Weighted Selection for the Multiset X+X with Application to R-Estimates and Associated Confidence Limits

Ian Robinson, Simon Sheather

Research output: Contribution to journalArticlepeer-review

Abstract

Let X={X(1),X(2),⃛, X(n)} a collection of real numbers, ordered from smallest to largest. In this paper we consider the problem of Finding the weighted kth smallest element of the multiset X + X= {X(i) + X(j):1≦ i≦ j ≦n}, in which X(i) + X(j)has weight w(i, j) = a(j-i+1) — a(j-i), where 0 = a(0)≦a(l)≦ ⃛ ≦a(n)≠0 are a set of integers called scores. We derive relations involving the weights w(i,j) which can in principle be used to eliminate a number of elements of X+X which cannot possibly be the kth smallest, regardless of the distribution of X. These results are applied to Wilcoxon scores and to winsorized Wilcoxon scores. We describe a fast algorithm for finding the weighted kth smallest element in both cases.

Original languageEnglish
Pages (from-to)19-35
Number of pages17
JournalJournal of Statistical Computation and Simulation
Volume31
Issue number1
DOIs
StatePublished - Jan 1 1989

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Statistics, Probability and Uncertainty
  • Applied Mathematics

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