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Summary Discrete power-law distributions have significant consequences for understanding many phenomena in practice, and have attracted much attention in recent decades. However, in many practical applications, there exists a natural upper bound for the probability tail. In this paper, we develop maximum likelihood estimates for truncated discrete power-law distributions based on the upper order statistics, and large sample properties are mentioned as well. Monte Carlo simulation is carried out to examine the finite sample performance of the estimates. Applications in real cyber attack data and peak gamma-ray intensity of solar flares are highlighted.

In this article, necessary conditions for comparing order statistics from distributions with regularly varying tails are discussed in terms of various stochastic orders. A necessary and sufficient condition for stochastically comparing tail behaviors of order statistics is derived. The main results generalize and recover some results in Kleiber (2002, 2004). Extensions to coherent systems are mentioned as well.

In this paper, the relative dependence of a linear regression model is studied. In particular, the dependence of autoregressive models in time series are investigated. It is shown that for the first-order non-stationary autoregressive model and the random walk with trend and drift model, the dependence between two states decreases with lag. Some numerical examples are presented as well.

Order statistics from heterogeneous samples have been extensively studied in the literature. However, most of the work focused on the effect of heterogeneity on the magnitude or dispersion of order statistics. In this paper, we study the skewness of order statistics from heterogeneous samples according to star ordering. The main results extend the corresponding results in Kochar and Xu (J. Appl. Probab. 46:342–352, 2009; J. Appl. Probab. 48:271–284, 2011). Examples and applications are highlighted as well.

In this paper, the sample range from a heterogeneous exponential sample is shown to be larger than that from a homogeneous exponential sample in the sense of the star ordering. Then, by using this result, some equivalent characterizations of stochastic comparisons of sample ranges with respect to various stochastic orders are established. In this process, two open problems mentioned in Mao and Hu (2010) [16] are solved. The main results established here extend and strengthen several known results in the literature including those of Khaledi and Kochar (2000) [8] , Zhao and Li (2009) [22] and Genest et al. (2009) [7] .

In this paper, a new sufficient condition for comparing linear combinations of independent gamma random variables according to star ordering is given. This unifies some of the newly proved results on this problem. Equivalent characterizations between various stochastic orders are established by utilizing the new condition. The main results in this paper generalize and unify several results in the literature including those of Amiri, Khaledi, and Samaniego [2], Zhao [18], and Kochar and Xu [9].

This paper studies capital allocation problems using a general loss function. Stochastic comparisons are conducted for general loss functions in several scenarios: independent and identically distributed risks; independent but non-identically distributed risks; comonotonic risks. Applications in optimal capital allocations and policy limits allocations are discussed as well.

In this paper we provide some sufficient conditions to stochastically compare linear combinations of independent random variables. The main results extend those given in Proschan (1965), Ma (1998), Zhao et al. (2011), and Yu (2011). In particular, we propose a new sufficient condition to compare the peakedness of linear combinations of independent random variables which may have heavy-tailed properties.

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.

Two sufficient conditions for comparing convolutions of heterogeneous gamma random variables in terms of star order are established. It is further shown that if the scale parameters of heterogeneous gamma random variables are more dispersed in the sense of majorization, then the convolutions are more dispersed according to the right spread order, which generalizes and strengthens the results in Diaconis and Perlman (1987), Kochar and Xu (2010) and Zhao and Balakrishnan (2009).

A sufficient condition for comparing convolutions of heterogeneous exponential random variables in terms of right spread order is established. As a consequence, it is shown that a convolution of heterogeneous independent exponential random variables is more skewed than that of homogeneous exponential random variables in the sense of NBUE order. This gives a new insight into the distribution theory of convolutions of independent random variables. A sufficient condition is also derived for comparing such convolutions in terms of Lorenz order.

In this article, mixture representations of survival functions of residual lifetimes of k-out-of-n systems are obtained when the components are independent but not necessarily identically distributed. Then we stochastically compare the residual lifetimes of k-out-of-n systems in one- and two-sample problems. In particular, the results extend some results in Li and Zhao [14], Khaledi and Shaked [13], Sadegh [17], Gurler and Bairamov [7] and Navarro, Balakrishnan, and Samaniego [16]. Applications in the proportional hazard rates model are presented as well.

