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In reliability theory, the kth order statistic represents the lifetime of a (n - k + 1) -out-of-n system. In particular, the smallest (largest) order statistics denotes the lifetime of a series (parallel) system that consist of n components; and the second smallest (second largest) order statistics is the lifetime of a (n - 1) -out-of-n (2-out-of-n) system. In this paper, sufficient conditions are provided to compare the smallest and the second smallest (largest and second largest) order statistics of dependent and heterogeneous random variables having the additive hazard model with the Archimedean copula in the sense of usual stochastic order and hazard rate order. Further, we compare the smallest order statistics of two sets of independent and heterogeneous random variables having the additive hazard model in the sense of dispersive order. Finally, our theoretical findings are evaluated by some numerical examples and counterexamples. [ABSTRACT FROM AUTHOR]

The problem of optimal estimation of location and scale parameters of absolutely continuous distributions, by means of two-dimensional confidence regions based on L-statistics, is considered. The case, when the sample size is random and tends to infinity, is studied. The paper can be considered as a supplement to Zaigraev and Alama-Bućko (Metrika 81:283–305, 2018) in case of samples of random size. [ABSTRACT FROM AUTHOR]

In this paper, we study properties of a series–parallel system. The component lifetimes may be dependent and non-identically distributed (DNID) discrete random variables. We consider the number of failed components upon system failure. We derive the probability mass function and the expected value of this quantity. In addition, we find the conditional probabilities corresponding to this variate given some partial information about the system failure. We also provide a numerical example to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider two k-out-of-n systems comprising heterogeneous dependent components under random shocks, with an Archimedean copula. We then provide sufficient conditions on the distributions of components' lifetimes and the generator of the Archimedean copula and on the random shocks for comparing the lifetimes of two systems with respect to the usual stochastic order. Finally, we present some examples to illustrate the established results. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to study how to predict the future failure times in a sample from the early failures (type II censored data). We consider both the case of independent and dependent lifetimes. In both cases we assume identically distributed random variables. To predict the future failures we use quantile regression techniques that also provide prediction regions for them. Some illustrative examples show how to apply the theoretical results to simulated and real data sets. [ABSTRACT FROM AUTHOR]

In this paper, we consider a coherent system composed of components whose lifetimes are independent and identically discretely distributed random variables. We study several aging and stochastic properties of the conditional residual lifetime of the system under the condition that some of its components have failed by time t. Moreover, we compare the conditional residual lifetimes of two coherent systems by using various stochastic orders. [ABSTRACT FROM AUTHOR]

In this article, we compare generalized order statistics in terms of increasing convex, mean residual life, hazard rate and reversed hazard rate orderings and establish some general results, including resolving an open problem. Some counterexamples are also provided for the cases wherein the results are not valid in general. Finally, some new applications of these results are demonstrated for sequential systems, k-record values and Pfeifer's records. [ABSTRACT FROM AUTHOR]

In the literature, the sharp positive upper mean-variance bounds on the expectations of order statistics based on independent identically distributed random variables with the decreasing and increasing failure rates, have been recently presented. In this paper we determine analogous evaluations in the dual cases when the parent distributions have monotone reversed failure rates. [ABSTRACT FROM AUTHOR]

In life-testing experiments, it is often of interest to predict unobserved future failure times based on observed early failure times. A point best linear unbiased predictor (BLUP) has been developed in this context by Kaminsky and Nelson (J Am Stat Assoc 70:145–150, 1975). In this article, we develop joint BLUPs of two future failure times based on early failure times by minimizing the determinant of the variance–covariance matrix of the predictors. The advantage of applying joint prediction is demonstrated by using two real data sets. The non-existence of joint BLUPs in certain setups is also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we study the number of failed components of a coherent system. We consider the case when the component lifetimes are discrete random variables that may be dependent and non-identically distributed. Firstly, we compute the probability that there are exactly i, i = 0 , ... , n - k , failures in a k-out-of-n system under the condition that it is operating at time t. Next, we extend this result to other coherent systems. In addition, we show that, in the most popular model of independent and identically distributed component lifetimes, the obtained probability corresponds to the respective one derived in the continuous case and existing in the literature. [ABSTRACT FROM AUTHOR]

