Non-asymptotic results for Cornish--Fisher expansions

The distribution and quantiles of functionals of weighted empirical distributions when observations have different distributions

This paper extends Edgeworth-Cornish-Fisher expansions for the distribution and quantiles of nonparametric estimates in two ways. Firstly it allows observations to have different distributions. Secondly it allows the observations to be weighted in a predetermined way. The use of weighted estimates has a long history including applications to regression, rank statistics and Bayes theory. However, asymptotic results have generally been only first order (the CLT and weak convergence). We give third order asymptotics for the distribution and percentiles of any smooth functional of a weighted empir…

A Sheet of Maple to Compute Second-Order Edgeworth Expansions and Related Quantities of any Function of the Mean of an iid Sample of an Absolutely Continuous Distribution

We designed a completely automated Maple ($\geqslant 15$) worksheet for deriving Edgeworth and Cornish-Fisher expansions as well as the acceleration constant of the bootstrap bias-corrected and accelerated technique. It is valid for non-parametric or parametric bootstrap, of any (studentized) statistics that is -a regular enough- function of the mean of an iid sample of an absolutely continuous distribution. This worksheet allowed us to point out one error in the second-order Cornish-Fisher expansion of the studentized mean stated in Theorem 13.5 by Das Gupta in [8, p. 194] as well as lay the …

高维情况下球性检验中两步单调不完全数据下的可计算误差界

Edgeworth expansions in sphericity testing under high-dimensional two-step monotone incomplete data and their computable error bounds

本文研究高维两步单调不完全数据下的一样本球性检验，推导了似然比检验统计量的Edgeworth展开并获得可计算的误差界，通过蒙特卡洛模拟验证了其优于现有渐近展开的精度。

Fisher transformation via Edgeworth expansion

We show how to calculate individual terms of the Edgeworth series to approximate the distribution of the Pearson correlation coefficient with the help of a simple Mathematica program. We also demonstrate how to eliminate the corresponding skewness, thus making the approximation substantially more accurate. This leads, in a rather natural way, to deriving a superior (in terms of its accuracy) version of Fisher's z transformation. The code can be easily modified to deal with any sample statistics defined as a function of several sample means, based on a random independent sample from a multivari…

关于频率论者的错误发现率：渐近展开

On the false discovery rates of a frequentist: Asymptotic expansions

考虑零假设$H_0:θ\inΘ_0$的检验问题。标准的频率论实践是当p值小于阈值$α$（通常为0.05）时拒绝零假设。我们要问的问题是，频率论者拒绝的零假设中有多少是真的。准确地说，我们在适当的先验密度$g(θ)$下研究贝叶斯错误发现率$δ_n=P_g(θ\inΘ_0|p-value<α)$。这取决于先前的$g$、样本量$n$、阈值$α$以及检验统计量的选择。我们表明，Benjamini-Hochberg FDR实际上收敛到$δ_n$几乎肯定下$g$任何固定的$n$。对于单侧零假设，当检验统计量是MLE时，我们推导出$δ_n$在连续指数族中的三阶渐近展开式，当检验统计量是样本中位数时，推导出在位置族中的三阶渐近展开式。我们还简要地提到了当检验统计量是极大似然估计时在均匀族中的展开。展开是通过将CDF的Edgeworth展开、分位数函数的Cornish-Fisher展开和各种Taylor展开放在一起而得到的。数值结果表明，即使对于小的$n$值（例如，$n=10$）。我们从这些扩展中得到了许多有用的结论，特别是频率论者不容易得到错误的发现，除非先前的$g$太尖。最后用实例说明了结果。

Fisher information and the central limit theorem

An Edgeworth-type expansion is established for the relative Fisher information distance to the class of normal distributions of sums of i.i.d. random variables, satisfying moment conditions. The validity of the central limit theorem is studied via properties of the Fisher information along convolutions.