Suppose that a statistic $S$ is asymptotically distributed as a distribution function $G(x)$ as some parameter $epsilono 0$. 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 $O(epsilon^{r-1})$. 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.
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机译：假设统计量$ S $作为一个分布函数$ G（x）$作为某些参数$ epsilon to 0 $渐近分布。我们考虑$ S $的单调变换，以改善渐近逼近。此处提出的变换保留单调性，并给出变换统计量$ T（S）$，其分布函数与$ G（x）$一致，直到阶次为$ O（ epsilon ^ {r-1}）$。可以观察到，提出的变换极大地改善了近似值。此外，我们还将考虑$ T（S）$分布的渐进展开的剩余项的误差范围。最后，针对某些测试统计数据证明了该发现的某些应用。
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