the dependence structure between lifetimes of components is the same. We also compare two series systems (and two parallel systems) with general one-parameter dependent components and with respect to the usual stochastic ordering. Examples are given to illustrate the theoretical findings.

Stochastic comparisons of second largest order statistics with dependent heterogeneous random variables

In the context of actuarial science, the second largest claim amount is crucial to insurance analysis since they provide useful information for determining annual premium. In this article, we provide sufficient conditions of the second largest claim amounts arising from two sets of dependent and heterogeneous individual risk models according to various stochastic orders. It is first shown that the reversed hazard order between the occurrence probabilities vectors implies the reversed hazard order of the second largest claim amounts under certain conditions. Second, sufficient conditions are established for the usual stochastic ordering of the second largest claim amounts arising from heterogeneous dependent individual risk models under copula dependence. Finally, several examples illustrating the theoretical results are presented as well.

Ordering results between two multiple-outlier finite [formula omitted]-mixtures

In this paper, we have obtained sufficient conditions for comparing two multiple-outlier (M-O) finite δ-mixtures based on the usual stochastic order and reversed hazard rate order. We have assumed a general parametric family of distributions for the subpopulations. Many distributions satisfying baseline-related conditions in the established results have also been provided as examples.

Stochastic comparison results between two finite mixture models with generalized Weibull distributed components

In this paper, we establish sufficient conditions for stochastic comparisons of two finite mixture models (FMMs) with respect to the usual stochastic order, hazard rate order, and likelihood ratio order when the mixing components have generalized Weibull family of distributions. The established (sufficient) conditions are mainly based on the majorization order, weak supermajorization order, and weak submajorization order. The stochastic comparisons are studied when there is heterogeneity in one (model) parameter, and then in two parameters (model parameter and mixing proportion). Further, the concept of unordered majorization order is employed to establish the usual stochastic order between two FMMs. To illustrate the theoretical results established here, several numerical examples and counterexamples are presented. Finally, we have generalized some of the results to the case of τ-mixture models.

Stochastic Orderings between Two Finite Mixtures with Inverted-Kumaraswamy Distributed Components

In this paper, we consider two finite mixture models (FMMs) with inverted-Kumaraswamy distributed components’ lifetimes. Several stochastic ordering results between the FMMs are obtained. Mainly, we focus on three different cases in terms of the heterogeneity of parameters. The usual stochastic order between the FMMs is established when heterogeneity presents in one parameter as well as two parameters. In addition, we also study ageing faster order in terms of the reversed hazard rate between two FMMs when heterogeneity is in two parameters. For the case of heterogeneity in three parameters, we obtain the comparison results based on reversed hazard rate and likelihood ratio orders. The theoretical developments are illustrated using several examples and counterexamples.

Standby redundancy allocation for series and parallel systems

. The allocation of redundant components to a system is an efficient technique to improve its lifetime. In this article, we study the problem of two standby redundancy allocations for series and parallel systems, consisting of n-component (n≥2), in the sense of various stochastic orderings. It is assumed that components and redundancies follow a general lifetime distribution. For the case of allocating two redundancies to a series (parallel) system, we show that allocating the relatively stronger redundancy to the weaker component and the weaker redundancy to the stronger components can result in a system lifetime that is longer (shorter) in terms of the usual stochastic order. Moreover, various results associated with the hazard rate and reversed hazard rate orders are established. The outcomes of various papers are strengthened and generalized through our results. In addition, some applications are provided to illustrate our findings.

Determining the optimal design for complex systems using a reliability signature

In reliability analysis, the structure-function is a commonly used mathematical representation of the studied system. A signature vector is used for systems with independently and identically distributed (i.i.d.) component lifetimes. Each element in the signature represents the probability that the failure of the corresponding component will fail the entire system. This paper aims to provide a comprehensive understanding of assessing the performance of two complex systems for optimal communication design. The study compares two systems with the same components using signatures, expected cost rate, survival signature, and sensitivity to determine which system is preferred. It also provides several sufficient conditions for comparing the lifetimes of two systems based on the usual stochastic order. The results are applied to two communication systems that have the same components. The mathematical properties presented in the study have been proven to enable efficient weighting of the optimal design.

