Evidence-Based Assessment and Application of Prognostic Markers: The Long Way from Single Studies to Meta-Analysis

In recent years, several attempts have been made to characterize the generalized Pareto distributions (GPD) based on the properties of order statistics and record values. In the present article, we give some characterization results on GPD based on order statistics and generalized order statistics. Some characterizations of uniform distribution based on expectation of some functions of order statistics are also given.     

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 on sample extremes of dependent and heterogenous observations

Stochastic comparison on order statistics from heterogeneous-dependent observations has been paid lots of attention recently. This paper devotes to investigating the ordering properties of order statistics from dependent observations. We derive the usual stochastic order for sample minimums and the second smallest order statistic, the dispersive order and the star order for minimums of samples having proportional hazards and Archimedean survival copulas. Similar ordering results are also obtained for maximums and the second largest order statistic of samples having proportional reversed hazards and Archimedean copulas. Several examples illustrating the main results are presented as well.     

Relationships among moments of order statistics in samples from two related outlier models and some applications

Balakrishnan (1987a) has recently shown that the moments of order statistics in samples drawn from a continuous population with pdf f(x) symmetric about zero comprising a single outlier with pdf g(x) also symmetric about zero can be expressed in terms of the moments of order statistics in samples drawn from the population obtained by folding the pdf f(x) at zero and the moments of order statistics in samples drawn from the population obtained by folding the pdf f(x) at zero comprising a single outlier with pdf obtained by folding g(x) at zero. The cumulative round off error involved in the numerical evaluation of the moments of order statistics from the symmetric outlier model, using a table of the moments of order statistics from the folded population and the moments of order statistics from the folded outlier model, has also been studied by Balakrishnan (1987a) and shown to be not serious. Making use of these results we study here the robustness of some estimators of th location and scale parameters of a double exponential distribution.     

Probability of ties and Markov property in discrete order statistics

It is well known that, in the continuous case, the probability that two consecutive order statistics are equal to zero, whereas it is not true when the distribution is discrete. It is, perhaps, for this reason that order statistics from discrete distributions has not been investigated in the literature as much as from a continuous distribution. The main purpose of this paper, therefore, is to obtain the probability of ties when the distribution is discrete. Also it is shown that, in the discrete case, the Markov property does not hold good. However, the order statistics from a geometric distribution forms a Markov chain.     

Recursive computation of the single and product moments of order statistics from the complementary exponential–geometric distribution

The complementary exponential–geometric distribution has been proposed recently as a simple and useful reliability model for analysing lifetime data. For this distribution, some recurrence relations are established for the single and product moments of order statistics. These recurrence relations enable the computation of the means, variances and covariances of all order statistics for all sample sizes in a simple and efficient recursive manner. By using these relations, we have tabulated the means, variances and covariances of order statistics from samples of sizes up to 10 for various values of the shape parameter θ. These values are in turn used to determine the best linear unbiased estimator of the scale parameter β based on complete and Type-II right-censored samples.     

On calculating the fisher information in order statistics

This article gives a simple result for the expression of the Fisher information in order statistics. This result enables us to calculate easily the Fisher information in any set of order statistics whose details have been known to be messy and complicated. We consider here its application in the optimal spacing problem where the exact Fisher information in order statistics has been approximated with the asymptotic information or the reciprocal of the variance of a suitable estimator. This work was supported by Korea Research Foundation Grant(KRF-2000-015-DP0056)     

Alternating sums of sub-diagonal product moments of order statistics

A relation satisfied by the product moments of order statistics from an arbitrary continuous distribution symmetric about the origin has been established in this paper. We have shown that in such a situation, for samples of even size, sum of the sub-diagonal product moments of order statistics with alternating plus and minus signs assumes a compact form.     

On a Characterization of Generalized Pareto Distribution Based on Generalized Order Statistics

In recent years, several attempts have been made to characterize the generalized Pareto distribution (GPD) based on the properties of order statistics and record values. In the present article, we give a characterization result on GPD based on the spacing of generalized order statistics.     

Moments of L-Statistics: A Divided Differences Approach

The derivation of the distributions of linear combinations of order statistics or L-statistics and the computation of their moments has been approached in the literature several ways. In this paper we use the properties of divided differences to obtain expressions for moments of some order statistics that arise as special cases of L-statistics. Expectations of some well-known L-statistics such as the trimmed mean and the winsorised mean for the pareto distribution are computed. The study also undertakes the computation of L-moments that are expectations of certain linear combinations of order statistics. The algorithms have been implemented using some well-known continuous distributions as examples.     

Distribution-free confidence intervals for quantiles in small samples

Estimators for quantiles based on linear combinations of order statistics have been proposed by Harrell and Davis(1982) and kaigh and Lachenbruch (1982). Both estimators have been demonstrated to be at least as efficient for small sample point estimation as an ordinary sample quantile estimator based on one or two order statistics: Distribution-free confidence intervals for quantiles can be constructed using either of the two approaches. By means of a simulation study, these confidence intervals have been compared with several other methods of constructing confidence intervals for quantiles in small samples. For the median, the Kaigh and Lachenbruch method performed fairly well. For other quantiles, no method performed better than the method which uses pairs of order statistics.     

