Usual Stochastic Order Research Articles

Isotonicity properties of generalized quantiles

We investigate whether several families of generalized quantiles (expectiles, Lp-quantiles and M-quantiles) respect various stochastic orders (the usual stochastic order, the convexity order, and the p-convexity orders). We employ techniques from monotone comparative statics developed by Topkis (1978) and Milgrom and Shannon (1994), in order to provide sufficient as well as necessary conditions for isotonicity. We show that expectiles with α≤1/2 are basically the only generalized quantiles that are isotonic with respect to the ≤icv ordering; more generally, the Lp-quantiles are isotonic with respect to the p-convex order.

LIKELIHOOD RATIO AND HAZARD RATE ORDERINGS OF THE MAXIMA IN TWO MULTIPLE-OUTLIER GEOMETRIC SAMPLES

In this paper, we study some stochastic comparisons of the maxima in two multiple-outlier geometric samples based on the likelihood ratio order, hazard rate order, and usual stochastic order. We establish a sufficient condition on parameter vectors for the likelihood ratio ordering to hold. For the special case whenn= 2, it is proved that thep-larger order between the two parameter vectors is equivalent to the hazard rate order as well as usual stochastic order between the two maxima. Some numerical examples are presented for illustrating the established results.

Stein's Method and Stochastic Orderings

A stochastic ordering approach is applied with Stein's method for approximation by the equilibrium distribution of a birth-death process. The usual stochastic order and the more general s-convex orders are discussed. Attention is focused on Poisson and translated Poisson approximations of a sum of dependent Bernoulli random variables, for example, k-runs in independent and identically distributed Bernoulli trials. Other applications include approximation by polynomial birth-death distributions.

Stein's Method and Stochastic Orderings

A stochastic ordering approach is applied with Stein's method for approximation by the equilibrium distribution of a birth-death process. The usual stochastic order and the more general s-convex orders are discussed. Attention is focused on Poisson and translated Poisson approximations of a sum of dependent Bernoulli random variables, for example, k-runs in independent and identically distributed Bernoulli trials. Other applications include approximation by polynomial birth-death distributions.

On maximum order statistics from heterogeneous geometric variables

Let X1,X2 be independent geometric random variables with parameters p1,p2, respectively, and Y1,Y2 be i.i.d. geometric random variables with common parameter p. It is shown that X2:2, the maximum order statistic from X1,X2, is larger than Y2:2, the second order statistic from Y1,Y2, in terms of the hazard rate order [usual stochastic order] if and only if \(p\geq \tilde{p}\), where \(\tilde{p}=(p_{1}p_{2})^{\frac{1}{2}}\) is the geometric mean of (p1,p2). This result answers an open problem proposed recently by Mao and Hu (Probab. Eng. Inf. Sci. 24:245–262, 2010) for the case when n=2.

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.

Ordinal Dominance Curve Based Inference for Stochastically Ordered Distributions

SummaryIn a variety of applications researchers collect data that are sampled under ordered experimental conditions. In such situations it is reasonable to assume that outcomes are ordered stochastically by the level of the treatment. For example a toxicologist may want to assess the effect of a toxin on the reproductive system by studying the number of offspring produced by an animal as a function of the dose that it received. A variety of scientific questions arise in such a context. The most basic is whether a dose–response relationship is present. If so, determining the minimal toxic dose is of great interest. The strength of the ordering between groups i<j may be assessed by the probabilities ℙXi⩽Xj or monotone functions thereof. Large probabilities indicate a strong ordering. We find that such (and other) questions are naturally addressed by studying the ordinal dominance curve and the area under the curve (AUC). We develop new, non-parametric, order-restricted estimators for the AUCs in K⩾2 samples and investigate them in detail. We show that the restricted estimators are strongly consistent and have lower mean-squared error compared with the unrestricted estimators. In the two-sample case estimators are considerably improved in those situations where the data are highly variable and/or sample sizes are small, and where the AUC is close to its boundary value. Note that these are precisely the conditions under which estimating the AUC is difficult. Simulations indicate that in the multisample case estimators are improved under a much broader set of conditions with substantial reductions in mean-squared error. These estimators provide a basis for a family of distribution-free tests for the usual stochastic order in K⩾2 samples. Limiting distributions are derived. An accurate and simple approximation to the tail area is given for K = 2. Simulation studies show that the new tests perform well compared with existing alternatives. The methodology proposed is illustrated by applying it to an aquatic toxicology study where it is used to determine the minimal toxic dose more accurately.

