Stein's method for functions of multivariate normal random vectors with application to the delta method

Abstract

Stein's method is a powerful technique which is used to obtain distributional approximations and characterisations in probability theory. Three great advantages of Stein's method are that it can be applied to many different distributions, it automatically gives a bound on the rate of convergence when proving probabilistic limit theorems, and it often deals well with dependence structures. In this thesis, we develop the theory of Stein's method for functions of multivariate normal random vectors and include a detailed application to the delta method. We also obtain a Stein characterisation of the distribution of the product of two correlated normal random variables with non-zero means, from which we deduce some basic distributional properties. We make a number of technical improvements to the theory of Stein's method for functions of multivariate normal random vectors. We obtain improved bounds for the derivatives of the solution of the multivariate normal Stein equation with unbounded test functions. We thus obtain improved bounds in terms of smaller constants and weaker moment assumptions, for a wide class of distributional approximations in which the limit distribution can be expressed as a function of a multivariate normal random vector. We apply our improved bounds to obtain general bounds on the rate of convergence in the multivariate delta method for a wide class of limit distributions. For normal limit distributions we obtain bounds with the optimal order $n^{-1/2}$ rate of convergence and for a wide class of non-normal limit distributions, we obtain bounds with a faster order $n^{-1}$ convergence rate ($n$ is the sample size) under additional assumptions. We consider an application of our general bounds to obtain explicit order $n^{-1}$ bounds for the chi-square approximation of a family of rank-based statistics and Pearson's statistic. In deriving our general bounds, we further generalise recent results on Stein's method for functions of multivariate normal random vectors to vector-valued functions and also to sums of independent random vectors whose components may be dependent.

Date of Award	1 Aug 2024
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Robert Gaunt (Supervisor)