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Contributions to asymptotic theory in nonparametric statistics


Type

Thesis

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Authors

Deo, Aditya 

Abstract

Nonparametric statistics, in which the parameter of interest is allowed to be infinite-dimensional, provides a theoretical lens through which to analyse the performance of modern data science and machine learning methods. The need for a rigorous understanding of these procedures has never been greater due to their widespread use throughout the sciences and beyond. This thesis studies a variety of nonparametric statistical models, and considers the problems of parameter estimation and uncertainty quantification.

Chapter 2 studies the problem of density estimation on the d-dimensional torus, using Wasserstein distances as the loss function. We consider the question of constructing adaptive honest confidence sets, which have uniform coverage guarantees but also shrink at an optimal rate depending on the smoothness of the true parameter, measured on the Besov scale. For the classically studied L^p-losses, it is known that such adaptive confidence sets only exist for a narrow range of smoothnesses; in the case of L^∞-loss, no adaptation is possible at all. For the Wasserstein distances W_p, 1 ≤ p ≤ 2, we uncover a new phenomenon: that in dimensions d ≤ 4, one may construct adaptive confidence sets which can adapt to any smoothness. In higher dimensions, there is a limited range of smoothnesses which can be adapted to, but this interval is wider than for the L^p-distances. The non-existence results are based on an analysis of a related composite hypothesis testing problem, while the construction of our confidence sets follows a risk estimation approach. We prove similar results for the W_1-distance when the sampling domain is R^d. These are the first results in a statistical approach to adaptive uncertainty quantification with Wasserstein distances. Our analysis extends to other weak loss functions such as negative order Sobolev norms.

In Chapter 3, we study a general non-linear inverse problem in which the parameter is assumed to have a compositional structure. Taking a Bayesian approach, we consider a deep Gaussian process (DGP) prior, which is designed to leverage the compositional structure of the parameter in order to achieve fast convergence rates. We show that the DGP prior does indeed consistently solve the inverse problem, proving a posterior contraction rate result; this contraction rate depends on the compositional structure of the true parameter, and the DGP prior is able to adapt to this unknown structure. We also show that Gaussian process priors are unable to exploit compositional structures in the same manner, by proving contraction rate lower bounds for a well-studied family of rescaled Whittle-Matérn process priors. When the dimension d is sufficiently large, these lower bounds are polynomially slower than the contraction rate achieved by the DGP prior. These results indicate that when the parameter has a complex structure other than straightforward Sobolev or Hölder smoothness, one must look beyond Gaussian priors to obtain good performance from the Bayesian method. Two examples of inverse problems encompassed by our general framework are the Darcy flow problem and the steady-state Schrödinger equation.

Chapter 4 studies the empirical process arising from a multi-dimensional diffusion process with periodic drift field and diffusivity. In particular, we address the question of proving the Donsker property for Besov classes of functions. Donsker classes have been completely characterised in the case of scalar diffusions, but these arguments rely on the use of local time, which does not exist in higher dimensions. Instead, we utilise the smoothing properties of the generator of the diffusion process, an elliptic second order partial differential operator, to show that the diffusion empirical process is smoother than its classical i.i.d. analogue. As an application, we deduce precise asymptotics for the Wasserstein-1 distance between the occupation measure and the invariant measure.

Description

Date

2023-07-17

Advisors

Nickl, Richard

Keywords

Bayesian nonparametrics, diffusion processes, mathematics, minimax rates, optimal transport, statistics, uncertainty quantification, Wasserstein distance

Qualification

Doctor of Philosophy (PhD)

Awarding Institution

University of Cambridge