Empirical Edgeworth Expansion for Finite Population Statistics. II

For symmetric asymptotically linear statistics based on simple random samples, we construct a one–term empirical Edgeworth expansion, where the moments defining the true Edgeworth expansion are replaced by their jackknife estimators. In order to establish the validity of the empirical Edgeworth expansion (in probability) we prove the consistency of the jackknife estimators.     

An Edgeworth Expansion for Studentized Finite Population Statistics

We show the validity of the one-term Edgeworth expansion for Studentized asymptotically linear statistics based on samples drawn without replacement from finite populations. Replacing the moments defining the expansion by their estimators we obtain an empirical Edgeworth expansion. We show the validity of the empirical Edgeworth expansion in probability.     

Edgeworth Expansions for Studentized Versions of Symmetric Finite Population Statistics

We show the validity of the one-term Edgeworth expansion for Studentized asymptotically linear statistics based on samples drawn without replacement from finite populations. Replacing the moments defining the expansion by their estimators, we obtain an empirical Edgeworth expansion. We show the validity of the empirical Edgeworth expansion in probability.     

One-Term Edgeworth Expansion for Finite Population U-Statistics of Degree Two

By means of Hoeffding"s decomposition, we represent a finite population U-statistic of degree two by the sum of a linear and a quadratic part. Assuming that the linear part is nondegenerate, we prove the validity of one-term Edgeworth expansion for the distribution function of the statistic under the optimal (minimal) conditions on the linear part and 2 + moment condition on the quadratic part. No condition is imposed on the ratio N / n, where N, respectively n, denotes the sample size respectively the population size.     

One- and Two-Term Edgeworth Expansions for Finite Population Sample Mean. Exact Results. II

We prove the validity of one- and two-term Edgeworth expansions under optimal conditions (a Cramer-type smoothness condition and the minimal moment conditions) and provide precise bounds for the remainders of expansions. The bounds depend explicitly on the ratio p=N/n, where N denotes the sample size and n the population size, respectively.     

Edgeworth Expansion of Densities of Order Statistics with Fixed Rank

We give an Edgeworth expansion for densities of order statistics with fixed rank k.The Edgeworth expansion for densities of extreme values is then obtained as a special case k=1.     

Continuity Corrections for Discrete Distributions Under the Edgeworth Expansion

The approximation of discrete distributions by Edgeworth expansion series for continuity points of a discrete distribution F _n implies that if t is a support point of F _n, then the expansion should be performed at a continuity point . When a value is selected to improve the approximation of , and especially when a single term of the expansion is used, the selected is defined to be a continuity correction. This paper investigates the properties of the approximations based on several terms of the expansion, when is the value at which the infimum of a residual term is attained. Methods of selecting the estimation and the residual terms are investigated and the results are compared empirically for several discrete distributions. The results are also compared with the commonly used approximation based on the normal distribution with . Some numerical comparisons show that the developed procedure gives better approximations than those obtained under the standard continuity correction technique, whenever is close to 0 and 1. Thus, it is especially useful for p-value computations and for the evaluation of probabilities of rare events.     

One- and two-term edgeworth expansions for a finite population sample mean. Exact results. I

We prove the validity of one- and two-term Edgeworth expansions under optimal conditions (a Cramer-type smoothness condition and the minimal moment conditions) and provide precise bounds for the remainders of expansions. The bounds depend explicitly on the ratiop=N/n, whereN andn denote the sample size and the population size, respectively. Supported by the Alexander von Humboldt Foundation. Published in Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 277–294, July–September, 2000.     

部分线性模型中的Edgeworth展开

本文在相当一般的条件下，首先给出了部分线性模型中有关参数β的标准化统计量的一阶Ｅｄｇｅｗｏｒｔｈ展开，然后构造了误差方差的一个非残差型相合估计，最后给出了相应的学生化统计量的Ｅｄｇｅｗｏｒｔｈ展开．     

半参数回归模型L-估计的渐近展开

对妆参数回归模型Ｙ＝x^Tβ＋g（ｔ）＋ε,构造了参数向量β的Ｌ－估计量－／λn,获得了－／λn的渐近正态性及分布的Ｅdgeworth展开,其速度可达到Ｏ(n^-1)。     

Berry–Esseen Bound for a Sample Sum from a Finite Set of Independent Random Variables

Let {X ₁,...,X _N} be a set of N independent random variables, and let S _n be a sum of n random variables chosen without replacement from the set {X ₁,...,X _N} with equal probabilities. In this paper we give an estimate of the remainder term for the normal approximation of S _n under mild conditions.     