In this article, we propose geometric vitality function introduced by Nair and Rajesh (IAPQR Trans 25(1):1–8, 2000) for the past lifetime of a random variable. This measure plays a vital role in analysing different characteristics of a system/component when it fails in the interval (0, t). The monotonic behaviour and some ordering properties in terms of the proposed measure were studied under certain conditions. Similar properties of the proposed measure were analysed for order statistics as well. Further, bounds were obtained for the past geometric vitality function of order statistics. Apart from this, characterizations of some lifetime probability distributions with respect to order statistics were also discussed. [ABSTRACT FROM AUTHOR]

This paper is devoted to some ordering results for the largest and the smallest order statistics arising from dependent heterogeneous exponentiated location-scale random observations. We assume that the sets of observations are sharing a common or different Archimedean copula(s). Sufficient conditions for which the usual stochastic order and the reversed hazard rate order between the extreme order statistics hold are derived. Various numerical examples are provided for the illustration of the proposed results. Finally, some applications of the comparison results in engineering reliability and auction theory are presented. [ABSTRACT FROM AUTHOR]

We provide in this paper sufficient conditions for comparing, in terms of the increasing concave order, some income random variables based on linear combinations of order statistics that are relevant in the framework of social welfare. The random variables under study are weighted average incomes of the poorest and, for some particular weights, their expectations are welfare measures whose integral representations are weighted areas underneath Bonferroni curves. [ABSTRACT FROM AUTHOR]

A multi-sample set-up of sequential order statistics from Weibull distribution functions with known scale parameters and a common unknown shape parameter is considered. The respective likelihood equation may have multiple roots even in the single-sample case, which is demonstrated by a simple example and illustrated with a simulation study. Uniqueness of the root of the likelihood equation and of the maximum likelihood estimator is examined with respect to different models of ordered data, sufficient conditions for uniqueness are shown, and the distribution of the number of roots of the likelihood equation is seen to be independent of the unknown shape parameter. [ABSTRACT FROM AUTHOR]

Measure of uncertainty in past lifetime plays an important role in different areas such as information theory, reliability theory, survival analysis, economics, business, forensic science and other related fields. In this paper, we propose a cumulative Tsallis entropy in past lifetime based on quantile function. We obtain different characterizations based on the proposed measure and quantile-based reliability measures. We also study the quantile-based cumulative Tsallis entropy of order statistics in past lifetime. [ABSTRACT FROM AUTHOR]

This paper investigates sufficient conditions for preservation property of the increasing convex order and the increasing concave order under the taking of maximum and minimum of statistically dependent random variables, respectively. As applications, we develop the preservation of NBUC and NBU(2) aging properties respectively under the parallel and series systems of components with statistically dependent lifetimes. Some copulas are presented as illustrations on statistical dependence structure satisfying the sufficient condition as well. [ABSTRACT FROM AUTHOR]

The study of stochastic comparisons of coherent systems with different structures is a relevant topic in reliability theory. Several results have been obtained for specific distributions. The present paper is focused on distribution-free comparisons, that is, orderings which do not depend on the component distributions. Different assumptions for the component lifetimes are considered which lead us to different comparison techniques. Thus, if the components are independent and identically distributed (IID) or exchangeable, the orderings are obtained by using signatures. If they are just ID (homogeneous components), then ordering results for distorted distributions are used. In the general case or in the case of independent (heterogeneous) components, a similar technique based on generalized distorted distributions is applied. In these cases, the ordering results may depend on the copula used to model the dependence between the component lifetimes. Some illustrative examples are included in each case. [ABSTRACT FROM AUTHOR]

The problem of optimal estimation of location and scale parameters of distributions, by means of two-dimensional confidence regions based on L-statistics, is considered. The case, when the sample size tends to infinity, is analyzed. [ABSTRACT FROM AUTHOR]

A distribution function F is a generalized distorted distribution of the distribution functions $$F_1,\ldots ,F_n$$ if $$F=Q(F_1,\ldots ,F_n)$$ for an increasing continuous distortion function Q such that $$Q(0,\ldots ,0)=0$$ and $$Q(1,\ldots ,1)=1$$ . In this paper, necessary and sufficient conditions for the stochastic (ST) and the hazard rate (HR) orderings of generalized distorted distributions are provided when the distributions $$F_1,\ldots ,F_n$$ are ordered. These results are used to obtain distribution-free ordering properties for coherent systems with heterogeneous components. In particular, we determine all the ST and HR orderings for coherent systems with 1-3 independent components. We also compare systems with dependent components. The results on distorted distributions are also used to get comparisons of finite mixtures. [ABSTRACT FROM AUTHOR]

