PASCAL - Pattern Analysis, Statistical Modelling and Computational Learning

Edgeworth expansions and rates of convergence for generalized self-normalized sums
Thorsten Dickhaus and Helmut Finner
In: 33rd Conference on Stochastic Processes and Their Applications (SPA 2009), 27-31 Jul 2009, Berlin, Germany.


Revisiting the fundamental articles by Hsu (\cite{hsu}) and Chung (\cite{chung}), we are concerned with Edgeworth expansions for self-normalized sums of iid real-valued random variables $(\zeta_n:n\in \mathbb{N})$ with $\mathbb{E}[\zeta_1] =0$ and $\mbox{Var}[\zeta_1] = 1$. Thereby, the self-normalized sum $S_n$ is defined by \[ S_n = \sqrt{n} \frac{\overline{\zeta}}{s} \mbox{ with } s = \left(\frac{1}{n} \sum_{i=1}^n (\zeta_i - \overline{\zeta})^2\right)^{1/2} \mbox{ and } \overline{\zeta} = \sum_{i=1}^n \zeta_i / n. \] In modern notation, the resulting expansions can be written as $$ F_{S_n} (x ) = \Phi(x) + \varphi(x) \sum_{i=1}^r n^{-i/2} q_i(x) + o (n^{-r/2}), $$ where $\Phi$ and $\varphi$ denote the cdf and pdf of the standard normal distribution. The $q_i$'s are polynomials of degree $3 i -1$ which depend on the moments $ \alpha_j = \mathbb{E} \zeta_1^j $, $ j=3, \ldots, i+2 $. In Chung's original paper \cite{chung}, only the terms contributing to $q_1$ 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 $ \Delta_n (x)= | F_{S_n} (x) - \Phi (x) | = O(n^{-1/2}) $ for non-vanishing skewness of $\zeta_1$ and $ \Delta_n (x)= O(n^{-1}) $ otherwise. We show that it is possible to improve convergence rates by replacing the norming sequence $1 / n$ in the definition of $s$. Therefore, we introduce generalized self-normalized sums $T_n$ with norming sequence $$ a_n = \frac{1}{n (1 - \sum_{j=1}^M C_j n^{-j/2})}, $$ where $M \in \mathbb{N}$ and $C_j \in \mathbb{R}$ for all $j = 1, \hdots, M$, and derive Edgeworth expansions for $T_n$. As a special case, this approach also covers studentized sums with $a_n = (n-1)^{-1}$ by setting $M=2$, $C_1 = 0$ and $C_2 = 1$. 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 $C_j$ depending on the moments of $\zeta_1$. Finally, we study Edgeworth-type expansions for $T_n$ in which we replace $\Phi$ and $\varphi$ by the cdf and pdf of Student's $t$-distribution with $n-1$ degrees of freedom and analyze rates of convergence in this case.

EPrint Type:Conference or Workshop Item (Talk)
Project Keyword:Project Keyword UNSPECIFIED
Subjects:Computational, Information-Theoretic Learning with Statistics
Learning/Statistics & Optimisation
ID Code:6808
Deposited By:Thorsten Dickhaus
Deposited On:08 March 2010