PASCAL - Pattern Analysis, Statistical Modelling and Computational Learning

How normal can the t-statistic possibly be?
Thorsten Dickhaus and Helmut Finner
In: Workshop on Validation in Statistics and Machine Learning, Berlin, Germany(2010).

Abstract

Revisiting the fundamental articles by Hsu (1945) and Chung (1946), we are concerned with Edgeworth expansions for self-normalized sums $S_n$ of iid real-valued random variables ($X_n$ : n integer) with mean zero and variance 1. In Chung's (1946) original paper, only the terms contributing to the first polynomial (appearing in the first-order Edgeworth expansion) are given correctly. After encapsulating and fixing a well-hidden error, we extended Chung's method to compute the expansion up to arbitrary order. Considering rates of convergence, we have that the cumulative distribution function (cdf) of S_n converges to the cdf of the standard normal distribution at rate $O(n^{-1/2})$ for non-vanishing skewness of X_1 and at rate $O(n^{-1})$ otherwise. We show that it is possible to improve convergence rates by replacing the norming sequence in the denominator of $S_n$ and thereby introducing "generalized self-normalized sums" $T_n$. It turns out that utilizing $T_n$ instead of $S_n$ can lead to a rate of convergence up to $O(n^{-2})$ for appropriate choices of the norming constants depending on the moments of $X_1$. Finally, we study Edgeworth-type expansions for $T_n$ in which we replace the standard normal distribution by Student's t-distribution with (n-1) degrees of freedom and analyze rates of convergence in this case.

EPrint Type:Conference or Workshop Item (Invited Talk)
Project Keyword:Project Keyword UNSPECIFIED
Subjects:Computational, Information-Theoretic Learning with Statistics
Learning/Statistics & Optimisation
ID Code:7877
Deposited By:Thorsten Dickhaus
Deposited On:17 March 2011