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Maximal dispersion of order statistics in dependent samples

Tomasz Rychlik

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Abstract

Consider n arbitrarily dependent identically distributed random variables. Rychlik [Extreme variances of order statistics in dependent samples. Statist Probab Lett. 2008;78:1577–1582] proved that the optimal bound for the ratio of variances of the mth order statistic and a single-parent variable is the maximum of n/m and n/(n+1−m). We show here that the evaluation is optimal for many other measures of dispersion of the order statistics and original variable.

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What this paper is about

Consider n arbitrarily dependent identically distributed random variables. Rychlik [Extreme variances of order statistics in dependent samples. Statist Probab Lett. 2008;78:1577–1582] proved that the optimal bound for the ratio of variances of the mth order statistic and a single-parent variable is the maximum of n/m and n/(n+1−m). We show here that the evaluation is optimal for many other measures of dispersion of the order statistics and original variable.

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Available abstract

Consider n arbitrarily dependent identically distributed random variables. Rychlik [Extreme variances of order statistics in dependent samples. Statist Probab Lett. 2008;78:1577–1582] proved that the optimal bound for the ratio of variances of the mth order statistic and a single-parent variable is the maximum of n/m and n/(n+1−m). We show here that the evaluation is optimal for many other measures of dispersion of the order statistics and original variable.

Key concepts: Mathematics, Independent and identically distributed random variables, Order statistic, Statistics, Statistic, Random variable, Dispersion (optics), Order (exchange)

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