Let R"n be the range of a random sample X"1,...,X"n of exponential random variables with hazard rate @l. Let S"n be the range of another collection Y"1,...,Y"n of mutually independent exponential random variables with hazard rates @l"1,...,@l"n whose average is @l. Finally, let r and s denote the reversed hazard rates of R"n and S"n, respectively. It is shown here that the mapping [email protected]?s(t)/r(t) is increasing on (0,~) and that as a result, R"n=X"("n")-X"("1") is smaller than S"n=Y"("n")-Y"("1") in the likelihood ratio ordering as well as in the dispersive ordering. As a further consequence of this fact, X"("n") is seen to be more stochastically increasing in X"("1") than Y"("n") is in Y"("1"). In other words, the pair (X"("1"),X"("n")) is more dependent than the pair (Y"("1"),Y"("n")) in the monotone regression dependence ordering. The latter finding extends readily to the more general context where X"1,...,X"n form a random sample from a continuous distribution while Y"1,...,Y"n are mutually independent lifetimes with proportional hazard rates.

A parallel system with heterogeneous exponential component lifetimes is shown to be more skewed (according to the convex transform order) than the system with independent and identically distributed exponential components. As a consequence, equivalent conditions for comparing the variabilities of the largest order statistics from heterogeneous and homogeneous exponential samples in the sense of the dispersive order and the right-spread order are established. A sufficient condition is also given for the proportional hazard rate model.

In this paper, we introduce a new copula-based dependence order to compare the relative degree of dependence between two pairs of random variables. Relationship of the new order to the existing dependence orders is investigated. In particular, the new ordering is stronger than the partial ordering, more monotone regression dependence as developed by Avérous et al. [J. Avérous, C. Genest, S.C. Kochar, On dependence structure of order statistics, Journal of Multivariate Analysis 94 (2005) 159–171]. Applications of this partial order to order statistics, k-record values and frailty models are given.

This note investigates the negative dependence in frailty models. First, we show that the frailty variable and the overall population variable are negatively likelihood ratio dependent and derive an upper bound for the survival function of the population with higher frailty. Secondly, we prove that the DFR property and the logconvex hazard rate of the baseline variable imply the DLR property of the population variable. Finally, we further prove that the likelihood ratio order, hazard rate order and reversed hazard rate order between two frailty variables imply the likelihood ratio order, reversed hazard rate order, and hazard rate order between the corresponding overall population variables, respectively.

This paper investigates the stochastic comparison of spacings of record values. The likelihood ratio order, the hazard rate order and the mean residual life order between spacings of record values from one sample sequence are developed. Furthermore, the increasing convex order and the mean residual life order between simple spacings of record values from two sample sequences are built as well.

Let X1, … , Xn be independent random variables with Xi having survival function Fλi, i = 1, … , n, and let Y1, … ,Yn be a random sample with common population survival distribution F, where c = ∑i=1nλi/n. Let Xn:n and Yn:n denote the lifetimes of the parallel systems consisting of these components, respectively. It is shown that Xn:n is greater than Yn:n in terms of likelihood ratio order. It is also proved that the sample range Xn:n − X1:n is larger than Yn:n − Y1:n according to reverse hazard rate ordering. These two results strengthen and generalize the results in Dykstra, Kochar, and Rojo [6] and Kochar and Rojo [11], respectively.

In this note, we further study the properties of excess wealth (or right spread) order and the location independent riskier order. It is proved that if X is less variable than Y according to excess wealth order, then X n : n − X k : n ≤ icx Y n : n − Y k : n for k = 0 , 1 , … , n − 1 , where X 0 : n = Y 0 : n ≡ 0 . Similar results are obtained for location independent riskier order. An application in k -price business auction models is presented as well.

This article deals with the multiple-outlier exponential model. The likelihood ratio order between m-spacings of the combined sample is developed, some results extend the conclusions on simple spacings in Wen et al. (2007).

This paper deals with preservation of the reversed hazard rate order between equilibrium random variables under formations of some reliability structures. We further investigate a new aging notion based upon the reversed hazard rate order between a random life and its equilibrium version. A nonparametric method is developed to test the exponentiality against such a strict aging property, some numerical results are presented as well.

For two non-negative random variables X and Y, it is shown that if X is smaller than Y in the sense of the likelihood ratio order and either X or Y is of decreasing likelihood ratio, then the m-spacing V k : n ( m ) of an X-sample is smaller than W k : n ( m ) , the m-spacing of a Y-sample, in terms of the likelihood ratio order. It is also proved that a similar result for the upshifted likelihood ratio order is valid. Finally, it is proved that the decreasing failure rate in average property of a population X is preserved by sample spacing V k : n ( 1 ) .