STOCHASTIC COMPARISONS OF LARGEST ORDER STATISTICS FROM MULTIPLE-OUTLIER EXPONENTIAL MODELS

In this paper, we carry out stochastic comparisons of largest order statistics from multiple-outlier exponential models according to the likelihood ratio order (reversed hazard rate order) and the hazard rate order (usual stochastic order). It is proved, among others, that the weak majorization order between the two hazard rate vectors is equivalent to the likelihood ratio order (reversed hazard rate order) between largest order statistics, and that the p-larger order between the two hazard rate vectors is equivalent to the hazard rate order (usual stochastic order) between largest order statistics. We also extend these results to the proportional hazard rate models. The results established here strengthen and generalize some of the results known in the literature.

The optimal allocation of active redundancies to k-out-of-n systems with respect to hazard rate ordering

Abstract This paper deals with the allocation of active redundancies to a k-out-of-n system with independent and identically distributed (i.i.d.) components in the sense of the hazard rate order. It is shown that the system's hazard rate may be decreased by balancing the allocation of active redundancies. This generalizes the main result of Singh and Singh (1997) and improves the corresponding one of Hu and Wang (2009) as well. As an application, we build the reversed hazard rate order on order statistics from sample having proportional hazard rates, which strengthens the usual stochastic order in Theorem 2.1 of Pledger and Proschan (1971) to the reversed hazard order in the situation that all components are of (rational) proportional hazard rates.

Order preserving property of moment estimators

Abstract Balakrishnan and Mi (2001) considered order preserving property of maximum likelihood estimators. In this paper there are given conditions under which the moment estimators have the property of preserving stochastic orders. The preserving property for the usual stochastic order as well as for likelihood ratio one is considered. Sufficient conditions are established for some parametric families of distributions.

Stochastic Orders Comparisons of Negative Binomial Distribution with Negative Binomial—Lindley Distribution

The purpose of this study is to compare a negative binomial distribution with a negative binomial—Lindley by using stochastic orders. We characterize the comparisons in usual stochastic order, likelihood ratio order, convex order, expectation order and uniformly more variable order based on theorem and some numerical example of comparisons between negative binomial random variable and negative binomial—Lindley random variable.

Probability inequalities for weighted distributions

Abstract In this paper, properties of weighted distributions for general weight functions are investigated. We establish the subadditivity [superadditivity] property of weighted distributions for log-concave [log-convex] weight functions in the sense of the usual stochastic order. The main result generalizes Lemma 2.3 in Brown (2006) . Several interesting moment inequalities are presented.

Stochastic orders and non-Gaussian risk factor models

The main results of this paper are monotonicity statements about the risk measures value-at-risk (VaR) and tail value-at-risk (TVaR) with respect to the parameters of single and multi risk factor models, which are standard models for the quantification of credit and insurance risk. In the context of single risk factor models, non-Gaussian distributed latent risk factors are allowed. It is shown that the TVaR increases with increasing claim amounts, probabilities of claims and correlations, whereas the VaR is in general not monotone in the correlation parameters. To compare the aggregated risks arising from single and multi risk factor models, the usual stochastic order and the increasing convex order are used in this paper, since these stochastic orders can be interpreted as being induced by the VaR-concept and the TVaR-concept, respectively. To derive monotonicity statements about these risk measures, properties of several further stochastic orders are used and their relation to the usual stochastic order and to the increasing convex order are applied.

Some stochastic inequalities for weighted sums

We compare weighted sums of i.i.d. positive random variables according to the usual stochastic order. The main inequalities are derived using majorization techniques under certain log-concavity assumptions. Specifically, let $Y_i$ be i.i.d. random variables on $\mathbf{R}_+$. Assuming that $\log Y_i$ has a log-concave density, we show that $\sum a_iY_i$ is stochastically smaller than $\sum b_iY_i$, if $(\log a_1,...,\log a_n)$ is majorized by $(\log b_1,...,\log b_n)$. On the other hand, assuming that $Y_i^p$ has a log-concave density for some $p>1$, we show that $\sum a_iY_i$ is stochastically larger than $\sum b_iY_i$, if $(a_1^q,...,a_n^q)$ is majorized by $(b_1^q,...,b_n^q)$, where $p^{-1}+q^{-1}=1$. These unify several stochastic ordering results for specific distributions. In particular, a conjecture of Hitczenko [Sankhy\={a} A 60 (1998) 171--175] on Weibull variables is proved. Potential applications in reliability and wireless communications are mentioned.

