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.     

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 moments of dual generalized order statistics from Topp-Leone distribution

In this paper, we have derived exact and explicit expressions for the ratio and inverse moments of dual generalized order statistics from Topp-Leone distribution. This result includes the single and product moments of order statistics and lower records . Further, based on n dual generalized order statistics, we have deduced the expression for Maximum likelihood estimator (MLE) and Uniformly minimum variance unbiased estimator (UMVUE) for the shape parameter of Topp-Leone distribution. Finally, based on order statistics and lower records, a simulation study is being carried out to check the efficiency of these estimators.     

Prediction intervals for ordinary and dual generalized order statistics from two independent sequences

ABSTRACT

Based on the observed dual generalized order statistics drawn from an arbitrary unknown distribution, nonparametric two-sided prediction intervals as well as prediction upper and lower bounds for an ordinary and a dual generalized order statistic from another iid sequence with the same distribution are developed. The prediction intervals for dual generalized order statistics based on the observed ordinary generalized order statistics are also developed. The coverage probabilities of these prediction intervals are exact and free of the parent distribution, F. Finally, numerical computations and real examples of the coverage probabilities are presented for choosing the appropriate limits of the prediction.     

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.     

Calculation of Scores for a Wilcoxon Generalization Applicable to Data Subject to Arbitrary Right Censorship

For data subject to right censoring it is suggested that the Wilcoxon ranking procedure can be generalized by scoring observations according to the expected values of order statistics from the uniform distribution subject to the same right censoring. This parallels the logrank scoring procedure in which scores correspond to the expected values of order statistics from the exponential distribution that have been subject to right censoring. A caveat is given that, in situations where the mechanism of censoring has been affected by treatment, the usual permutational analysis of ranking scores would be inappropriate. But a jackknife approach could be remedial.     

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.     

Characterizations of geometric distribution through progressively Type-II right-censored order statistics

In this paper, we consider characterizations of geometric distribution based on some properties of progressively Type-II right-censored order statistics. Specifically, we establish characterizations through conditional expectation, identical distribution, and independence of functions of progressively Type-II right-censored order statistics. Moreover, extensions of these results to generalized order statistics are also sketched. These generalize the corresponding results known for the case of ordinary order statistics.     

Simultaneous Pitman closeness of progressively type-II right-censored order statistics to population quantiles

In this work, we extend prior results concerning the simultaneous Pitman closeness of order statistics (OS) to population quantiles. By considering progressively type-II right-censored samples, we derive expressions for the simultaneous closeness probabilities of the progressively censored OS to population quantiles. Explicit expressions are deduced for the cases when the underlying distribution has bounded and unbounded supports. Illustrations are provided for the cases of exponential, uniform and normal distributions for various progressive type-II right-censoring schemes and different quantiles. Finally, an extension to the case of generalized OS is outlined.     

Entropy-based goodness-of-fit tests for the Pareto I distribution

In this article, having observed the generalized order statistics in a sample, we construct a test for the hypothesis that the underlying distribution is the Pareto I distribution. The Shannon entropy of generalized order statistics is used to test the null hypothesis.     

On distributions Of generalized order statistics

In a wide subclass of generalized order statistics, representations of marginal density and distribution functions are developed. The results are applied to obtain several relations, such as recurrence relations, and explicit expressions for the moments of generalized order statistics from Pareto, power function and Weibull distributions Moreover, characterizations of exponential distributions are shown by means of a distributional identity as well as by* an identity of expectations involving a subrange and a corresponding generalized order statistic.     

Characterization based on generalized entropy of order statistics

ABSTRACT

In the present study, several characterizations of order statistics are obtained on the basis of the generalized entropy. Under some conditions, it is shown that the parent distribution can be uniquely determined by equality of generalized entropy of order statistics.     

Point Processes with a Generalized Order Statistic Property

Mixed Poisson processes are characterized by a well-known order statistic property: their occurrence times are distributed like ordinary uniform order statistics, given the state of the process at a certain time. We study a generalization of this property using a generalized model of ordered random variables, including a sequence of real parameters.     

