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Optimal Inference With a Multidimensional Multiscale Statistic- [electronic resource]

서명/저자: Optimal Inference With a Multidimensional Multiscale Statistic - [electronic resource]

주기사항: Source: Dissertations Abstracts International, Volume: 85-04, Section: B.

초록/해제: 요약We observe a stochastic process Y on [0,1]d (d ≥ 1) satisfying dY(t)=n1/2f(t)dt + dW(t), t ∈ [0,1]d, where n ≥ 1 is a given scale parameter (`sample size'), W is the standard Brownian sheet on [0,1]d and f ∈ L1([0,1]d) is the unknown function of interest. We propose a multivariate multiscale statistic in this setting and prove that the statistic attains a subexponential tail bound; this extends the work of 'Dumbgen and Spokoiny (2001)' who proposed the analogous statistic for d = 1. In the process, we generalize Theorem 6.1 of 'Dumbgen and Spokoiny (2001)' about stochastic processes with sub-Gaussian increments on a pseudometric space, which is of independent interest. We use the proposed multiscale statistic to construct optimal tests (in an asymptotic minimax sense) for testing f = 0 versus (i) appropriate Holder classes of functions, and (ii) alternatives of the form f = μn IBn, where Bn is an axis-aligned hyperrectangle in [0,1]d and μn ∈ R; μn and Bn unknown. In Chapter 3 we use this proposed multiscale statistics to construct honest confidence bands for multivariate shape-restricted regression including monotone and convex functions.