Resampling-Based Confidence Regions and Multiple Tests for a Correlated Random Vector

Abstract

We derive non-asymptotic confidence regions for the mean of a random vector whose coordinates have an unknown dependence structure. The random vector is supposed to be either Gaussian or to have a symmetric bounded distribution, and we observe n i.i.d copies of it. The confidence regions are built using a data-dependent threshold based on a weighted bootstrap procedure. We consider two approaches, the first based on a concentration approach and the second on a direct boostrapped quantile approach. The first one allows to deal with a very large class of resampling weights while our results for the second are restricted to Rademacher weights. However, the second method seems more accurate in practice. Our results are motivated by multiple testing problems, and we show on simulations that our procedures are better than the Bonferroni procedure (union bound) as soon as the observed vector has sufficiently correlated coordinates.

The false discovery rate for statistical pattern recognition

“…Blanchard and Fleuret present an extension of the Occam's razor principle for generalization error analysis in classification, and apply it to derive p-value adjustment procedures for controlling FDR [5]. Arlot et al develop concentration inequalities that apply to multiple testing with correlated observations [2]. None of these works consider FDR/FNDR as performance criteria for classification.…”

Adaptive non-asymptotic confidence balls in density estimation

“…We use two different ingredients to compute ρ(m). The first one is a resampling estimator of s m − ŝm 2 , where s m denotes the projection of s onto S m . It is naturally derived from Efron's heuristic (see Efron [10]), in the same way as Arlot, Blanchard & Roquain [2].…”

“…The first one is a resampling estimator of s m − ŝm 2 , where s m denotes the projection of s onto S m . It is naturally derived from Efron's heuristic (see Efron [10]), in the same way as Arlot, Blanchard & Roquain [2]. This allows us in particular to keep all the sample to build ŝm .…”

“…2 , thus expρ 2 2n/([d n /2]) − 1 ≤ ρ 2 2n/([d n /2]) + 3.2 ρ 2 n/([d n /2]) ≤ 1.6ρ 4 n 2 /([d n /2]) ≤ d n − 1 2[d n /2] ln(1 + η 2 ) ≤ ln(1 + η 2 ).…”

“…2 = sup t∈B(Λ)P s (t − P s t)α i 2 ≤ v 2 s,Λ P s α 2 i . P s ((ψ λ − P s ψ λ )α i )) 2 ≤ nv 2 s,Λ .The same inequality holds for β j , thus we obtainEvaluation of B 3 : For all x in R, E[(U (x, X 2 )) 2 ] is the variance of the function t x = λ∈Λ (ψ λ (x)− P s ψ λ )ψ λ .…”