Hazard rate ordering of the second-order statistics from multiple-outlier PHR samples

In this paper, we compare the hazard rate functions of the second-order statistics arising from two sets of independent multiple-outlier proportional hazard rates (PHR) samples. It is proved that the submajorization order between the sample size vectors together with the supermajorization order between the hazard rate vectors imply the hazard rate ordering between the corresponding second-order statistics from multiple-outlier PHR random variables. The results established here provide theoretical guidance both for the winner's price for the bid in the second-price reverse auction in auction theory and fail-safe system design in reliability. Some numerical examples are also provided for illustration.     

Ordering properties of largest order statistics from independent and heterogeneous Dagum populations

Abstract

The Dagum distribution has been extensively used to model income data, and its features have been appreciated in economics and financial studies. In this article, we discuss ordering properties of largest order statistics from independent and heterogeneous Dagum populations. We present some sufficient conditions for stochastic comparisons between largest order statistics in terms of the reversed hazard rate order, the usual stochastic order, the convex order, the likelihood ratio order and the dispersive order. Several numerical examples are presented to illustrate the results established here.     

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.     

Stochastic comparisons between extreme order statistics from scale models

Stochastic ordering relations between extreme order statistics from exponential, Weibull and gamma distributions have been studied extensively by many researchers in recent years. In this work, we obtain various ordering results for the comparisons of two extreme order statistics from scale models when one set of scale parameters majorizes the other. The new results obtained here are applied when the baseline distributions are exponentiated Weibull or generalized gamma distributions. In this way, we generalize and extend some results established recently in the literature.     

Comparisons of order statistics from heterogeneous negative binomial variables with applications

This paper studies the ordering properties of extreme order statistics arising from independent negative binomial random variables. Employing the useful tool of majorization-type orders, sufficient conditions are given for comparing extreme negative binomial order statistics according to the usual stochastic order. Some numerical examples are provided to illustrate the theoretical results. Applications in Poisson-Gamma shock model in reliability engineering and claim frequency in insurance are presented to show the practicability of our results as well.     

Likelihood ratio comparisons among spacings related to both one or two samples

In this paper, we investigate some stochastic comparisons in terms of likelihood ratio ordering between spacings from independent random variables exponentially distributed with different scale parameters. We partially solve some open problems in [Wen S, Lu Q, Hu T. Likelihood ratio orderings of spacings of heterogeneous exponential random variables. J Multivariate Anal. 2007;98:743–756] for a one-sample problem and in [Hu T, Lu Q, Wen S. Stochastic comparisons and dependence of spacings from two samples of exponential random variables. Commun Stat – Theory Methods 2006;35:979–988] for a two-sample problem. Specifically, we prove that the second spacing is always smaller than the third spacing in terms of the likelihood ratio order and we provide the ordering among all spacings in the case n=4. In the two-sample case, we establish comparisons between the second spacings related to each sample under certain conditions.     

A note on hazard rates of order statistics

Let F_k:m be the cumulative disribution function of the kth order statistic in a sample of size n from a distribution

F(x) with density function f(x).The primary objective of this paper is to show that F_k+1mis IHR(increasing hazard rate) if F_km(x)is IHH and that F_k-1:n(x)is DHR.(decreasing hazard rate) if F_km(x) is DHR.     

Likelihood ratio order of parallel systems under multiple-outlier models

This 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.     

Stochastic comparisons of order statistics,with applications in relibility

This paper reviews recent developments in the stochastic comparison of order statistics. The results discussed are basically: (l) Stochastic comparisons of linear combinations of order statistics from distributions F and G where G^?1 F is convex or starshaped. (2) Stochastic comparisons of individual order statistics and of vectors of order statistics from underlying heterogeneous distributions by the use of majorization and Schur function theory. (3) Stochastic comparison of random processes. Applications to reliability problems are presented illustrating the use and value of the theoretical results described     

A zero-one law for large order statistics

Fix r ≥ 1, and let {M_nr} be the rth largest of {X₁,X₂,…X_n}, where X₁,X₂,… is a sequence of i.i.d. random variables with distribution function F. It is proved that P[M_nr ≤ u_n i.o.] = 0 or 1 according as the series Σ^∞_n=3Fⁿ(u_n)(log log n)^r/n converges or diverges, for any real sequence {u_n} such that n{1 -F(u_n)} is nondecreasing and divergent. This generalizes a result of Bamdorff-Nielsen (1961) in the case r = 1.     