Edgeworth expansions for stochastic approximation theory

To estimate the root ϑ of an unknown regression function f: ℝ → ℝ the iterative Robbins-Monro method X _n+1 = X _n − a/nY _n with noisy observations Y _n = f(X _n ) + V _n of f(X _n ) can be used. It is well known that X _n − ϑ can be approximated by a weighted sum of the observation errors V _n . As recently shown this approximation can be improved by adding quadratic and cubic forms in the observation errors. This paper presents valid Edgeworth expansions of the distribution function of the approximating sequence up to a remainder term of order o(1/√n) or even o(1/n).      

Bootstrap Approximation to Distributions of Finite Population U-statistics

We show that the without replacement bootstrap of Booth, Butler and Hall (J. Am. Stat. Assoc. 89, 1282–1289, 1994) provides second order correct approximation to the distribution function of a Studentized U-statistic based on simple random sample drawn without replacement. In order to achieve similar approximation accuracy for the bootstrap procedure due to Bickel and Freedman (Ann. Stat. 12, 470–482, 1984) and Chao and Lo (Sankhya Ser. A 47, 399–405, 1985) we introduce randomized adjustments to the resampling fraction.      

Dependence Between Order Statistics in Samples from Finite Population and its Application to Ranked Set Sampling

Let X1, X2,..., Xm, Y1, Y2,..., Yn be a simple random sample without replacement from a finite population and let X(1) X(2) ... X(m) and Y(1) Y(2) ... Y(n) be the order statistics of X1, X2,..., Xm and Y1, Y2,..., Yn, respectively. It is shown that the joint distribution of X(i) and X(j) is positively likelihood ratio dependent and Y(j) is negatively regression dependent on X(i). Using these results, it is shown that when samples are drawn without replacement from a finite population, the relative precision of the ranked set sampling estimator of the population mean, relative to the simple random sample estimator with the same number of units quantified, is bounded below by 1.     

General Regular Variation of n-th Order and the 2nd Order Edgeworth Expansion of the Extreme Value Distribution （Ⅰ）

In Part Ⅰ the concept of the general regular variation of n-th order is proposed and its construction is discussed. The uniqueness of the standard expression and the higher order regularity of the auxiliary functions are proved.     

有Edgeworth展式的分布的随机加权逼近的重对数律

本文研究了未知分布的逼近问题，利用随机加权法，给出了有Edgeworth展式的一类（未知）分布的模拟分布，证明了在一定条件下，模拟分布与未知分布的逼近精度达到O(n^-1√lnlnn），称之为随机加权逼近的重对数律。     

L-统计量的Edgeworth展开和随机加权逼近

文［１］讨论了Ｌ－统计量的一种能达到Ｏ１√ｎ精确度的随机加权逼近,本文则给出了Ｌ－统计量的Ｅｄｇｅｗｏｒｔｈ展开和一种能达到ｏ１√ｎ精确性的新的随机加权逼近     

Edgeworth expansion for the pre-averaging estimator

In this paper, we study the Edgeworth expansion for a pre-averaging estimator of quadratic variation in the framework of continuous diffusion models observed with noise. More specifically, we obtain a second order expansion for the joint density of the estimators of quadratic variation and its asymptotic variance. Our approach is based on martingale embedding, Malliavin calculus and stable central limit theorems for continuous diffusions. Moreover, we derive the density expansion for the studentized statistic, which might be applied to construct asymptotic confidence regions.     

Edgeworth expansion for circular distribution

ＥＤＧＥＷＯＲＴＨＥＸＰＡＮＳＩＯＮＦＯＲＣＩＲＣＵＬＡＲＤＩＳＴＲＩＢＵＴＩＯＮ￥ＷＵＣＨＡＯＢＩＡＯＡＮＤＤＥＮＧＷＥＩＣＡＩ（Ｄｅｐｔ．ｏｆＳｔａｔｉｓｔ．,ＥａｓｔＣｈｉｎａＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ,Ｓｈａｎｇｈａｉ２０００６２．）（...     

The Edgeworth expansion for distributions of extreme values

We present necessary and sufficient conditions of Edgeworth expansion for distributions of extreme values. As a corollary, rates of the uniform convergence for distributions of extreme values are obtained.