This article is devoted to characterize several ordering properties of the maximum order statistic of heterogenous random variables with an Archimedean copula. Some examples are also included to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

A finite sequence of binary random variables is called a weak exchangeable sequence of order m if the sequence consists of m random vectors such that the elements within each random vector are exchangeable in the usual sense and the different random vectors are dependent. The exact and asymptotic joint distributions of the m-dimensional random vector whose elements include the number of successes in each exchangeable sequence are derived. Potential applications of the concept of weak exchangeability are discussed with illustrative examples. [ABSTRACT FROM AUTHOR]

The study on the inactivity times is useful in evaluating the aging and reliability properties of coherent systems in reliability engineering. In the present paper, we investigate the inactivity time of a coherent system consisting of n i.i.d. components. We drive some mixture representations for the reliability function of conditional inactivity times of coherent systems under two specific conditions on the status of the system components. Some ageing and stochastic properties of the proposed conditional inactivity times are also explored. [ABSTRACT FROM AUTHOR]

The residual lifetime is of significant interest in reliability and survival analysis. In this article, we obtain a mixture representation for the reliability function of the residual lifetime of a coherent system with heterogeneous components in terms of the reliability functions of residual lifetimes of order statistics. Some stochastic comparisons are made on the residual lifetimes of the systems. Some examples are also given to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, we compare the largest order statistics arising from independent heterogeneous Weibull random variables based on the likelihood ratio order. Let $$X_{1},\ldots ,X_{n}$$ be independent Weibull random variables with $$X_{i}$$ having shape parameter $$0<\alpha \le 1$$ and scale parameter $$\lambda _{i}$$ , $$i=1,\ldots ,n$$ , and $$Y_{1},\ldots ,Y_{n}$$ be a random sample of size n from a Weibull distribution with shape parameter $$0<\alpha \le 1$$ and a common scale parameter $$\overline{\lambda }=\frac{1}{n}\sum \nolimits _{i=1}^{n}\lambda _{i}$$ , the arithmetic mean of $$\lambda _{i}^{'}s$$ . Let $$X_{n:n}$$ and $$Y_{n:n}$$ denote the corresponding largest order statistics, respectively. We then prove that $$X_{n:n}$$ is stochastically larger than $$Y_{n:n}$$ in terms of the likelihood ratio order, and provide numerical examples to illustrate the results established here. [ABSTRACT FROM AUTHOR]

Two types of records in multivariate sequences are considered in this paper. According to the first definition, a multivariate observation is accepted as a record if it is not dominated in at least one of the coordinates of previous record and the first observation is a record. Some basic straightforward results concerning the distributions of record times and records according to this definition are given. The development of distribution theory for these types of record and also providing examples with available analytical results still involves challenging unsolved problems. Second, we consider records of bivariate sequences according to conditionally N-ordering, introduced in Bairamov (J Multivar Anal 97:797-809, 2006). The joint distributions of record times and distributions of record values are derived. Some examples, with particular underlying distributions demonstrating the availability of obtained formulae are provided. [ABSTRACT FROM AUTHOR]

In this paper, the robust Bayesian methodology has been developed in the sense of Pitman measure of closeness. To do this, the definition of Pitman posterior closeness, introduced by Ghosh and Sen (Commun Stat Theory Methods 20:3659-3678, 1991) and simultaneous closeness are integrated. First, the $$\varGamma $$ -minimax problem is developed in the sense of Pitman's criterion and the basic results and definitions are provided. Then, several results regarding Pitman $$\varGamma $$ -minimax have been proved. Some examples have been presented to illustrate the application of the findings. Finally, other aspect of robust Bayesian methodology such as: Pitman stable rules and Pitman regret type estimators have been proposed. [ABSTRACT FROM AUTHOR]

