Expectations Of Order Statistics Research Articles

The quantile-based flattened logistic distribution: Some properties and applications

In this paper, the quantile-based flattened logistic distribution has been studied. Some classical and quantile-based properties of the distribution have been obtained. Closed form expressions of L-moments, L-moment ratios and expectation of order statistics of the distribution have been obtained. A quantile-based analysis concerning the method of matching L-moments estimation is employed to estimate the parameters of the proposed model. We further derive the asymptotic variance–covariance matrix of the matching L-Moments estimators of the proposed model. Finally, we apply the proposed model to simulated as well as two real life datasets and compare the fit with the logistic distribution.

Refined solution to upper-bound problem for the expectations of order statistics from decreasing density on the average distributions

ABSTRACTRychlik [Metrika 77, 539–557, 2014] described sharp upper negative bounds for the expectations of low-rank order statistics, centered about the population mean and measured in the mean absolute deviation from the mean units, for the i.i.d. sequences with common distribution possessing decreasing density function on the average. The bounds coincide with the negatives of maximal values of complicated functions on the unit interval. Here, we provide more precise solutions to the maximization problems.

Inequalities for order statistics of samples from distributions with monotone hazard rate

A number of inequalities relating the mathematical expectations of order statistics or functions of order statistics in the case where the sample is taken from a distribution with monotone hazard rate function are presented. These inequalities can be applied in reliability theory and other areas of probability theory and mathematical statistics. In particular, in the paper, they are used to obtain new characterizations of the exponential distribution.

Optimal bounds on expectations of order statistics and spacings from nonparametric families of distributions generated by convex transform order

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.

Non-positive upper bounds on expectations of small order statistics from DDA and DFRA populations

Rychlik [Appl Math (Warsaw) 29:15–32, 2002] presented positive sharp upper bounds on the expectations of order statistics with sufficiently large ranks, based on i.i.d. samples from the decreasing density and failure rate populations (DDA and DFRA, for short). They were expressed in terms of the population mean and standard deviation. Here we provide respective non-positive upper tight evaluations for expected small order statistics centered about the population mean, measured in various scale units.

Characterization of Mixtures of Weibull and Pareto Distributions Based on Conditional Expectation of Order Statistics

The mixture of Weibull random variables X 1 and X 2 are identified in terms of relations between the conditional expectation of given X 1: 2 (or ( given X 1: 2, ∀ k ≤ r) and hazard rate function of the distribution, where X 1: 2 and X 2: 2 denote the corresponding order statistics, r is a positive integer and α > 0. In addition, by using the natural logarithm transformation, we also characterize and mixture of Pareto distributions based on the conditional expectation of order statistics. Furthermore, when the sample size is n, the above results are also valid, and we also give an application to Multi-Hit models of carcinogenesis (Parallel Systems). Finally, we also characterize mixture of Weibull and Pareto distributions based on the conditional expectation of upper record values.

On the Order Statistics of Standard Normal-Based Power Method Distributions

This paper derives a procedure for determining the expectations of order statistics associated with the standard normal distribution () and its powers of order three and five ( and ). The procedure is demonstrated for sample sizes of . It is shown that and have expectations of order statistics that are functions of the expectations for and can be expressed in terms of explicit elementary functions for sample sizes of . For sample sizes of the expectations of the order statistics for , , and only require a single remainder term.

Tests for Stochastic Orders and Mean Order Statistics

The aim of this article is to emphasize the fact, not observed previously in the literature, that many discrepancy measures used in tests related to different stochastic orders can be expressed as expectations of order statistics. In this way, we provide a new meaning to the corresponding test statistics which allows us to understand better, and potentially improve, the testing procedures. As illustration, we consider tests to detect overdispersion with respect to a specific probability model. In this setting, a test for the Weibull distribution is discussed in detail.

Least Error Sample Distribution Function

Johannes Kepler University of Linz, Austria The empirical distribution function (ecdf) is unbiased in the usual sense, but shows certain order bias. Pyke suggested discrete ecdf using expectations of order statistics. Piecewise constant optimal ecdf saves 200%/N of sample size N. Results are compared with linear interpolation for U(0, 1), which require up to sixfold shorter samples at the same accuracy. Key words: Unbiased, order statistics, approximation, optimal. Introduction Natural sciences search for regularities in the chaos of real world events at different levels of complexity. As a rule, the regularities become apparent after statistical analysis of noisy data. This defines the fundamental role of statistical science, which collects causally connected facts for subsequent quantitative analysis. There are two kinds of probabilistic interface between statistical analysis and empirical observations. In differential form this corresponds to histograms, and in integral form to the so-called sample distribution function or empirical distribution function (edf or ecdf in Matlab notation), c.f. Pugachev (1984), Feller (1971), Press, et al. (1992), Abramowitz & Stegun (1970), Cramer (1971), Gibbons & Chakraborti (2003). If histogram bins contain sufficiently big numbers of points, the usual concept of ecdf is more or less satisfactory. The focus of this paper is on short samples, where a histogram approach is not possible and an optimal integral approach is welcome. Consider i.i.d. sample X with N elements, numbered according to their appearance on the x-axis statistics, Gibbons & Chakraborti (2003). In X= [X

