Transformations with improved asymptotic approximations and their accuracy

Abstract

Suppose that a statistic $S$ is asymptotically distributed as a distribution function $G(x)$ as some parameter $ϵ\to 0$. We consider monotone transformations of $S$ in order to improve the asymptotic approximation. The transformations proposed here preserve monotonicity and give transformed statistics $T(S)$ whose distribution function is coincident with $G(x)$ up to the order $O({ϵ}^{r-1})$. It may be observed that the proposed transformations give a significant improvement to approximations. Further, we shall also consider error bounds for the remainder term of an asymptotic expansion for the distribution of $T(S)$. Finally, some applications of the findings are demonstrated for some test statistics.