Optimal bounds on l-statistics based on samples drawn with replacement from finite populations

We describe a method of establishing optimal bounds on the expectations of arbitrary linear combinations of order statistics based on iid samples drawn with replacement from finite populations of a fixed size. The bounds are expressed in terms of the population size, mean, central absolute moments, and coefficients of the combination. The bounds are precisely determined for the trimmed means and their differences, and single order statistics and their differences in particular. We also show that with increase in population size, our bounds approach the respective universal ones for arbitrary iid samples.     

Nonparametric prediction of future order statistics

Prediction of censored order statistics from a Type-II censored sample can be done with trivial bounds having perfect confidence. However, given independent samples from the same absolutely continuous distribution, improved bounds can be attained. In this regard, we develop here point prediction based on L-statistics for predicting order statistics (OS) from a future sample as well as for predicting censored OS from a Type-II censored sample. An example is taken to illustrate these ideas, and the limiting case wherein a single independent sample is arbitrarily large is also discussed.     

Recurrence relations mid identities for moments of order statistics,i: arbitrary continuous distribution

In this paper, we review several recurrence relations and identities established for the single and product moments of order statistics from an arbitrary continuous distribution. We point out the interrelationships between many of these recurrence relations. We discuss the results giving the bounds for the number of single and double integrals needed to be evaluated in order to compute the first, second and product moments of order statistics in a sample of size n from an arbitrary continuous distribution, given these moments in samples of sizes n-1 and less. Improvements of these bounds for the case of symmetric continuous distributions are also discussed     

Evaluations of generalized order statistics from bounded populations

We consider generalized order statistics with arbitrary model parameters based on distributions supported on finite intervals. We determine optimal bounds on the expectations of arbitrary linear combinations of centered generalized order statistics gauged in support length scale units. More precise representations of bounds are obtained for single generalized order statistics and respective differences.     

Lower bounds on expectations of positive L-statistics from without-replacement models

We consider samples drawn without replacement from finite populations. We establish optimal lower non-negative and upper non-positive bounds on the expectations of linear combinations of order statistics centered about the population mean in units generated by the population central absolute moments of various orders. We also specify the general results for important examples of sample extremes, Gini mean differences and sample range. The paper completes the results of Papadatos and Rychlik [2004. Bounds on expectations of L-statistics from without replacement samples. J. Statist. Plann. Inference 124, 317–336], where sharp negative lower and positive upper bounds on the expectations of the combinations were presented for the without-replacement samples.     

Lower bounds for confidence coefficients for confidence intervals for finite population quantiles

Assuming stratified simple random sampling, a confidence interval for a finite population quantile may be desired. Using a confidence interval with endpoints given by order statistics from the combined stratified sample, several procedures to obtain lower bounds (and approximations for the lower bounds) for the confidence coefficients are presented. The procedures differ with respect to the amount of prior information assumed about the var-iate values in the finite population, and the extent to which sample data is used to estimate the lower bounds.     

Optimal bounds on the bias of quasimidranges

We derive sharp upper and lower bounds on expectations of sample quasimidranges, that is, arithmetic means of two fixed order statistics of the sample, expressed in various scale units. They can be considered as the bounds on the bias of estimating unknown mean of the parent distribution by the above statistics. While determining the appropriate projection, we consider two new auxiliary functions whose usage provides analytical conditions determining the form of corresponding greatest convex minorant. The results are illustrated with numerical examples.     

A Note on the Upper Bound to Variance of the Sample Extreme from a Finite Population

We construct a new upper bound for the variance of the sample minimum and maximum in the case of a simple random sample drawn without replacement. The bound is optimal in the form provided. Similar bounds are shown for the other order statistics.     

Berry-Esseen-type bounds for the kernel estimator of conditional distribution and conditional quantiles

Berry-Esseen bounds of order O(n^−1/2) have been obtained for several classes of statistics. In this paper, the rates of convergence in central limit theorem for conditional empirical functions and conditional sample quantiles based on kernel estimators are studied for both conditional and unconditional distributions.     

Upper and lower bounds for the correlation ratio of order statistics from a sample without replacement

Using properties of the Hahn Polynomials, we give upper and lower bounds for the correlation ratio between order statistics of a sample without replacement from a finite population. The upper bound is attained when the population contains a lattice set. Except for particular cases, the lower bound is not sharp.     

