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We 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 Hölder classes of functions, and (ii) alternatives of the form 𝑓 = 𝜇_𝑛𝕀_{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 Chapter 3 we use this proposed multiscale statistics to construct honest confidence bands for multivariate shape-restricted regression including monotone and convex functions.

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