Ordering properties of sample minimum from Kumaraswamy-G random variables

In this paper we compare the minimums of two independent and heterogeneous samples each following Kumaraswamy (Kw)-G distribution with the same and the different parent distribution functions. The comparisons are carried out with respect to usual stochastic ordering and hazard rate ordering with majorized shape parameters of the distributions. The likelihood ratio ordering between the minimum order statistics is established for heterogeneous multiple-outlier Kw-G random variables with the same parent distribution function.     

Random translation of discrete order statistics

We investigate the existence and uniqueness of a discrete parent distribution supported on the integers whose order statistics are related by a random translation. We also provide some examples using the constructive method that we propose.     

On extreme order statistics from heterogeneous beta distributions with applications

This paper treats the problem of stochastic comparisons for the extreme order statistics arising from heterogeneous beta distributions. Some sufficient conditions involved in majorization-type partial orders are provided for comparing the extreme order statistics in the sense of various magnitude orderings including the likelihood ratio order, the reversed hazard rate order, the usual stochastic order, and the usual multivariate stochastic order. The results established here strengthen and extend those including Kochar and Xu (2007 Kochar, S.C., Xu, M. (2007). Stochastic comparisons of parallel systems when components have proportional hazard rates. Probab. Eng. Inf. Sci. 21:597–609.[Crossref], [Web of Science ®] , [Google Scholar]), Mao and Hu (2010 Mao, T., Hu, T. (2010). Equivalent characterizations on orderings of order statistics and sample ranges. Probab. Eng. Inf. Sci. 24:245–262.[Crossref], [Web of Science ®] , [Google Scholar]), Balakrishnan et al. (2014 Balakrishnan, N., Barmalzan, G., Haidari, A. (2014). On usual multivariate stochastic ordering of order statistics from heterogeneous beta variables. J. Multivariate Anal. 127:147–150.[Crossref], [Web of Science ®] , [Google Scholar]), and Torrado (2015 Torrado, N. (2015). On magnitude orderings between smallest order statistics from heterogeneous beta distributions. J. Math. Anal. Appl. 426:824–838.[Crossref], [Web of Science ®] , [Google Scholar]). A real application in system assembly and some numerical examples are also presented to illustrate the theoretical results.     

Sequential order statistics from dependent random variables

Abstract

This paper provides an extension for “sequential order statistics” (SOS) introduced by Kamps. It is called “developed sequential order statistics” (DSOS) and is useful for describing lifetimes of engineering systems when component lifetimes are dependent. Explicit expressions for the joint density function, the marginal distributions and the means of DSOS are derived. Under the well known “conditional proportional hazard rate” (CPHR) model and the Gumbel families of copulas for dependency among component lifetimes, some findings are reported. For example, it is proved that the joint density functions of DSOS and SOS have the same structure. Various illustrative examples are also given.     

On the skewness of extreme order statistics from heterogeneous samples

The effect of heterogeneity on order statistics has attracted much attention in recent decades. In this paper, we study the skewness of extreme order statistics from heterogeneous samples in the scale models according to star ordering. It is shown that, without any restriction on the scale parameters, the skewness of extreme order statistics from heterogeneous samples is larger than that from homogeneous samples. We further extend this study to the multiple-outlier scale models. Some examples and applications are highlighted as well.     

Multivariate testing and model-checking for generalized order statistics with applications

The exponential family structure of the joint distribution of generalized order statistics is utilized to establish multivariate tests on the model parameters. For simple and composite null hypotheses, the likelihood ratio test (LR test), Wald's test, and Rao's score test are derived and turn out to have simple representations. The asymptotic distribution of the corresponding test statistics under the null hypothesis is stated, and, in case of a simple null hypothesis, asymptotic optimality of the LR test is addressed. Applications of the tests are presented; in particular, we discuss their use in reliability, and to decide whether a Poisson process is homogeneous. Finally, a power study is performed to measure and compare the quality of the tests for both, simple and composite null hypotheses.     

Prediction intervals for ordinary and dual generalized order statistics from two independent sequences

ABSTRACT

Based on the observed dual generalized order statistics drawn from an arbitrary unknown distribution, nonparametric two-sided prediction intervals as well as prediction upper and lower bounds for an ordinary and a dual generalized order statistic from another iid sequence with the same distribution are developed. The prediction intervals for dual generalized order statistics based on the observed ordinary generalized order statistics are also developed. The coverage probabilities of these prediction intervals are exact and free of the parent distribution, F. Finally, numerical computations and real examples of the coverage probabilities are presented for choosing the appropriate limits of the prediction.     

A note on moments of order statistics from exchangeable variates

Joshi (1973) and Balakrishnan and Malik (1985) have derived some some identities for the moments of order statistics from independent and identically distributed random variables. In this paper, we make use of a basic result due to David and Joshi (1968) and show that these identities for the moments also hold when the order statistics arise from exchangeable variables.