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 nonzero 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 nonnormal 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 chisquare approximation of a family of rankbased 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 vectorvalued 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) 

 Stein's method
 distributional approximation
 functions of multivariate normal random vectors
 multivariate normal approximation
 delta method
 rate of convergence
 product of correlated normal random variables
Stein's method for functions of multivariate normal random vectors with application to the delta method
Sutcliffe, H. (Author). 1 Aug 2024
Student thesis: Phd