We consider i.i.d. random variables $$X_1,\ldots , X_n$$ with a distribution function F preceding the exponential distribution function V in the convex transform order which means that F has an increasing failure rate. We determine sharp upper bounds on the expectations of order statistics and spacings based on $$X_1,\ldots , X_n$$ , expressed in the population standard deviation units. We also specify the distributions for which all these bounds are attained. Finally, we indicate some reliability applications. [ABSTRACT FROM AUTHOR]

Two new tests for exponentiality, of integral- and Kolmogorov-type, are proposed. They are based on a recent characterization and formed using appropriate V-statistics. Their asymptotic properties are examined and their local Bahadur efficiencies against some common alternatives are found. A class of locally optimal alternatives for each test is obtained. The powers of these tests, for some small sample sizes, are compared with different exponentiality tests. [ABSTRACT FROM AUTHOR]

Fisher information (FI) forms the backbone for many parametric inferential procedures and provides a useful metric for the design of experiments. The purpose of this paper is to suggest an easy way to compute the FI in censored samples from an unfolded symmetric distribution and its folded version with minimal computation that involves only the expectations of functions of order statistics from the folded distribution. In particular we obtain expressions for the FI in a single order statistic and in Type-II censored samples from an unfolded distribution and the associated folded distribution. We illustrate our results by computing the FI on the scale parameter in censored samples from a Laplace (double exponential) distribution in terms of the expectations of special functions of order statistics from exponential samples. We discuss the limiting forms and illustrate applications of our results. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the concomitants of order statistics. We study asymptotic properties of the concomitants of largest order statistics and we pay special attention to strong limit results. The strong limit results of this work are derived by applying the Borel-Cantelli lemma and some of its recent generalizations. The theoretical results of this paper are illustrated with examples. [ABSTRACT FROM AUTHOR]

In this paper, we study the number of near minimum-concomitant observations for Progressively Type-II Censored Order Statistics (PCOS). We first define the concomitants of PCOS and the number of near minimum-concomitant observations. We then investigate distributional and asymptotic properties of these random variables. Finally, we propose simulation techniques for generating the concomitants of PCOS. [ABSTRACT FROM AUTHOR]

Assume that $$X_1,\ldots , X_n$$ are i.i.d. random variables with a common distribution function $$F$$ which precedes a fixed distribution function $$W$$ in the convex transform order. In particular, if $$W$$ is either uniform or exponential distribution function, then $$F$$ has increasing density and failure rate, respectively. We present sharp upper bounds on the expectations of single order statistics and spacings based on $$X_1,\ldots , X_n$$ , expressed in terms of the population mean and standard deviation, for the family of all parent distributions preceding $$W$$ in the convex transform order. We also characterize the distributions which attain the bounds, and specify the general results for the distributions with increasing density function. [ABSTRACT FROM AUTHOR]

We consider a problem of characterization of continuous distributions for which linearity of regression of overlapping order statistics, $$\mathbb {E}(X_{i:m}|X_{j:n})=aX_{j:n}+b$$ , $$m\le n$$ , holds. Due to a new representation of conditional expectation $$\mathbb {E}(X_{i:m}|X_{j:n})$$ in terms of conditional expectations $$\mathbb {E}(X_{l:n}|X_{j:n})$$ , $$l=i,\ldots ,n-m+i$$ , we are able to use the already known approach based on the Rao-Shanbhag version of the Cauchy integrated functional equation. However this is possible only if $$j\le i$$ or $$j\ge n-m+i$$ . In the remaining cases the problem essentially is still open. [ABSTRACT FROM AUTHOR]

The representation of the entropy in terms of the hazard function and its extensions have been studied by many authors including Teitler et al. (IEEE Trans Reliab 35:391-395, ). In this paper, we consider a representation of the Kullback-Leibler information of the first $$r$$ order statistics in terms of the relative risk (Park and Shin in Statistics, ), the ratio of hazard functions, and extend it to the progressively Type II censored data. Then we study the change in Kullback-Leibler information of the first $$r$$ order statistics according to $$r$$ and discuss its relation with Fisher information in order statistics. [ABSTRACT FROM AUTHOR]

We study here the distributions of order statistics and linear combinations of order statistics from a multivariate unified skew-elliptical distribution. We show that these distributions can be expressed as mixtures of unified skew-elliptical distributions, and then use these mixture forms to study some distributional properties and moments. [ABSTRACT FROM AUTHOR]