Stochastic ordering of variability measure estimators

Variability measures, such as the variance or the Gini mean difference, are widely used to summarize the dispersion of random variables. In the statistical setting, it is quite natural to assume that if a random quantity has more variability, then the estimators of its variability measures should be greater in some stochastic sense. Stochastic orders can be used to give inequalities in this regard, confirming, or not, this suitable property. This paper is devoted to the stochastic comparison of some variability measure estimators; conditions such that some of these estimators are comparable in the usual stochastic order and in the increasing convex order whenever the involved random variables have different variability are provided. Special attention is devoted to the cases of sample variance and Gini mean differences, and to the case of simple random samples.

Some results on the odds rate order along with its statistical testing

Recently, Lando et al. [J. Statist. Plann. Inference 221 (2022) 313–325] define the odds rate order and show that it is stronger than the usual stochastic order, but weaker than the hazard rate order. The purpose of this paper is to investigate the other properties of this new stochastic order. Some characterization and preservation results and its relationship with other well-known stochastic orders are given. In addition, two relative ordering based on the odds rate functions are also defined and their relation with the odds rate order is investigated. Applications in statistical theory of reliability and a statistical test for the odds rate order are also included.

Some new results involving residual Renyi's information measure for $ k $-record values

<abstract><p>This article dealt with further properties of the Renyi entropy and the residual Renyi entropy of $ k $-record values. First, we discussed the Renyi entropy order and its connection with the usual stochastic and dispersive orders. We then addressed the monotonicity properties of the residual Renyi entropy of $ k $-records, focusing on the aging properties of the component lifetimes. We also expressed the residual $ n $th upper $ k $-records in terms of Renyi entropy when the first dataset exceeded a certain threshold, and then studied various properties of the given formula. Finally, we conducted a parametric estimation of the Renyi entropy of the $ n $th upper $ k $-records. The estimation was performed using both real COVID-19 data and simulated data.</p></abstract>

Stochastic comparisons of second-order statistics from dependent and heterogeneous modified proportional (reversed) hazard rates scale models

<abstract><p>In this paper, we investigate the problem of stochastically comparing the second-order statistics from dependent and heterogeneous samples following modified proportional hazard rates scale (MPHRS) and modified proportional reversed hazard rates scale (MPRHRS) models under Archimedean copula. We built some sufficient conditions for the usual stochastic order whenever the samples have different parameter vectors. Finally, some numerical examples were provided to illustrate the theoretical results.</p></abstract>

Estimation of a likelihood ratio ordered family of distributions

Consider bivariate observations (X1,Y1),…,(Xn,Yn)∈R×R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(X_1,Y_1), \\ldots , (X_n,Y_n) \\in {\\mathbb {R}}\ imes {\\mathbb {R}}$$\\end{document} with unknown conditional distributions Qx\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q_x$$\\end{document} of Y, given that X=x\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X = x$$\\end{document}. The goal is to estimate these distributions under the sole assumption that Qx\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q_x$$\\end{document} is isotonic in x with respect to likelihood ratio order. If the observations are identically distributed, a related goal is to estimate the joint distribution L(X,Y)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {L}(X,Y)$$\\end{document} under the sole assumption that it is totally positive of order two. An algorithm is developed which estimates the unknown family of distributions (Qx)x\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(Q_x)_x$$\\end{document} via empirical likelihood. The benefit of the stronger regularization imposed by likelihood ratio order over the usual stochastic order is evaluated in terms of estimation and predictive performances on simulated as well as real data.

Stochastically ordered aggregation operators

In aggregation theory, there exists a large number of aggregation functions that are defined in terms of rearrangements in increasing order of the arguments. Prominent examples are the Ordered Weighted Operator and the Choquet and Sugeno integrals. Following a probability approach, ordering random variables by means of stochastic orders can be also a way to define aggregations of random variables. However, stochastic orders are not total orders, thus pairs of incomparable distributions can appear. This paper is focused on the definition of aggregations of random variables that take into account the stochastic ordination of the components of the input random vectors. Three alternatives are presented, the first one by using expected values and admissible permutations, then a modification for multivariate Gaussian random vectors and a third one that involves a transformation of the initial random vectors in new ones whose components are ordered with respect to the usual stochastic order. A deep theoretical study of the properties of all the proposals is made. A practical example regarding temperature prediction is provided

Ordering Results of Aggregate Claim Amounts from Two Heterogeneous Portfolios

In this paper, we discuss the stochastic comparison of two classical surplus processes in an one-year insurance period. Under the Marshall-Olkin extended Weibull random aggregate claim amounts, we establish some sufficient conditions for the comparison of aggregate claim amounts in the sense of the usual stochastic order. Applications of our results to the Value-at-Risk and ruin probability are also given. The obtained results show that the heterogeneity of the risks in a given insurance portfolio tends to make the portfolio volatile, which in turn leads to requiring more capital. We also obtain some sufficient conditions for comparing aggregate non-random claim amounts with different occurrence frequency vectors in terms of increasing convex order.

