Projection Mean–Variance Bounds on Expectations of kth Record Values from Restricted Families

We present sharp mean–variance bounds for expectations of kth record values based on distributions coming from restricted families of distributions. These families are defined in terms of convex or star ordering with respect to generalized Pareto distribution. The bounds for expectations of kth record values from DD, DFR, DDA, and DFRA families are special cases of our results. The bounds are derived by application of the projection method.     

SHARP BOUNDS FOR EXPECTATIONS OF kTH RECORD INCREMENTS

This paper presents sharp bounds for expectations of non‐adjacent increments of kth record statistics, measured in various scale units, for a sequence of independent identically distributed random variables with continuous cumulative distribution function. The results for kth record spacings are considered as special cases. The paper also characterizes probability distributions for which the bounds are attained.     

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.     

Lower bounds on the expectations of upper record values

For infinite sequences of independent random variables with identical continuous distributions, we establish optimal lower bounds on the deviations of the expectations of record values from population means in units generated by the central absolute moments of various orders. The bounds are non-negative for the classic record values, and non-positive for the other kth records with k?2. We also provide analogous bounds for the record increments.     

Extensions of Samuelson's Inequality

In this note we obtain upper and lower bounds for the kth largest number in a set of real numbers in terms of their mean and standard deviation. For each inequality necessary and sufficient conditions for equality are given.     

RECURRENCE RELATIONS FOR SINGLE AND PRODUCT MOMENTS OF GENERALIZED ORDER STATISTICS FROM PARETO,GENERALIZED PARETO,AND BURR DISTRIBUTIONS

We give recurrence relations for single and product moments of generalized order statistics under the concept of Kamps from Pareto, generalized Pareto and Burr distributions. The results include as particular cases the above relations for moments of k–th record values.     

Maximum variance of order statistics from symmetric populations revisited

We consider i.i.d. samples of size n with symmetric non-degenerate parent distributions and finite variances. Papadatos [A note on maximum variance of order statistics from symmetric populations, Ann. Inst. Statist. Math. 48 (1997), pp. 117–121] proved that the maximal variance of each non-extreme order statistic, expressed in the population variance units, is attained in a one-parametric family of symmetric two- and three-point distributions. The parameters of the extreme variance distributions coincide with the arguments maximizing some polynomials of degree 2n?1 over a finite interval. The bounds for variances are equal to the maximal values of the polynomials. We present a more precise solution to the problem by applying the variation diminishing property of Bernstein polynomials.     

On characterization results based on the number of observations near the k-records

In this paper, using the completeness properties of the sequence of functions {h_n(x)=(?log x)ⁿ, 0<x<1, n≥1}, some characterization results are established. The results are based on the number of observations near the k-records. It is shown that the equality of the moment of the appropriate subsequence of the number of observations near to upper and lower k-records is a characteristic property of symmetric distributions. Since ordinary record values are contained in the k-records, the results hold for usual records as a particular case.     

Estimation of parameters of bivariate normal distribution using concomitants of record values

In this paper, we discuss the concomitants of record values arising from the well-known bivariate normal distribution BVND(μ₁, μ₂,σ₁,σ₂, ρ). We have obtained the best linear unbiased estimators of μ₂ and σ₂ when ρ is known and derived some unbiased linear estimators of ρ when μ₂ and σ₂ are known, based on the concomitants of first n record values. The variances of these estimators have been obtained.     

Comparing the Fisher information in record data and random observations

We compare the Fisher information (FI) contained in the firstn record values and record times with the FI inn i. i. d. observations. General results are established for exponential family and Weibull type setups, and a summary table is provided listing several common distributions. We show that the FI in record data improves notably once the record times are included, often changing from being less to being equal or greater than the FI in a random sample of the same size. The behavior in the Weibull case is surprising. There it depends onn, whether the record or the i.i. d. observations have more FI. We propose new estimators based on record data. The results may be of interest in some life testing situations. Supported in part by Fondo Nacional de Desarrollo Cientifico y Tecnologico (FONDECYT) grant # 1010222 of Chile.     