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.     

Bounds on expected generalized order statistics

We establish the upper nonpositive and all the lower bounds on the expectations of generalized order statistics based on a given distribution function with the finite mean and central absolute moment of a fixed order. We also describe the distributions for which the bounds are attained. The methods of deriving the lower nonpositive (upper nonnegative) and lower nonnegative (upper nonpositive) bounds are totally different. The first one, the greatest convex minorant method is the combination of the Moriguti and well-known Hölder inequalities and the latter one is based on the maximization of some norm on the properly chosen convex set. The paper completes the results of Cramer et al. [Evaluations of expected generalized order statistics in various scale units. Appl Math. 2002;29:285–295].     

Sharp bounds on the expectations of L-statistics expressed in the Gini mean difference units

We describe a method of calculating sharp lower and upper bounds on the expectations of arbitrary, properly centered L-statistics expressed in the Gini mean difference units of the original i.i.d. observations. Precise values of bounds are derived for the single-order statistics, their differences, and some examples of L-estimators. We also present the families of discrete distributions which attain the bounds, possibly in the limit.     

Comparison of Records and Maxima

We establish best upper bounds on the expected differences of records and sample maxima, and kth records and kth maxima based on sequences of independent random variables with identical continuous distribution and finite variance. The bounds are expressed in terms of the standard deviation units of the parent distribution. We also provide conditions for attaining the bounds.     

TESTING THE LARGEST OF A SET OF CORRELATION COEFFICIENTS

A previous paper which studied the distribution of the smallest distance between N independent random points on the surface of a sphere is generalised to higher dimensions in order to study the distribution of the largest sample correlation coefficient between a set of independent normally distributed variables. Inclusion-exclusion arguments provide accurate bounds for the tail of this distribution, and by another argument more exact bounds are also found, one of which is an improvement on the result in the previous paper. Bounds are also found for the power of the test against the alternative hypothesis that one only of the population correlation coefficients is non-zero. The test is also shown to be the likelihood ratio test against the latter alternative.     

Multi-sample inference for simple-tree alternatives with ranked-set samples

This paper develops a non‐parametric multi‐sample inference for simple‐tree alternatives for ranked‐set samples. The multi‐sample inference provides simultaneous one‐sample sign confidence intervals for the population medians. The decision rule compares these intervals to achieve the desired type I error. For the specified upper bounds on the experiment‐wise error rates, corresponding individual confidence coefficients are presented. It is shown that the testing procedure is distribution‐free. To achieve the desired confidence coefficients for multi‐sample inference, a nonparametric confidence interval is constructed by interpolating the adjacent order statistics. Interpolation coefficients and coverage probabilities are provided, along with the nominal levels.     

Projection bounds on expectations of generalized order statistics from DD and DDA families

We present sharp upper mean-variance bounds for expectations of generalized order statistics based on distributions coming from restricted families of distributions. Two families are considered: distributions with decreasing density and with density decreasing on the average. The bounds are derived by application of the projection method.     

A NOTE ON MOMENTS OF PROGRESSIVELY TYPE II CENSORED ORDER STATISTICS

ABSTRACT

Upper and lower bounds for moments of progressively Type II censored order statistics in terms of moments of (progressively Type II censored) order statistics are derived. In particular, this yields conditions for the existence of moments of progressively Type II censored order statistics based on an absolutely continuous distribution function.     

Entropy properties of record statistics

Record values can be viewed as order statistics from a sample whose size is determined by the values and the order of occurrence of observations. They are closely connected with the occurrence times of a corresponding non-homogenous Poisson process and reliability theory. In this paper, the information properties of record values are presented based on Shannon information. Several upper and lower bounds for the entropy of record values are obtained. It is shown that, the mutual information between record values is distribution free and is computable using the distribution of the record values of the sequence from the uniform distribution.     

Maximal Expectations of Extreme Order Statistics from Increasing Density and Failure Rate Populations

We determine sharp bounds for the expectations of the first and second and last order statistics for the i.i.d. samples from the increasing density and failure rate populations. These bounds are expressed by means of the population mean and standard deviation. The results are determined by use of the projections of elements of Hilbert spaces onto convex cones.