Minimum variance unbiased estimation in a modified power series distribution and some of its applications

In this paper we study the minimum variance unbiased estimation in the modified power series distribution introduced by the author (1974a). Necessary and sufficient conditions for the existence of minimum variance unbiased estimate (MVUE) of the parameter based on sufficient statistics are obtained. These results are, then, applied to obtain MVUE of θ^r (r ≥ 1) for the generalized negative binomial and the decapitated generalized negative binomial distributions (Jain and Consul, 1971). Similar estimates are obtained for the generalized Poisson (Consul and Jain, 1973a) and the generalized logarithmic series distributions (Jain and Gupta, 1973). Several of the well-known results follow trivially from the results obtained here.     

Structure-preserving mappings of uniform order statistics

There are several well-known mappings which transform the first r common order statistics in a sample of size n from a standard uniform distribution to a full vector of dimension r of order statistics in a sample of size r from a uniform distribution. Continuing the results reported in a previous paper by the authors, it is shown that transformations of these types do not lead to order statistics from an i.i.d. sample of random variables, in general, when being applied to order statistics from non-uniform distributions. By accepting the loss of one dimension, a structure-preserving transformation exists for power function distributions.     

Estimation of parameters of a pareto distribution by generalized order statistics

In this paper we address the problem of estimating the parameters of Pareto II distribution based on generalized order statistics. The estimators based on order statistics and record values are shown to be special cases of these estimators.     

On small sample properties of the generalized signed rank and generalized sign tests

The generalized signed rank (GSR) and generalized sign (GS) tests were recently proposed for matched pair studies with censored observations (Woolson and Lechenbruch, 1980). The results provided in that paper were asymptotic, and no indicatin of small sample behavior was given. In this paper we report on simulation studied of these statistics for a variety of distributions. We find that the GSR is more powerful than the GS, and that censoring does not affect power greatly. In the original paper, we assumed each member of the pair has the same censoring time. We consider a variant of this in which each member of the pair has a censoring time chosen from a uniform distribution, and the minimum of these times is selected as the censoring time for the pair. It is found that the power of the test is slightly reduced because the number of doubly censored pairs is increased.     

Concomitants of generalized order statistics from Farlie–Gumbel–Morgenstern distributions

Generalized order statistics constitute a unified model for ordered random variables that includes order statistics and record values among others. Here, we consider concomitants of generalized order statistics for the Farlie–Gumbel–Morgenstern bivariate distributions and study recurrence relations between their moments. We derive the joint distribution of concomitants of two generalized order statistics and obtain their product moments. Application of these results is seen in establishing some well known results given separately for order statistics and record values and obtaining some new results.     

Further Results on Order Statistics Generated by Two Simulation Methods

Uniform order statistics generated by two simulation methods are compared by means of Pitman’s measure of closeness. This measure, as a probability, is shown to be asymptotically 1/2. Some results are also established for fixed points of the cumulative distribution function (CDF) for a uniform order statistic. These fixed points are important for calculations involving the joint distribution of these order statistics.     

Recurrence relations for single and product moments of progressively Type-II censored order statistics from a generalized half-logistic distribution with application to inference

In this paper, we establish several recurrence relations for the single and product moments of progressively Type-II right-censored order statistics from a generalized half-logistic distribution. The use of these relations in a systematic recursive manner enables the computation of all the means, variances, and covariances of progressively Type-II right-censored order statistics from the generalized half-logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R ₁, …, R _m ). The results established here generalize the corresponding results for the usual order statistics due to Balakrishnan and Sandhu [Recurrence relations for single and product moments of order statistics from a generalized half-logistic distribution with applications to inference, J. Stat. Comput. Simul. 52 (1995), pp. 385–398.]. The moments so determined are then utilized to derive the best linear unbiased estimators of the scale and location–scale parameters of the generalized half-logistic distribution. The best linear unbiased predictors of censored failure times are discussed briefly. Finally, a numerical example is presented to illustrate the inferential method developed here.