In this paper, we review some recent results on the stochastic comparison of order statistics and related statistics from independent and heterogeneous proportional hazard rates models, gamma variables, geometric variables, and negative binomial variables. We highlight the close connections that exist between some classical stochastic orders and majorization-type orders.

N. Balakrishnan and Peng Zhao have prepared an outstanding survey of recent results that stochastically compare various order statistics and some ranges based on two collections of independent heterogeneous random variables. Their survey focuses on results for heterogeneous exponential random variables and their extensions to random variables with proportional hazard rates. In addition, some results that stochastically compare order statistics based on heterogeneous gamma, Weibull, geometric, and negative binomial random variables are also given. In particular, the authors of have listed some stochastic comparisons that are based on one heterogeneous collection of random variables, and one homogeneous collection of random variables. Personally, I find these types of comparisons to be quite fascinating. Balakrishnan and Zhao have done a thorough job of listing all the known results of this kind.

Professors Balakrishnan and Zhao have written an excellent survey on the recent developments of stochastic comparisons of order statistics, which cover almost every aspect of ordering properties of order statistics from both continuous and discrete heterogeneous populations. My discussion will be limited to the skewness of order statistics and order statistics from heterogeneous populations with different shape parameters.

Balakrishnan and Zhao does an excellent job in this issue at reviewing the recent advances on stochastic comparison between order statistics from independent and heterogeneous observations with proportional hazard rates, gamma distribution, geometric distribution, and negative binomial distributions, the relation between various stochastic order and majorization order of concerned heterogeneous parameters is highlighted. Some examples are presented to illustrate main results while pointing out the potential direction for further discussion.

I thank the authors for providing another excellent review on stochastic comparisons of order statistics after the review paper written by Kochar and Xu (2007).

At first, I would like to thank the authors for giving the detailed survey on the current state of stochastic orderings of order statistics from heterogeneous populations. Their review illustrates the fast development of research in this area during the recent years, in particular including many important contributions by the authors themselves. Besides the concrete open problems given in the different sections of the review, the following general topics may be also of interest for further studies.

First of all, we express our sincere thanks to all the discussants for their valuable comments and suggestions as well as for their own significant contributions to the area of order statistics in general, and to the topic of stochastic comparison in particular. We shall now provide our response to the comments and suggestions of all the discussants.

When waiting times cannot be observed directly, Little's law can be applied to estimate the average waiting time by the average number in system divided by the average arrival rate, but that simple indirect estimator tends to be biased significantly when the arrival rates are time-varying and the service times are relatively long. Here it is shown that the bias in that indirect estimator can be estimated and reduced by applying the time-varying Little's law (TVLL). If there is appropriate time-varying staffing, then the waiting time distribution may not be time-varying even though the arrival rate is time varying. Given a fixed waiting time distribution with unknown mean, there is a unique mean consistent with the TVLL for each time t. Thus, under that condition, the TVLL provides an estimator for the unknown mean wait, given estimates of the average number in system over a subinterval and the arrival rate function. Useful variants of the TVLL estimator are obtained by fitting a linear or quadratic function to arrival data. When the arrival rate function is approximately linear (quadratic), the mean waiting time satisfies a quadratic (cubic) equation. The new estimator based on the TVLL is a positive real root of that equation. The new methods are shown to be effective in estimating the bias in the indirect estimator and reducing it, using simulations of multi-server queues and data from a call center.

Consider a portfolio of n identically distributed risks X1, …, Xn with dependence structure modelled by an Archimedean survival copula. It is known that the probability of a large aggregate loss of $\sum\nolimits_{i=1}^{n} X_{i}$ is in proportion to the probability of a large individual loss of X1. The proportionality factor depends on the dependence strength and the tail behavior of the individual risk. In this paper, we establish analogous results for an aggregate loss of the form g(X1, …, Xn) under the more general model in which the Xi's have different but tail-equivalent distributions and the copula remains unchanged, where g is a homogeneous function of order 1. Properties of these factors are studied, and asymptotic Value-at-Risk behaviors of functions of dependent risks are also given. The main results generalize those in Wüthrich [16], Alink, Löwe, and Wüthrich [2], Barbe, Fougères, and Genest [4], and Embrechts, Nešlehová, and Wüthrich [9].

This work focuses on Exchangeable Occupancy Models (EOMs) and their relations with the Uniform Order Statistics Property (UOSP) for point processes in discrete time. As our main purpose, we show how definitions and results presented in Shaked, Spizzichino, and Suter [8] can be unified and generalized in the frame of occupancy models. We first show some general facts about EOMs. Then we introduce a class of EOMs, called ℳ(a)-models, and a concept of generalized Uniform Order Statistics Property in discrete time. For processes with this property, we prove a general characterization result in terms of ℳ(a)-models. Our interest is also focused on properties of closure w.r.t. some natural transformations of EOMs.

Recently, proportional mean residual life model has received a lot of attention after the importance of the model was discussed by Zahedi [17]. In this paper, we define dynamic proportional mean residual life model and study its properties for different aging classes. The closure of this model under different stochastic orders is also discussed. Many examples are presented to illustrate different properties of the model.