Theses Doctoral

Optimal Inference with a Multidimensional Multiscale Statistic

Datta, Pratyay

We observe a stochastic process π‘Œ on [0,1]^𝑑 (𝑑 β‰₯ 1) satisfying π‘‘π‘Œ(𝑑)=𝑛¹/²𝑓(𝑑)𝑑𝑑 + π‘‘π‘Š(𝑑), 𝑑 ∈ [0,1]^𝑑, where 𝑛 β‰₯ 1 is a given scale parameter (`sample size'), π‘Š 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 = πœ‡_𝑛𝕀_{𝐡_𝑛}$, where 𝐡_𝑛 is an axis-aligned hyperrectangle in [0,1]^𝑑 and πœ‡_𝑛 ∈ ℝ; πœ‡_𝑛 and 𝐡_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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More About This Work

Academic Units
Thesis Advisors
Sen, Bodhisattva
Zheng, Tian
Ph.D., Columbia University
Published Here
October 18, 2023