Stochastic comparisons of series and parallel systems with randomized independent components

Abstract Consider a series or parallel system of independent components and assume that the components are randomly chosen from two different batches, with the components of the first batch being more reliable than those of the second. In this note it is shown that the reliability of the system increases, in usual stochastic order sense, as the random number of components chosen from the first batch increases in increasing convex order. As a consequence, we establish a result analogous to the Parrondo’s paradox, which shows that randomness in the number of components extracted from the two batches improves the reliability of the series system.

ORDER STATISTICS FROM HETEROGENOUS NEGATIVE BINOMIAL RANDOM VARIABLES

In this article, we study the order statistics from heterogenous negative binomial random variables. Sufficient conditions are provided for comparing the extreme order statistics according to the usual stochastic order. For the special case of geometric distribution, a sufficient condition is established for comparing order statistics in the sense of multivariate stochastic order. Applications in the Poisson–Gamma shock model and redundant systems have been described as well.

Bounds for mixtures of order statistics from exponentials and applications

This paper deals with the stochastic comparison of order statistics and their mixtures. For a random sample of size n from an exponential distribution with hazard rate λ , and for 1 ≤ k ≤ n , let us denote by F k : n ( λ ) the distribution function of the corresponding k t h order statistic. Let us consider m random samples of same size n from exponential distributions having respective hazard rates λ 1 , … , λ m . Assume that p 1 , … , p m > 0 , such that ∑ i = 1 m p i = 1 , and let U and V be two random variables with the distribution functions F k : n ( λ ) and ∑ i = 1 m p i F k : n ( λ i ) , respectively. Then, V is greater in the hazard rate order (or the usual stochastic order) than U if and only if λ ≥ ∑ i = 1 m p i λ i k k , and V is smaller in the hazard rate order (or the usual stochastic order) than U if and only if λ ≤ min 1 ≤ i ≤ m λ i , for all k = 1 , … , n . These properties are used to find the best bounds for the survival functions of order statistics from independent heterogeneous exponential random variables. For the proof, we will use a mixture type representation for the distribution functions of order statistics.

Stochastic comparisons for allocations of policy limits and deductibles with applications

Abstract In this paper, we study the problem of comparing losses of a policyholder who has an increasing utility function when the form of coverage is policy limit and deductible. The total retained losses of a policyholder ∑ i = 1 n ( X i − l i ) + are ordered in the usual stochastic order sense when X i ( i = 1 , … , n ) are ordered with respect to the likelihood ratio order. The parallel results for the case of deductibles are obtained in the same way. It is shown that the ordering of the losses are related to the characteristics (log-concavity or log-convexity) of distributions of the risks. As an application of the comparison results, the optimal problems of allocations of policy limits and deductibles are studied in usual stochastic order sense and the closed-form optimal solutions are obtained in some special cases.

A Unified Approach to Likelihood Inference on Linear Inequality Constraints in Contingency Tables

In contingency table analysis, a likelihood ratio test for linear inequality constraints is discussed. The restriction condition considered in this article is much more general than usual stochastic order restriction conditions. Asymptotic property of test statistic is shown. Simulation study is conducted to compare the empirical power and size performed via the proposed method and others. Several real data sets are used to illustrate our theoretical result. The idea used in this article can be applied to test the problems with nonlinear inequality constraints.

Additive factors and stages of mental processes in task networks

Abstract To perform a task a subject executes mental processes. An experimental manipulation, such as a change in stimulus intensity, is said to selectively influence a process if it changes the duration of that process leaving other process durations unchanged. For random process durations a definition of a factor selectively influencing a process by increments is given in terms of stochastic dominance (also called “the usual stochastic order”). A technique for analyzing reaction times, Sternberg’s Additive Factor Method, assumes all the processes are in series. When all processes are in series, each process is called a stage. With the Additive Factor Method, if two experimental factors selectively influence two different stages by increments, the factors will have additive effects on reaction time. An assumption of the Additive Factor Method is that if two experimental factors interact, then they influence the same stage. We consider sets of processes in which some pairs of processes are sequential and some are concurrent (i.e., the processes are partially ordered). We propose a natural definition of a stage for such sets of processes. For partially ordered processes, with our definition of a stage, if two experimental factors selectively influence two different processes by increments, each within a different stage, then the factors have additive effects. If each process selectively influenced by increments is in the same stage, then an interaction is possible, although not inevitable.