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.     

Inference Based on Order Statistics from the Generalized Pareto Distribution and Application

ABSTRACT

In this article, we derive exact explicit expressions for the single, double, triple, and quadruple moments of order statistics from the generalized Pareto distribution (GPD). Also, we obtain the best linear unbiased estimates of the location and scale parameters (BLUE's) of the GPD. We then use these results to determine the mean, variance, and coefficients of skewness and kurtosis of certain linear functions of order statistics. These are then utilized to develop approximate confidence intervals for the generalized Pareto parameters using Edgeworth approximation and compare them with those based on Monte Carlo simulations. To show the usefulness of our results, we also present a numerical example. Finally, we give an application to real data.     

Generalized order statistics from two parameter uniform distribution

In this paper some distributional properties of the generalized order statistics from uniform distribution are given. The minimum variance linear unbiased as well best ( in the sense of minimum mean squared error) invariant estimators of the parameters of the two parameter uniform distribution based on the first m generalized order statistics are presented.     

On Distributions of Exceedances Based on Generalized Order Statistics

Distributions of exceedance statistics based on generalized order statistics are obtained for a random threshold model. The ordinary order statistics, progressively Type-II right censored order statistics and record values are considered as special cases. The results obtained in the article imply many results on exceedance statistics for the variety of models of ordered random variables.     

Generalized order statistics from arbitrary distributions and the Markov chain property

The joint and marginal distributions of generalized order statistics based on an arbitrary distribution function are established in terms of the lexicographic distribution function. Furthermore, we show that generalized order statistics and the corresponding number of ties form a two-dimensional Markov chain.     

NEYMAN SMOOTH TESTS FOR THE GENERALIZED PARETO DISTRIBUTION

ABSTRACT

The generalized Pareto distribution (GPD) is commonly used as extreme values's distribution. We present goodness of fit tests for the GPD based on Neyman's smooth tests statistics. The methods of maximum likelihood, moments and probability-weighted moments are used for estimating the GPD's parameters. Simulations are done to study the power of these tests.     

Two-Sample Bayesian Prediction Intervals of Generalized Order Statistics Based on Multiply Type II Censored Data

In this article, two-sample Bayesian prediction intervals of generalized order statistics (GOS) based on multiply Type II censored data are derived. To illustrate these results, the Pareto, Weibull, and Burr-Type XII distributions are used as examples. Finally, a numerical illustration of the sequential order statistics from the Pareto distribution is presented.     

Generalized order statistic processes and Pfeifer records

We introduce a uniform generalized order statistic process. It is a simple Markov process whose initial segment can be identified with a set of uniform generalized order statistics. A standard marginal transformation leads to a generalized order statistic process related to non-uniform generalized order statistics. It is then demonstrated that the nth variable in such a process has the same distribution as an nth Pfeifer record value. This process representation of Pfeifer records facilitates discussion of the possible limit laws for Pfeifer records and, in some cases, of sums thereof. Because of the close relationship between Pfeifer records and generalized order statistics, the results shed some light on the problem of determining the nature of the possible limiting distributions of the largest generalized order statistic.     

On the Characterization of a Mixture of Generalized Power Function Distributions by Conditional Expectation of Order Statistics

In the present article, we give some theorems to characterize the mixture of two generalized power function distributions based on conditional expectation of order statistics.     

Linear Estimators and Predictors Based on Generalized Order Statistics from Generalized Pareto Distributions

Linear estimation and prediction based on several samples of generalized order statistics from generalized Pareto distributions is considered. Representations of best linear unbiased estimators (BLUEs) and best linear equivariant estimators in location-scale families are derived, as well as corresponding optimal linear predictors. Moreover, we study positivity of the linear estimators of the scale parameter. An example illustrates that the BLUE may attain negative values with positive probability in certain situations.     

Multivariate Excess Wealth Ordering of Generalized Order Statistics

The concept of generalized order statistics (GOSs) was introduced as a unified approach to a variety of models of ordered random variables. The purpose of this paper is to investigate conditions on the underlying distribution function and the parameters on which GOSs are based, to establish multivariate excess wealth ordering of GOSs from one sample and two samples, respectively.     

Generalized exponential distribution: Moments of order statistics

In this paper, we consider the generalized exponential distribution (GED) with shape parameter α. We establish several recurrence relations satisfied by the single and the product moments for order statistics from the GED. The relationships can be written in terms of polygamma and hypergeometric functions and used in a simple recursive manner in order to compute the single and the product moments of all order statistics for all sample sizes.     

