Partial sorting: A large vector technique and its application

Abstract

Let vectorA containN elements. When vectorA is sorted the relationA(1) ⩽A(2) ⩽ ⋯ ⩽A(N) holds.A partial sort is a procedure by which theJth, 1 ⩽J ⩽N, element ofA,A(J), is found without completely sorting vectorA.A(J) is the Jth-order statistic ofA. A partial sort technique based on the Quicksort algorithm requires storage ofA. In this article an alternate technique based on confidence intervals of ranks and a rejection procedure findsA(J). This procedure only requires a small working storage and is suited to the case when the elements ofA, for largeN, are serially computed. The minimum number of working storage locations needed for the algorithm to terminate successfully with set probability for specifiedJ andN was computed. Partial sorting, especially this technique, is useful for computing critical values for statistics for which the sampling distributions are analytically known. Error bounds, which can be estimated from a segment of the ordered sample, are given for the estimated critical value. The technique is applied in several examples to estimate the mathematical constant π.