In terms of stochastic orders, the purpose of this paper is to show how the random environment can affect the number of working components of a system with heterogeneous components sharing a common random environment. Applications to a class of semiparametric mixture models, stress-strength model and warm standby system are presented. [ABSTRACT FROM AUTHOR]

In the context of life-testing, progressive censoring has been studied extensively. But, all the results have been developed under the key assumption that the units under test are independently distributed. In this paper, we consider progressively Type-II censored order statistics (PCOS-II) arising from dependent units that are jointly distributed according to an Archimedean copula. Density and distribution functions of dependent general PCOS-II (GPCOS-II) are derived under this set-up. These results include those in Kamps and Cramer (Statistics 35:269–280, 2001 ) as special cases. Some bounds for the mean of PCOS-II from dependent data are then established. Finally, through an example, a special case of PCOS-II from $$N$$ dependent components is illustrated. [ABSTRACT FROM AUTHOR]

In this note, we consider a coherent system with the property that, upon failure of the system, some of its components remain unfailed in the system. Under this condition, we study the residual lifetime of the live components of the system. Signature based mixture representation of the joint and marginal reliability functions of the live components are obtained. Various stochastic and aging properties of the residual lifetime of such components are investigated. Some characterization results on exponential distributions are also provided. [ABSTRACT FROM AUTHOR]

Given a large sample from a location-scale population we estimate the unknown parameters by means of confidence regions constructed on the basis of two order statistics. The problem of the best choice of those statistics to obtain good estimates, as $$n\rightarrow \infty ,$$ is considered. [ABSTRACT FROM AUTHOR]

In the two-sample prediction problem, record values from the present sample may be used as predictors of order statistics from a future sample. In this paper, we investigate the nearness of record statistics (upper and lower) to order statistics from a location-scale family of distributions in the sense of Pitman closeness and discuss the corresponding monotonicity properties. We then determine the closest record value to a specific order statistic from a future sample. Even though in general it depends on the parent distribution, exact and explicit expressions are derived for the required probabilities in the case of exponential and uniform distributions, and some computational results are presented as well. Finally, we consider the mean squared error criterion and examine the corresponding results in the exponential case. [ABSTRACT FROM AUTHOR]

In this work the ranked set sampling technique has been applied to estimate the scale parameter $$\alpha $$ of a log-logistic distribution under a situation where the units in a sample can be ordered by judgement method without any error. We have evaluated the Fisher information contained in the order statistics arising from this distribution and observed that median of a random sample contains the maximum information about the parameter $$\alpha $$. Accordingly we have used median ranked set sampling to estimate $$\alpha $$. We have further carried out the multistage median ranked set sampling to estimate $$\alpha $$ with improved precision. Suppose it is not possible to rank the units in a sample according to judgement method without error but the units can be ordered based on an auxiliary variable $$Z$$ such that $$(X, Z)$$ has a Morgenstern type bivariate log-logistic distribution (MTBLLD). In such a situation we have derived the Fisher information contained in the concomitant of rth order statistic of a random sample of size $$n$$ from MTBLLD and identified those concomitants among others which possess largest amount of Fisher information and defined an unbalanced ranked set sampling utilizing those units in the sample and thereby proposed an estimator of $$\alpha $$ using the measurements made on those units in this ranked set sample. [ABSTRACT FROM AUTHOR]

We consider the mixed systems composed of a fixed number of components whose lifetimes are i.i.d. with a known distribution which has a positive and finite variance. We show that a certain of the k-out-of- n systems has the minimal lifetime variance, and the maximal one is attained by a mixture of series and parallel systems. The number of the k-out-of- n system, and the probability weights of the mixture depend on the first two moments of order statistics of the parent distribution of the component lifetimes. We also show methods of calculating extreme system lifetime variances under various restrictions on the system lifetime expectations, and vice versa. [ABSTRACT FROM AUTHOR]

In the present study we extend and unify some existing results in the literature on characterization of the generalized Pareto distributions based on generalized order statistics. [ABSTRACT FROM AUTHOR]