Reliability properties of record values from non-identically distributed random variables

In reliability theory, order statistics and record values are used for statistical modelling. The r-th order statistic in a sample of size n represents the life—length of a (n?r+l)-out-of-n system, and record values are used in shock models. In recent years, reliability properties of order statistics and record values have been investigated. The two models are included in Pfeifer's concept of record values from non-identically distributed random variables. Here, some results on the transmission of distributional properties, such as increasing failure rate, are shown for such records, which contain the results for order statistics and ordinary record values as particular cases.     

A skew logistic distribution as an alternative to the model of van Staden and King

The logistic distribution is a simple distribution possessing many useful properties and has been used extensively for analyzing growth. Recently, van Staden and King proposed a quantile-based skew logistic distribution. In this paper, we introduce an alternative skew logistic distribution. We then establish recurrence relations for the computation of the single and product moments of order statistics from the standard skew logistic distribution by using the moments of order statistics from the standard half logistic distribution. These enable an efficient computation of means, variances and covariances of order statistics from the skew logistic distibution for all sample sizes. The results become useful in determining the best linear unbiased estimators of the location and scale paramters of the skew logistic distribution. Finally, we provide an example to illustrate the usefulness of the developed model and then compare its fit with that provided by the model of van Staden and King.     

History and role of order statistics

This paper starts with an itroduction which gives definitions of order statistics,makes a clear distinction between order statistics and rank statistics, and gives an overview of the problems in which methods based on order statstics are useful.The following sections deal with the history and role of order sta¬tistics in selected areas: central tendency, dispersion, and regression; treatment of outliers and robust estimation; maximum likelihood estimators; best linear unbiased estimators; extreme-values; multiple comparisons and studentized range. The last two sections discuss a basic coverage property and some of its conse¬quences and describe the author's chronological annotated bibliography of order statistics. Finally, there is a list of references.     

Exact nonparametric confidence,prediction and tolerance intervals based on multi-sample Type-II right censored data

Exact nonparametric inference based on ordinary Type-II right censored samples has been extended here to the situation when there are multiple samples with Type-II censoring from a common continuous distribution. It is shown that marginally, the order statistics from the pooled sample are mixtures of the usual order statistics with multivariate hypergeometric weights. Relevant formulas are then derived for the construction of nonparametric confidence intervals for population quantiles, prediction intervals, and tolerance intervals in terms of these pooled order statistics. It is also shown that this pooled-sample approach assists in achieving higher confidence levels when estimating large quantiles as compared to a single Type-II censored sample with same number of observations from a sample of comparable size. We also present some examples to illustrate all the methods of inference developed here.     

On generalized order statistics from generalized pareto distribution

The aim of this paper is to estimate parameters of generalized Pareto distribution based on generalized order statistics. Some non-Bayesian methods, such as MLE, bootstrap and unbiased estimators have been obtained to develop point and interval estimations. Bayesian estimations have also been derived under LSE and LINEX loss functions. To compare the performances of the employed methods, numerical results have been computed. To illustrate dependence and association properties of generalized order statistics, correlation coefficient and some informational measures in closed form have been obtained.     

On the distribution of the maximum of bivariate normal random variables with general means and variances

A great amount of effort has been devoted to achieving exact expressions for moments of order statistics of independent normal random variables, as well as the dependent case with the same correlation coefficients, means and variances. It does not seem as if there are handy formulae for the order statistics of even the simple bivariate normal random variables when the means and variances are allowed to be different. In this paper we give an explicit formula for the Lanl ace-Stielties Transform of the maximum of bivariate normal random variables by which we obtain formulae for the first two moments in the standard way.     

Unbiased Estimation of the Distribution Function of a Two-Parameter Exponential Population Using Order Statistics

In this article, we consider the problem of unbiased estimation of the distribution function of a two-parameter exponential population using order statistics based on a random sample from the population. We give necessary and sufficient conditions for the existence of an unbiased estimator based on an arbitrary set of order statistics and suggest unbiased estimators in some situations where unbiased estimators exist. A few properties of the suggested estimators for some special cases have also been discussed.     

Partial Ordering and Aging Properties of Order Statistics When the Sample Size is Random: A Brief Review

In many practical situations, order statistics arise naturally with random sample size. In this article, we review results on partial orderings and aging properties of such order statistics. The comparison of order statistics for different sample sizes is also discussed here.     

An identity for the joint distribution of order statistics and its applications

It is shown that the joint distribution function of several order statistics can be expressed in terms of the distribution functions of the maximum (or the minimum) order statistics of suitable subsamples. The result is used to derive explicit expressions for the expectations of functions of order statistics from certain families of distributions which include the exponential distribution and the power function distribution. The results generalize earlier work by the authors for a single order statistic.