TY - JOUR

T1 - Nonequivariant simultaneous confidence intervals less likely to contain zero

AU - Benjamini, Yoav

AU - Stark, Philip B.

N1 - Funding Information:
Yoav Benjamini is Senior Lecturer, Department of Statistics, Tel Aviv University, Israel. Philip B. Stark is Associate Professor, Department of Statistics and Space Sciences Laboratory, University of California, Berkeley, CA 94720. The authors are grateful to Peter Bickel, Stephen Evans, Yosef Hochberg, Erich Lehmann, Juliet Shafkr, and David Steinberg for helpful conversations, suggestions and comments on an earlier draft. Stark gratefully acknowledges support from the National Science Foundation through PYI Award DMS-8957573 and Grant DMS-9404276, and from the National Aeronautics and Space Administration through Grant NAGW

PY - 1996/3/1

Y1 - 1996/3/1

N2 - 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.

AB - 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.

KW - Conditional procedure

KW - Multiple comparisons

KW - Nonequivariant hypothesis test

KW - Nonlinear procedure

KW - Stepwise test

KW - Variable-length confidence interval

UR - http://www.scopus.com/inward/record.url?scp=0042624277&partnerID=8YFLogxK

U2 - 10.1080/01621459.1996.10476692

DO - 10.1080/01621459.1996.10476692

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AN - SCOPUS:0042624277

SN - 0162-1459

VL - 91

SP - 329

EP - 337

JO - Journal of the American Statistical Association

JF - Journal of the American Statistical Association

IS - 433

ER -