We consider a ( n − k + 1)-out-of- n system with independent and nonidentical components. Under the condition that at time t the system has failed we study the past lifetime of the components of the system. The mean past lifetime of the components is defined and some of its properties are investigated. Stochastic comparisons are also made between the past lifetime of different systems. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the statistical inference of the lifetime distribution of components based on observing the system lifetimes when the system structure is known. A general proportional hazard rate model for the lifetime of the components is considered, which includes some commonly used lifetime distributions. Different estimation methods-method of moments, maximum likelihood method and least squares method-for the proportionality parameter are discussed. The conditions for existence and uniqueness of method of moments and maximum likelihood estimators are presented. Then, we focus on a special case when the lifetime distributions of the components are exponential. Computational formulas for point and interval estimations of the unknown mean lifetime of the components are provided. A Monte Carlo simulation study is used to compare the performance of these estimation methods and recommendations are made based on these results. Finally, an example is provided to illustrate the methods proposed in this paper. [ABSTRACT FROM AUTHOR]

Consider an i.i.d. sample $${X^*_{1},X^*_{2},\ldots,X^*_{n}}$$ from a location-scale family, and assume that the only available observations consist of the partial maxima (or minima) sequence, $${X^*_{1:1},X^*_{2:2},\ldots,X^*_{n:n}}$$, where $${X^*_{j:j}=\max\{ X^*_1, \ldots,X^*_j \}}$$. This kind of truncation appears in several circumstances, including best performances in athletics events. In the case of partial maxima, the form of the BLUEs (best linear unbiased estimators) is quite similar to the form of the well-known Lloyd's (in Biometrica 39:88-95, 1952) BLUEs, based on (the sufficient sample of) order statistics, but, in contrast to the classical case, their consistency is no longer obvious. The present paper is mainly concerned with the scale parameter, showing that the variance of the partial maxima BLUE is at most of order O(1/ log n), for a wide class of distributions. [ABSTRACT FROM AUTHOR]

Let X, . . . , X be independent exponential random variables with respective hazard rates λ, . . . , λ, and Y, . . . , Y be independent and identically distributed random variables from an exponential distribution with hazard rate λ. Then, we prove that X, the second order statistic from X, . . . , X, is larger than Y, the second order statistic from Y, . . . , Y, in terms of the dispersive order if and only if We also show that X is smaller than Y in terms of the dispersive order if and only if Moreover, we extend the above two results to the proportional hazard rates model. These two results established here form nice extensions of the corresponding results on hazard rate, likelihood ratio, and MRL orderings established recently by Pǎltǎnea (J Stat Plan Inference 138:1993-1997, 2008), Zhao et al. (J Multivar Anal 100:952-962, 2009), and Zhao and Balakrishnan (J Stat Plan Inference 139:3027-3037, 2009), respectively. [ABSTRACT FROM AUTHOR]

Some results on the relationships between distributions of order statistics and of spacings are presented. These results are then used to establish a characterization of the uniform distribution extending some existing results in this direction. [ABSTRACT FROM AUTHOR]

In the present paper, we consider a ( n − k + 1)-out-of- n system with identical components where it is assumed that the lifetimes of the components are independent and have a common distribution function F. We assume that the system fails at time t or sometime before t, t > 0. Under these conditions, we are interested in the study of the mean time elapsed since the failure of the components. We call this as the mean past lifetime (MPL) of the components at the system level. Several properties of the MPL are studied. It is proved that the relation between the proposed MPL and the underlying distribution is one-to-one. We have shown that when the components of the system have decreasing reversed hazard then the MPL of the system is increasing with respect to time. Some examples are also provided. [ABSTRACT FROM AUTHOR]

It is shown that if ( X1, X2, . . . , X n) is a random vector with a logconcave (logconvex) joint reliability function, then X P = min i∈ P X i has increasing (decreasing) hazard rate. Analogously, it is shown that if ( X1, X2, . . . , X n) has a logconcave (logconvex) joint distribution function, then X P = max i∈ P X i has decreasing (increasing) reversed hazard rate. If the random vector is absolutely continuous with a logconcave density function, then it has a logconcave reliability and distribution functions and hence we obtain a result given by Hu and Li (Metrika 65:325–330, 2007). It is also shown that if ( X1, X2, . . . , X n) has an exchangeable logconcave density function then both X P and X P have increasing likelihood ratio. [ABSTRACT FROM AUTHOR]