On k-th record times,record values and their moments

We give new formula for moments of k-th record values in terms of Stirling numbers of the first kind. In particular, the formulae allow to derive the explicit formulae for moments of k-th lower record values from exponential distribution which have not been known yet. Moreover, some interesting identities involving harmonic numbers are also obtained as corollaries to presented results.     

Nonparametric confidence intervals for quantiles and quantile intervals from inversely sampled record breaking data in a multi-sample plan

There are a number of situations in which an observation is retained only if it is a record value, which include studies in industrial quality control experiments, destructive stress testing, meteorology, hydrology, seismology, athletic events and mining. When the number of records is fixed in advance, the data are referred to as inversely sampled record-breaking data. In this paper, we study the problems of constructing the nonparametric confidence intervals for quantiles and quantile intervals of the parent distribution based on record data. For a single record-breaking sample, the confidence coefficients of the confidence intervals for the pth quantile cannot exceed p and 1?p, on the basis of upper and lower records, respectively; hence, replication is required. So, we develop the procedure based on k independent record-breaking samples. Various cases have been studied and in each case, the optimal k and the exact nonparametric confidence intervals are obtained, and exact expressions for the confidence coefficients of these confidence intervals are derived. Finally, the results are illustrated by numerical computations.     

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.     

On tail behaviour of kth upper order statistics under fixed and random sample sizes via tail equivalence

For a fixed positive integer k, limit laws of linearly normalized kth upper order statistics are well known. In this article, a comprehensive study of tail behaviours of limit laws of normalized kth upper order statistics under fixed and random sample sizes is carried out using tail equivalence which leads to some interesting tail behaviours of the limit laws. These lead to definitive answers about their max domains of attraction. Stochastic ordering properties of the limit laws are also studied. The results obtained are not dependent on linear norming and apply to power norming as well and generalize some results already available in the literature. And the proofs given here are elementary.     

Measures of inaccuracy in record values

In this paper , we consider a measure of inaccuracy between distributions of the nth record value and parent random variable. We also propose the measure of residual inaccuracy of record values and study characterization results of dynamic cumulative residual inaccuracy measure. We discuss some properties of the proposed measures.     

SELECTING THE t BEST POPULATIONS — SCALE PARAMETER CASE

Selection from k independent populations of the t (< k) populations with the smallest scale parameters has been considered under the Indifference Zone approach by Bechhofer k Sobel (1954). The same problem has been considered under the Subset Selection approach by Gupta & Sobel (1962a) for the normal variances case and by Carroll, Gupta & Huang (1975) for the more general case of stochastically increasing distributions. This paper uses the Subset Selection approach to place confidence bounds on the probability of selecting all “good” populations, or only “good” populations, for the Case of scale parameters, where a “good” population is defined to have one of the t smallest scale parameters. This is an extension of the location parameter results obtained by Bofinger & Mengersen (1986). Special results are obtained for the case of selecting normal populations based on variances and the necessary tables are presented.     

Testing for equivalence of variances using Hartley's ratio

Hartley's test for homogeneity of k normal‐distribution variances is based on the ratio between the maximum sample variance and the minimum sample variance. In this paper, the author uses the same statistic to test for equivalence of k variances. Equivalence is defined in terms of the ratio between the maximum and minimum population variances, and one concludes equivalence when Hartley's ratio is small. Exact critical values for this test are obtained by using an integral expression for the power function and some theoretical results about the power function. These exact critical values are available both when sample sizes are equal and when sample sizes are unequal. One related result in the paper is that Hartley's test for homogeneity of variances is no longer unbiased when the sample sizes are unequal. The Canadian Journal of Statistics 38: 647–664; 2010 © 2010 Statistical Society of Canada     

Single-stage sampling procedure of the t best populations under heteroscedasticity

Given k( ? 3) independent normal populations with unknown means and unknown and unequal variances, a single-stage sampling procedure to select the best t out of k populations is proposed and the procedure is completely independent of the unknown means and the unknown variances. For various combinations of k and probability requirement, tables of procedure parameters are provided for practitioners.