Nonparametric Statistical Theory



Nonparametric Statistical Theory - (L16) A. Bull

In parametric statistics, it is assumed the data comes from a known finite-dimensional family of distributions. While that assumption is often convenient, it may not always be true; in this course, we will ask whether it is possible to do statistics without it. We will see that, in many cases, the standard maximumlikelihood approach fails, and we must instead use procedures designed specifcally for nonparametric settings.


We will focus on the fundamental problems of estimating a distribution, density, or regression function, and describe techniques including empirical distribution functions, kernels, and wavelets. We will see that, while there are inherent limits to the nonparametric approach, we can nevertheless obtain some impressive results, and thereby perform statistics in much greater generality.


Distribution functions Basic empirical process theory, uniform laws of large numbers, Donsker and Kolmogorov-Smirnov theorems.
Minimax lower bounds Reduction to testing problems.
Approximation theory Convolution with kernels, series approximations, wavelets.
Density estimation and regression Kernel, local polynomial and wavelet estimators.
Choice of smoothing parameters Cross-validation, variable bandwidths, wavelet thresholding.

Pre-requisite Mathematics
Basic knowledge of probability, statistics and analysis is required. Measure theory and linear analysis are also useful, but the relevant material can be learnt as needed. This course complements the Michaelmas term course on Statistical Theory.
Literature
Complete notes will be available online; other relevant works include the following.
1. Tsybakov, A.B. Introduction to Nonparametric Estimation, Springer, 2009.
2. Van der Vaart, A.W. Asymptotic Statistics, Cambridge University Press, 1998.

 

 

 

 


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