An exponential nonuniform Berry-Esseen bound for self-normalized sums
Nonuniform Berry-Esseen bounds for self-normalized sums of independent random variables, with application to the Student \(t\)-statistic, are obtained. They are substantially better than, in fact in sharp contrast with, those available for simply standardized sums in that the dependence on the range point \(x\) is exponentially instead of polynomial, even under only finiteness of the third moments. These results apply in particular to moderate deviations. Here is one of the corollaries: If \(X_i\) are independent identically distributed symmetric random variables with \(E|X_i|< \infty\) for all \(i\), and if we let \(L_{3n}:= n^{-1/2}E|X_1|^3/(EX^2_1)^{3/2}\), then the absolute deviation of the distribution function \(F_n(x)\) of \(\sum^n_{i=1} X_i/(\sum^n_{i=1} X^2_i)^{1/2}\) from that of the normal, \(\Phi(x)\), satisfies \[ |F_n(x)- \Phi(x)|\leq A\min\{(1+|x|^3) L_{3n},1\} \exp(- x^2/2), \] where \(A\) is an absolute constant.
scientific article; zbMATH DE number 1496121