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The Distribution of Order Statistics for Discrete Random Variables with Applications to Bootstrapping

Department of Mathematics, Rose–Hulman Institute of Technology, 5500 Wabash Avenue, Terre Haute, Indiana 47803, USA
Department of Mathematics, The College of William & Mary, P.O. Box 8795, Williamsburg, Virginia 23187–8795, USA
Department of Mathematics, The College of William & Mary, P.O. Box 8795, Williamsburg, Virginia 23187–8795, USA
diane.evans{at}rose-hulman.edu
leemis{at}math.wm.edu
jhdrew{at}math.wm.edu

An algorithm for computing the PDF of order statistics drawn from discrete parent populations is presented, along with an implementation of the algorithm in a computer algebra system. Several examples and applications, including exact bootstrapping analysis, illustrate the utility of this algorithm. Bootstrapping procedures require that B bootstrap samples be generated in order to perform statistical inference concerning a data set. Although the requirements for the magnitude of B are typically modest, a practitioner would prefer to avoid the resampling error introduced by choosing a finite B, if possible. The part of the order-statistic algorithm for sampling with replacement from a finite sample can be used to perform exact bootstrapping analysis in certain applications, eliminating the need for replication in the analysis of a data set.

Key words: combinatorial algorithms; computer algebra systems; probability; probability distributions; statistics
History: received June 2000; revised August 2001; revised October 2003; revised April 2004; accepted June 2004.