Lower and upper stochastic bounds for the joint stationary distribution of a non-preemptive priority retrial queueing system

Consider a single-server retrial queueing system with non-preemptive priority service, where customers arrive in a Poisson process with a rate of $\lambda_1$ for high-priority customers (class 1) and $\lambda_2$ for low-priority customers (class 2). If a high-priority customer is blocked, they are queued, while a low-priority customer must leave the service area and return after some random period of time to try again. In contrast with existing literature, we assume different service time distributions for the two customer classes. This investigation proposes a stochastic comparison method based on the general theory of stochastic orders to obtain lower and upper bounds for the joint stationary distribution of the number of customers at departure epochs in the considered model. Specifically, we discuss the stochastic monotonicity of the embedded Markov queue-length process in terms of both the usual stochastic and convex orders. We also perform a numerical sensitivity analysis to study the effect of the arrival rate of high-priority customers on system performance measures.

Ordering properties of parallel and series systems with a general lifetime family of distributions for independent components under random shocks

Often, reliability systems suffer shocks from external stress factors, stressing the system at random. These random shocks may have non ignorable effects on the reliability of the system. In this article, we provide sufficient (and necessary) conditions on components’ lifetimes and their survival probabilities from random shocks for comparing the lifetimes of two parallel systems with independent components by means of usual stochastic and reversed hazard rate orders, when the matrix of parameters changes to another matrix in a mathematical sense. We also discuss stochastic comparisons of series systems with independent components under random shocks in the sense of usual stochastic and hazard rate orders, by using the concept of vector majorization and related orders. The results developed in this article generalize some known results in the literature.

Stochastic comparisons of the largest and smallest claim amounts with heterogeneous survival exponentiated location-scale distributed claim severities

Suppose X 1 , . , X n are independent survival exponentiated location-scale random variables, and I p 1 , … , I p n are independent Bernoulli random variables, independently of X i ’s, i = 1 , … , n . Let Y i = I p i X i , for i = 1 , … , n . Then, in actuarial context, Y i corresponds to the claim amount in a portfolio of heterogeneous risks. In this work, we compare the largest and smallest order statistics arising from two heterogeneous portfolios in the sense of usual stochastic order. The results obtained here are based on multivariate chain majorization with heterogeneity in different parameters, and generalize some of the results known in the literature. Some examples and counterexamples are also presented for illustrating the results established here.

Stochastic ordering results on the duration of the gambler’s ruin game

AbstractIn the classical gambler’s ruin problem, the gambler plays an adversary with initial capitals z and $a-z$ , respectively, where $a>0$ and $0< z < a$ are integers. At each round, the gambler wins or loses a dollar with probabilities p and $1-p$ . The game continues until one of the two players is ruined. For even a and $0<z\leq {a}/{2}$ , the family of distributions of the duration (total number of rounds) of the game indexed by $p \in [0,{\frac{1}{2}}]$ is shown to have monotone (increasing) likelihood ratio, while for ${a}/{2} \leq z<a$ , the family of distributions of the duration indexed by $p \in [{\frac{1}{2}}, 1]$ has monotone (decreasing) likelihood ratio. In particular, for $z={a}/{2}$ , in terms of the likelihood ratio order, the distribution of the duration is maximized over $p \in [0,1]$ by $p={\frac{1}{2}}$ . The case of odd a is also considered in terms of the usual stochastic order. Furthermore, as a limit, the first exit time of Brownian motion is briefly discussed.

Preservation of mean inactivity time ordering for coherent systems

AbstractPreservation of stochastic orders through the system signature has captured the attention of researchers in recent years. Signature-based comparisons have been made for the usual stochastic order, hazard rate order, and likelihood ratio orders. However, for the mean residual life (MRL) order, it has recently been proved that the preservation result does not hold true in general, but rather holds for a particular class of distributions. In this paper, we study whether or not a similar preservation result holds for the mean inactivity time (MIT) order. We prove that the MIT order is not preserved from signatures to system lifetimes with independent and identically distributed (i.i.d.) components, but holds for special classes of distributions. The relationship between these classes and the order statistics is also highlighted. Furthermore, the distribution-free comparison of the performance of coherent systems with dependent and identically distributed (d.i.d.) components is studied under the MIT ordering, using diagonal-dependent copulas and distorted distributions.