Nonequivariant simultaneous confidence intervals less likely to contain zero

Yoav Benjamini*, Philip B. Stark

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We present a procedure for finding simultaneous confidence intervals for the expectations μ = (μj)nj=1 of a set of independent random variables, identically distributed up to their location parameters, that yields intervals less likely to contain zero than the standard simultaneous confidence intervals for many μ ≠ 0. The procedure is defined implicitly by inverting a nonequivariant hypothesis test with a hyperrectangular acceptance region whose orientation depends on the unsigned ranks of the components of μ, then projecting the convex hull of the resulting confidence region onto the coordinate axes. The projection to obtain simultaneous confidence intervals implicitly involves solving n! sets of linear inequalities in n variables, but the optima are attained among a set of at most n2 such sets and can be found by a simple algorithm. The procedure also works when the statistics are exchangeable but not independent and can be extended to cases where the inference is based on statistics for μ that are independent but not necessarily identically distributed, provided that there are known functions of μ that are location parameters for the statistics. In the general case, however, it appears that all n! sets of linear inequalities must be examined to find the confidence intervals.

Original languageEnglish
Pages (from-to)329-337
Number of pages9
JournalJournal of the American Statistical Association
Volume91
Issue number433
DOIs
StatePublished - 1 Mar 1996

Funding

FundersFunder number
National Science FoundationDMS-9404276, DMS-8957573
National Aeronautics and Space Administration

    Keywords

    • Conditional procedure
    • Multiple comparisons
    • Nonequivariant hypothesis test
    • Nonlinear procedure
    • Stepwise test
    • Variable-length confidence interval

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