Higher Order Moments of Order Statistics from the Pareto Distribution and Edgeworth Approximate Inference

Abstract

In this paper, we first derive exact explicit expressions for the triple and quadruple moments of order statistics from the Pareto distribution. Also, we establish recurrence relations for single, double, triple and quadruple moments of order statistics from the Pareto distribution. These relations will enable one to find all moments (of order up to four) of order statistics for all sample sizes in a simple recursive manner. We then use these results to determine the mean, variance, and coefficients of skewness and kurtosis of certain linear functions of order statistics. These are then utilized to develop approximate confidence intervals for the Pareto parameters using the Edgeworth approximation. Finally, we extend the recurrence relations to the case of the doubly truncated Pareto distribution.Keywords and phrasesOrder statisticsexact momentssingle momentsdouble momentstriple momentsquadruple momentsPareto distributiondoubly truncated distributionrecurrence relationsEdgeworth approximationcoefficients of skewness and kurtosisapproximate confidence intervalpivotal quantity