Generalized elliptical distributions

A family of distributions generated by an operator acting on generalized normal density is introduced. This family contains as particular cases many known distributions, including the generalized normal, generalized t, and generalized gamma distributions. Several mathematical properties of the family (including expansions, characteristic function, moments, cumulants, and order statistics properties) are derived. Estimation procedures are derived too by the method of moments, method of maximum likelihood, and the method of empirical characteristic function. A real data application is presented. Finally, extensions to the multivariate case are outlined.     

Characterizations via regression of generalized order statistics

In this paper, we present some characterizations of distributions based on the regression of generalized order statistics. In the case of adjacent generalized order statistics, the conditional expectation of one generalized order statistic given the other one completely characterizes distributions depending on the type of regression function. In the case of non-adjacent generalized order statistics, the characterization of distributions using conditional expectations becomes more complicated. The results presented in the paper unify and extend some of the existing results involving order statistics and record values.     

A matching prior for extreme quantile estimation of the generalized Pareto distribution

Extreme quantile estimation plays an important role in risk management and environmental statistics among other applications. A popular method is the peaks-over-threshold (POT) model that approximate the distribution of excesses over a high threshold through generalized Pareto distribution (GPD). Motivated by a practical financial risk management problem, we look for an appropriate prior choice for Bayesian estimation of the GPD parameters that results in better quantile estimation. Specifically, we propose a noninformative matching prior for the parameters of a GPD so that a specific quantile of the Bayesian predictive distribution matches the true quantile in the sense of Datta et al. (2000).     

Approximation of the distribution of excesses through a generalized probability-weighted moments method

The POT (peaks-over-threshold) approach consists in using the generalized Pareto distribution (GPD) to approximate the distribution of excesses over a threshold. In this paper, we consider this approximation using a generalized probability-weighted moments (GPWM) method. We study the asymptotic behaviour of our new estimators and also the functional bias of the GPD as an estimate of the distribution function of the excesses. A simulation study is provided in order to appreciate the efficiency of our approach.     

Joint Moment Generating Functions of Nonadjacent Dual Generalized Order Statistics from Reflected Generalized Pareto Distributions

In this article, a class of reflected generalized Pareto distributions (cf. Burkschat et al., 2003 Burkschat , M. , Cramer , E. , Kamps , U. ( 2003 ). Dual generalized order statistics . Metron LXI ( 1 ): 13 – 26 . [Google Scholar]) is considered. Recurrence relations for joint moment generating functions of higher non adjacent dual generalized order statistics based on a random sample drawn from the considered class are derived. Higher joint moments of non adjacent dual generalized order statistics (reversed ordered order statistics and lower k-records as special cases) are obtained. Recurrence relations for single and product moment generating functions and moments of higher non adjacent dual generalized order statistics are derived. Some results of higher moments of non adjacent generalized order statistics from generalized Pareto distributions (cf. Johnson et al., 1995 Johnson , N. L. , Kotz , S. , Balakrishnan , N. ( 1995 ). Continuous Univariate Distributions. , 2nd ed. Vol. 2. New York : Wiley & Sons . [Google Scholar]), are obtained by using a relation connecting higher moments of generalized order statistics and its dual.     

Asymptotic Normality of Extreme Quantile Estimators Based on the Peaks-Over-Threshold Approach

The POT (Peaks-Over-Threshold) approach consists of using the generalized Pareto distribution (GPD) to approximate the distribution of excesses over thresholds. In this article, we establish the asymptotic normality of the well-known extreme quantile estimators based on this POT method, under very general assumptions. As an illustration, from this result, we deduce the asymptotic normality of the POT extreme quantile estimators in the case where the maximum likelihood (ML) or the generalized probability-weighted moments (GPWM) methods are used. Simulations are provided in order to compare the efficiency of these estimators based on ML or GPWM methods with classical ones proposed in the literature.     

Dependence structure of generalized order statistics

In this article, simple expressions for marginal density functions of multiply censored generalized order statistics based on continuous distribution functions are obtained. Moreover, it is shown that generalized order statistics are multivariate totally positive and, thus, associated. This property is applied to show that regressions of generalized order statistics are nondecreasing under weak conditions.     

Prediction Intervals for Future Order Statistics from a Generalized Modified Weibull

Viewing the future order statistics as latent variables at each Gibbs sampling iteration, several Bayesian approaches to predict future order statistics based on type-II censored order statistics, X₍₁₎, X₍₂₎, …, X_(r), of a size n( > r) random sample from a four-parameter generalized modified Weibull (GMW) distribution, are studied. Four parameters of the GMW distribution are first estimated via simulation study. Then various Bayesian approaches, which include the plug-in method, the Monte Carlo method, the Gibbs sampling scheme, and the MCMC procedure, are proposed to develop the prediction intervals of unobserved order statistics. Finally, four type-II censored samples are utilized to investigate the predictions.