Characterization of the mixtures of Rayleigh distributions by conditional expectation of order statistics

The mixture of Rayleigh random variables X 1and X 2 are identified in terms of relations between the conditional expectation of \({\left( {X_{2:2}^2 -X_{1:2}^2}\right)^{r}}\) given X 1:2 (or \({X_{2:2}^{2k}}\) given \({X_{1:2},\forall k\leq r)}\) and hazard rate function of the distribution, where X 1:2 and X 2:2 denote the corresponding order statistics, r is a positive integer. In addition, we also mention some related theorems to characterize the mixtures of Rayleigh distributions. Finally, we also give an application to Multi-Hit models of carcinogenesis (Parallel Systems) and a simulated example is used to illustrate our results.

Non-positive upper bounds on expectations of low rank order statistics from DFR populations

Danielak (Sharp upper mean-variance bounds on trimmed means from restricled families, Statistics 37 (2003), pp. 305–324) established strictly positive optimal upper mean-variance bounds on the expectations of order statistics with relatively large ranks coming from decreasing failure rate populations. She also proved that the respective evaluations of low rank order statistics cannot be positive. Here, we show that the zero bounds on the deviation of the expected small order statistic from the population mean expressed in the scale units based on the pth absolute central moments with p>1 cannot be improved, and determine strictly negative sharp bounds in terms of the mean absolute deviation units.

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.

Characterization of the Mixtures of Gompertz Distributions by Conditional Expectation of Order Statistics

Characterization of the Mixtures of Gompertz Distributions by Conditional Expectation of Order Statistics

Characterization of discrete mixtures of exponential distribution based on conditional expectation of order statistics

In a random sample of size two, we use the relation between the conditional expectation of order statistics and the hazard rate function of the distribution to characterize the discrete mixtures of exponential distribution. In addition, we also n ention some related theorems to characterize the discrete mixtures of exponential distribution. Moreover, when the sample size is n, the above results are also valid. Finally, we give a relative result of the different formula for Nassar [14] and an application to parallel system in engineering practice.

Characterization of discrete populations through conditional expectations of order statistics

In this paper we give some properties of the expected values of any order statistic when one of its adjacent order statistics is known (order mean function) from a sequence of sizen of independent and identically distributed random variables with discrete distribution. Furthermore, we obtain the explicit expressions of the distribution from these order mean functions, and finally, we show the necessary and sufficient conditions for any real function to be an order mean function. We also add some examples of characterization of discrete distributions from the order mean functions.

On distributions of random closed sets and expected convex hulls

The problem posed by Vitale (1987, Acta Appl. Math. 9) on determination of random sets distributions by expectations of order statistics is discussed. It is proved that the distribution of a random convex closed set can be determined by means of the nested sequence of the expected convex hulls of support functions, defined by independent copies of the random set in question.

Bounds on expectations of order statistics via extremal dependences

Using the concept of r-extremal dependence, which generalizes Lai and Robbins (1976) maximal dependence, we propose alternative proofs and some new results concerning expectations of order statistics with any rank, from possibly dependent variates. In particular, new, distribution-free and tight bounds are given for the expectations of order statistics from i.d. variates whose common distribution is symmetrical. Sharp approximations of the tight bounds are also given for the standard normal distribution.

Characterization of Mixtures of Geometric Distributions by Means of a Predictor and Expectations of Order Statistics

Characterization of Mixtures of Geometric Distributions by Means of a Predictor and Expectations of Order Statistics

A note on quantile estimation by the kernel method

Estimators of the expectations of order statistics, suggested by Harrell & Davis, are used in place of the order statistics in a quantile estimator, proposed by Kappenman , to produce a modification of Kappenman's procedure. Simulation studies indicate that the modification generally results in a reduction of mean squared error

Characterizing the geometric distribution using expectations of order statistics

The constancy of the conditional expectation of some appropriate functions of order statistics on some others, is used to characterize the geometric distribution among the discrete distributions.