Optimal inference with a multidimensional multiscale statistic

Abstract

We observe a stochastic process Y on ${[0,1]}^d$ ($d\ge 1$) satisfying $dY(t)=n^{1∕2}f(t)dt+dW(t)$, $t\in {[0,1]}^d$, where $n\ge 1$ is a given scale parameter (‘sample size’), W is the standard Brownian sheet on ${[0,1]}^d$ and $f\in L_1({[0,1]}^d)$ 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 Dümbgen and Spokoiny [11] who proposed the analogous statistic for $d=1$. In the process, we generalize Theorem 6.1 of Dümbgen and Spokoiny [11] 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 Hölder classes of functions, and (ii) alternatives of the form $f={\mathrm{\mu }}_n{\mathbb{I}}_{B_n}$, where $B_n$ is an axis-aligned hyperrectangle in ${[0,1]}^d$ and ${\mathrm{\mu }}_n\in \mathbb{R}$; ${\mathrm{\mu }}_n$ and $B_n$ unknown.