We investigate some new properties of mean inactivity time (MIT) order and increasing MIT (IMIT) class of life distributions. The preservation property of MIT order under increasing and concave transformations, reversed preservation properties of MIT order, and IMIT class of life distributions under the taking of maximum are developed. Based on the residual life at a random time and the excess lifetime in a renewal process, stochastic comparisons of both IMIT and decreasing mean residual life distributions are conducted as well.

Linear combinations of independent random variables have been extensively studied in the literature. However, most of the work is based on some specific distribution assumptions. In this paper, a companion of (J. Appl. Probab. 48 (2011) 1179-1188), we unify the study of linear combinations of independent nonnegative random variables under the general setup by using some monotone transforms. The results are further generalized to the case of independent but not necessarily identically distributed nonnegative random variables. The main results complement and generalize the results in the literature including (In Studies in Econometrics, Time Series, and Multivariate Statistics (1983) 465-489 Academic Press; Sankhy\={a} Ser. A 60 (1998) 171-175; Sankhy\={a} Ser. A 63 (2001) 128-132; J. Statist. Plann. Inference 92 (2001) 1-5; Bernoulli 17 (2011) 1044-1053)., Comment: Published in at http://dx.doi.org/10.3150/12-BEJ429 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

In this paper, some ordering properties of convolutions of heterogeneous Bernoulli random variables are discussed. It is shown that, under some suitable conditions, the likelihood ratio order and the reversed hazard rate order hold between convolutions of two heterogeneous Bernoulli sequences. The results established here extend and strengthen the previous results of Pledger and Proschan (1971) and Boland, Singh and Cukic (2002).

Kochar and Xu (2009) proved that a parallel system with heterogeneous exponential component lifetimes is more skewed (according to the convex transform order) than the system with independent and identically distributed exponential components. In this paper we extend this study to the general k-out-of-n systems for the case when there are only two types of component in the system. An open problem proposed in Pǎltǎnea (2008) is partially solved.

In this paper we review some of the recently obtained results in the area of stochastic comparisons of sample spacings when the observations are not necessarily identically distributed. A few new results on necessary and sufficient conditions for various stochastic orderings among spacings are also given. The paper is concluded with some examples and applications.

The statistics literature has mostly focused on the case when the data available is in the form of a random sample. In many cases, the observations are not identically distributed. Such samples are called heterogeneous samples. The study of heterogeneous samples is of great interest in many areas, such as statistics, econometrics, reliability engineering, operation research and risk analysis. Stochastic orders between probability distributions is a widely studied concept. There are several kinds of stochastic orders that are used to compare different aspects of probability distributions like location, variability, skewness, dependence, etc. In this dissertation, most of the work is devoted to investigating the properties of statistics based on heterogeneous samples with the aid of stochastic orders. We will see the effect of the change in the stochastic properties of various functions of observations as their parameters change. The theory of majorization will be used for this purpose. First, order statistics from heterogeneous samples will be investigated. Order statistics appear everywhere in statistics and related areas. The k-out-of-n systems are building blocks of a coherent system. The lifetime of such a system is the same as that of the (n−k+1)th order statistic in a sample size of n. Stochastic comparisons between order statistics have been studied extensively in the literature in case the parent observations are independent and identically distributed. i However, in practice this assumption is often violated as different components in a system may not have the same distribution. Comparatively less work has been done in the case of heterogeneous random variables, mainly because of the reason that their distribution theory is very complicated. Some open problems in the literature have been solved in the dissertation. Some new problems associated with order statistics have been investigated in the thesis. Next, stochastic properties of spacings based on heterogeneous observations are studied. Spacings are of great interest in many areas of statistics, in particular, in the characterizations of distributions, goodness-of-fit tests, life testing and reliability models. In particular, the stochastic properties of the sample range are investigated in detail. Applications in reliability theory are highlighted. The relative dependence between extreme order statistics will be investigated in Chapter 4. In particular, the open problem discussed in Dolati, et al. [18] is solved in this Chapter. In the last Chapter, convolutions of random variables from heterogeneous samples will be investigated. Convolutions have been widely used in many areas to model many practical situations. For example, in reliability theory, it arises as the lifetime of a redundant standby system; in queuing theory, it is used to model the total service time by an agent in a system; in insurance, it is used to model total claims on a number of policies in the individual risk model. I will compare the dispersion and skewness properties of convolutions of different heterogeneous

In our article [3] we have found a gap in the middle of the proof of Theorem 3.2. Therefore, we do not know whether Theorem 3.2 is true for the reverse hazard rate order. However, we could prove the following weaker result for the stochastic order.