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Let $V_{(r,n,\tilde {m}_n,k)}^{(p)}$ and $W_{(r,n,\tilde {m}_n,k)}^{(p)}$ be the $p$ -spacings of generalized order statistics based on absolutely continuous distribution functions $F$ and $G$ , respectively. Imposing some conditions on $F$ and $G$ and assuming that $m_1=\cdots =m_{n-1}$ , Hu and Zhuang (2006. Stochastic orderings between p -spacings of generalized order statistics from two samples. Probability in the Engineering and Informational Sciences 20: 475) established $V_{(r,n,\tilde {m}_n,k)}^{(p)} \leq _{{\rm hr}} W_{(r,n,\tilde {m}_n,k)}^{(p)}$ for $p=1$ and left the case $p\geq 2$ as an open problem. In this article, we not only resolve it but also give the result for unequal $m_i$ 's. It is worth mentioning that this problem has not been proved even for ordinary order statistics so far. [ABSTRACT FROM AUTHOR]

Let $X_1, \ldots, X_n$ be mutually independent exponential random variables with distinct hazard rates $\lambda _1, \ldots, \lambda _n$ and let $Y_1, \ldots, Y_n$ be a random sample from the exponential distribution with hazard rate $\bar \lambda = \sum _{i=1}^{n} \lambda _i/n$. Also let $X_{1:n} \lt \cdots \lt X_{n:n}$ and $Y_{1:n} \lt \cdots \lt Y_{n:n}$ be their associated order statistics. It is proved that for $1\le i \lt j \le n$ , the generalized spacing $X_{j:n} - X_{i:n}$ is more dispersed than $Y_{j:n} - Y_{i:n}$ according to dispersive ordering and for $2\le i \le n$ , the dependence of $X_{i:n}$ on $X_{1:n}$ is less than that of $Y_{i:n}$ on $Y_{1 :n}$ , in the sense of the more stochastically increasing ordering. This dependence result is also extended to the proportional hazard rates (PHR) model. This extends the earlier work of Genest et al. [(2009)]. On the range of heterogeneous samples. Journal of Multivariate Analysis 100: 1587–1592] who proved this result for $i =n$. [ABSTRACT FROM AUTHOR]

We show that the kth order statistic from a heterogeneous sample of n ≥ k exponential random variables is larger than that from a homogeneous exponential sample in the sense of star ordering, as conjectured by Xu and Balakrishnan [14]. As a consequence, we establish hazard rate ordering for order statistics between heterogeneous and homogeneous exponential samples, resolving an open problem of Pǎltǎnea [11]. Extensions to general spacings are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the moderate deviation principle of sample quantiles and order statistics for stationary m -dependent random variables. The results obtained in this paper extend the corresponding ones for an independent and identically distributed sequence to a stationary m -dependent sequence. [ABSTRACT FROM AUTHOR]

Let X1, …, Xn be non-negative, independent and identically distributed random variables with a common distribution function F, and denote by X1:n ≤ ··· ≤ Xn:n the corresponding order statistics. In this paper, we investigate the second-order regular variation (2RV) of the tail probabilities of Xk:n and Xj:n − Xi:n under the assumption that $\bar {F}$ is of the 2RV, where 1 ≤ k ≤ n and 1 ≤ i < j ≤ n. [ABSTRACT FROM PUBLISHER]

Let F be a bathtub-shaped (BT) hazard rate distribution function. It has been shown that the hazard rate function of the order statistics may be BT, increasing, etc. Then, we have carried out a graphical study for some useful lifetime models.Moreover, we are interested to compare the time that maximizes the mean residual life (MRL) function of F with the one related to a general weighted model in terms of their locations. Also, the times maximizing the conditional reliability proposed by Mi [13] of F have been compared with the corresponding times of a general weighted model. As special cases, we consider order statistics and the proportional hazard rate model. [ABSTRACT FROM AUTHOR]

Sequential order statistics describe the ordered failure times in a k-out-of-n system, where the failures of components might affect the performance of remaining working components. In this article aging properties of sequential order statistics are examined and conditions are given such that the distribution of a sequential order statistic is ILR, IFR, IFRA, or NBU. Moreover, conditions for aging properties of spacings of sequential order statistics are obtained. [ABSTRACT FROM AUTHOR]

This article deals with Rényi entropies for the residual life and the inactivity time. Monotonic properties of the entropy in order statistics, record values, and weighted distributions are investigated, and the comparison on weighted random variables is studied in terms of residual Rényi entropy as well. [ABSTRACT FROM PUBLISHER]

In this article, we investigate the sufficient and/or necessary conditions in order to stochastically compare the order statistics and their spacing vectors of two random vectors X and Y with special symmetric distributions. The conditions are imposed on the sample ranges Xn:n-X1:n and Yn:n-Y1:n or on (X1:n, Xn:n-X1:n) and (Y1:n, Yn:n-Y1:n). In particular, we consider the multivariate usual stochastic order, the convex order, the increasing convex order, and the directionally convex order. Several examples are also given to illustrate the power of the main results. [ABSTRACT FROM AUTHOR]

In this article we investigate less restrictive conditions on the model parameters and the underlying distribution function upon which the generalized order statistics are based, which enable one to establish the likelihood ratio and the hazard ratio orderings for p-spacings of generalized order statistic. Some previous works in the literature are extended. [ABSTRACT FROM AUTHOR]

The Normal and Gamma distributions are the most popular models for analyzing symmetric and skewed data, respectively. In this article, a new multimodal distribution is introduced that contains the Normal and Gamma distributions as particular cases and thus could be a better model for both symmetric and skewed data. Various structural properties of this distribution are derived, including its moment-generating function, characteristic function, moments, entropy, asymptotic distribution of the extreme order statistics, method of moment estimates, maximum likelihood estimates, Fisher information matrix, and simulation issues. The superiority of the new distribution is illustrated by means of two real datasets. [ABSTRACT FROM AUTHOR]

In this article we investigate conditions on the underlying distribution functions on which the sequential order statistics are based, to obtain stochastic comparisons of sequential order statistics in the multivariate likelihood ratio, the multivariate hazard rate, and the usual multivariate stochastic orders. Some applications of the main results are also given. [ABSTRACT FROM AUTHOR]