EPSRC Reference: 
EP/Y008650/1 
Title: 
Rates of Convergence in Multivariate Normal Approximation by Stein's Method 
Principal Investigator: 
Gaunt, Dr RE 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
University of Manchester, The 
Scheme: 
New Investigator Award 
Starts: 
01 December 2023 
Ends: 
30 November 2026 
Value (£): 
392,666

EPSRC Research Topic Classifications: 
Statistics & Appl. Probability 


EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
The central limit theorem is one of the most important results in probability and statistics. In simple terms, it states that a suitable normalisation of the sample mean is approximately normally distributed for large sample sizes. Amongst its numerous applications, it plays a crucial role in statistical hypothesis tests used throughout science. Quantifying the error in this distributional approximation was a major problem in the early part of 20th century, which culminated in the celebrated BerryEsseen theorem (194142) that gives a precise quantification of this error. Such results are of interest in statistical inference, as they can be used to derive conservative confidence intervals or provide rigorous justification of rulesofthumb used in the implementation of statistical tests.
More generally, distributional approximations are a central theme in probability theory. Here one is interested in approximating the (possibly intractable) distribution of a quantity of interest by a simpler, wellunderstood distribution, which allows one to make inference. There is now a vast literature on distributional approximations that has found application throughout the mathematical sciences and beyond. However, many important research challenges have remained out of reach.
The proposed research concerns the following fundamental problem:
How large is the error in approximating the distribution of a statistic of interest by a limit distribution that can be expressed as a function of a multivariate normal random vector?
This problem is of interest because many of the most important distributional approximations take this form, such as the central limit theorem and the chisquare approximation of Pearson's statistic, and assessing the quality of these approximations is key, for example, when sample sizes are small. Most techniques for proving distributional approximations are not wellsuited to this problem, because often significant extra work is required to extract a convergence rate; they cannot disentangle the complex dependence structures found in applications; or are highly specialised for a small class of distributions.
I have recently introduced a breakthrough approach to this problem via a powerful probabilistic technique called Stein's method. The main focus of this research is to develop the theory so that it can be used to prove quantitative distributional approximations in which the limit distribution is expressible as a function of a multivariate normal random vector in the increasingly complex settings encountered in applications. The research will explore some fascinating theoretical problems, such as the identification of (surprising) necessary and sufficient conditions under which `faster than expected' convergence rates occur in substantial generalisations of the central limit theorem. Knowing these conditions will be useful to statisticians in the design of statistical tests, as it will allow them to identify statistics with desirable convergence properties efficiently. Indeed, a major part of the research will involve the application of this general theory to obtain bounds on the convergence rates of widely used statistics for hypothesis testing and popular statistics from alignmentfree sequence comparison.

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Organisation Website: 
http://www.man.ac.uk 