Optimal Inference with a Multidimensional Multiscale Statistic
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We observe a stochastic process Y on [0,1]^d (d≥ 1) satisfying dY(t)=n^1/2f(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 ∈ L_1([0,1]^d) is the unknown function of interest. We propose a multivariate multiscale statistic in this setting and prove its almost sure finiteness; this extends the work of Dümbgen and Spokoiny (2001) who proposed the analogous statistic for d=1. We use the proposed multiscale statistic to construct optimal tests for testing f=0 versus (i) appropriate Hölder classes of functions, and (ii) alternatives of the form f=μ_n I_B_n, where B_n is an axis-aligned hyperrectangle in [0,1]^d and μ_n ∈R; μ_n and B_n unknown. In the process we generalize Theorem 6.1 of Dümbgen and Spokoiny (2001) about stochastic processes with sub-Gaussian increments on a pseudometric space, which is of independent interest.
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