On the effect of random norming on the rate of convergence in the central limit theorem
Let \(\bar X_ n\) be the sample mean based on i.i.d. nondegenerate random variables \(X_ 1,X_ 2,...,X_ n\). Consider the conditions: \[ (1)\quad E [p(y)\{1-p(y)\}]>0\quad where\quad p(y)=P(X-y>0| | X- y|); \] \[ (2)\quad x^ 2P(| X_ 1| >x)=O\{f(x^ 2)\}\quad as\quad x\to \infty, \] where f is a nonincreasing function f: [0,\(\infty)\to (0,\infty)\) such that \(x^{1-\epsilon}f(x^ 2)\) is eventually nondecreasing for some \(\epsilon >0\); \[ (3)\quad E(X^ 2_ 1G(| X_ 1|)<\infty,\quad where\quad G(x)=\int^{x}_{1}u^{- 1}g(u)du, \] g being a measurable function such that \(x^{\epsilon - 1}g(x)\) is eventually nonincreasing for some \(\epsilon >0.\) The main theorems of the paper concern rates of convergence of \[ \Delta_ n=| P\{n^{1/2}{\hat \sigma}^{-1}(\bar X_ n-E X_ 1)\leq x\}-\Phi (x)| \] where \({\hat \sigma}{}^ 2\) is the sample variance and \(\Phi\) (\(\cdot)\) is the standard normal distribution function. As consequences one observes that (a) under (1) and (2): \(\Delta_ n=O\{f(n)\}\), and (b) under (1) and (3): \(\sum n^{- 1}G(n^{1/2})\Delta_ n<\infty.\) As the conditions (2) or (3) are known to lead to the fastest rate of convergence in the central limit theorem for \(n^{1/2}(var X_ 1)^{- 1/2}(\bar X_ n-E X_ 1)\) using special norming constants, one concludes that ``studentizing'', i.e. normalizing by the sample s.d. rather than the population s.d., can improve the rate of convergence in the central limit theorem.
scientific article; zbMATH DE number 4126391