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.     

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.     

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.     

Interpretation and manipulation of Edgeworth expansions

With a given Edgeworth expansion sequences of i.i.d. r.v.'s are associated such that the Edgeworth expansion for the standardized sum of these r.v.'s agrees with the given Edgeworth expansion. This facilitates interpretation and manipulation of Edgeworth expansions. The theory is applied to the power of linear rank statistics and to the combination of such statistics based on subsamples. Complicated expressions for the power become more transparent. As a consequence of the sum-structure it is seen why splitting the sample causes no loss of first order efficiency and only a small loss of second order efficiency.     

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．     

An edgeworth expansion for finite-population L-statistics

We consider the one-term Edgeworth expansion for finite-population L-statistics. We provide an explicit formula for the Edgeworth correction term and give sufficient conditions for the validity of the expansion that are expressed in terms of the weight function defining the statistics and moment conditions.     

Edgeworth expansion of the Studentized product-limit estimator for truncated and censored data

Based on random left truncated and right censored data we investigate the one-term Edgeworth expansion for the Studentized product-limit estimator, and show that the Edgeworth expansion is close to the exact distribution of the Studentized product-limit estimator with a remainder of On(su-1/2).     

An Asymptotic Expansion for the Distribution of Hotelling'sT-Statistic under Nonnormality

In this paper we obtain an asymptotic expansion for the distribution of Hotelling'sT²-statisticT²under nonnormality when the sample size is large. In the derivation we find an explicit Edgeworth expansion of the multivariatet-statistic. Our method is to use the Edgeworth expansion and to expand the characteristic function ofT².     

Empirical Edgeworth Expansion for Finite Population Statistics. I

For symmetric asymptotically linear statistics based on simple random samples, we construct the 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.     

High-dimensional Edgeworth expansion of a test statistic on independence and its error bound

In this paper, we calculate Edgeworth expansion of a test statistic on independence when some of the parameters are large, and simulate the goodness of fit of its approximation. We also calculate an error bound for Edgeworth expansion. Some tables of the error bound are given, which show that the derived bound is sufficiently small for practical use.     

Edgeworth expansion for the survival function estimator in the Koziol-Green model

In the Koziol-Green or proportional hazards random censorship model, the asymptotic accuracy of the estimated one-term Edgeworth expansion and the smoothed bootstrap approximation for the Studentized Abdushukurov-Cheng-Lin estimator is investigated. It is shown that both the Edgeworth expansion estimate and the bootstrap approximation are asymptotically closer to the exact distribution of the Studentized Abdushukurov-Cheng-Lin estimator than the normal approximation.     

Développement d'Edgeworth d'ordre 1 pour des M-estimateurs dans le cas de chaînes V-géométriquement ergodiques

Conditions under which a first-order Edgeworth expansion for M-estimators of V-geometrically ergodic Markov chains exists are proposed in this Note. The expansion is explicit.     

Edgeworth expansion in censored linear regression model

For the censored simple linear regression model, we establish a oneterm Edgeworth expansion for the Koul, Susarla and Van Ryzin type estimator of the regression coefficient. Our approach is to represent the estimator of the regression coefficient as an asymptoticU-statistic plus some ignorable terms and hence apply the known results on the Edgeworth expansions for asymptoticU-statistic. The counting process and martingale techniques are used to provide the proof of the main results.     

An implicit function approach to constrained optimization with applications to asymptotic expansions

In this article, an unconstrained Taylor series expansion is constructed for scalar-valued functions of vector-valued arguments that are subject to nonlinear equality constraints. The expansion is made possible by first reparameterizing the constrained argument in terms of identified and implicit parameters and then expanding the function solely in terms of the identified parameters. Matrix expressions are given for the derivatives of the function with respect to the identified parameters. The expansion is employed to construct an unconstrained Newton algorithm for optimizing the function subject to constraints.Parameters in statistical models often are estimated by solving statistical estimating equations. It is shown how the unconstrained Newton algorithm can be employed to solve constrained estimating equations. Also, the unconstrained Taylor series is adapted to construct Edgeworth expansions of scalar functions of the constrained estimators. The Edgeworth expansion is illustrated on maximum likelihood estimators in an exploratory factor analysis model in which an oblique rotation is applied after Kaiser row-normalization of the factor loading matrix. A simulation study illustrates the superiority of the two-term Edgeworth approximation compared to the asymptotic normal approximation when sampling from multivariate normal or nonnormal distributions.     

Onm-dependence and Edgeworth expansions

This paper contains two results. The first establishes, under mild assumptions, the validity of an Edgeworth expansion with remaindero(N ^–1/2) for aU-statistic with a kernel of degree two using observations from anm-dependent shift. The second result gives a necessary and sufficient condition for the distribution of a sum ofm-dependent random variables to possess an Edgeworth expansion. This generalizes a result of Bickel and Robinson from the i.i.d. case to them-dependent case.This research was supported in part by National Science Foundation, Grant DMS 89-23071.     

Improved confidence intervals for quantiles

We derive the Edgeworth expansion for the studentized version of the kernel quantile estimator. Inverting the expansion allows us to get very accurate confidence intervals for the pth quantile under general conditions. The results are applicable in practice to improve inference for quantiles when sample sizes are moderate.     

A strong large deviation theorem

We prove a strong large deviation theorem for an arbitrary sequence of random variables, that is, we establish a full asymptotic expansion of large deviation type for the tail probabilities. An Edgeworth expansion is required to derive the result. We illustrate our theorem with two statistical applications: the sample variance and the kernel density estimator.     

RANDOM WEIGHTING APPROXIMATION IN LINEAR REGRESSION MODELS

ＲＡＮＤＯＭＷＥＩＧＨＴＩＮＧＡＰＰＲＯＸＩＭＡＴＩＯＮＩＮＬＩＮＥＡＲＲＥＧＲＥＳＳＩＯＮＭＯＤＥＬＳＳＨＩＪＩＡＮ（ＤｅｐａｒｔｍｅｎｔｏｆＰｒｏｂａｂｉｌｉｔｙａｎｄＳｔａｔｉｓｔｉｃｓ，ＰｅｋｉｎｇＵｎｉｖｅｒｓｉｔｙ，Ｂｅｉｊｉｎｇ１００８...     

On the First-Order Edgeworth Expansion for a Markov Chain

We consider the first-order Edgeworth expansion for summands related to a homogeneous Markov chain. Certain inaccuracies in some earlier results by Nagaev are corrected and the expansion is obtained under relaxed conditions. An application of our result to the distribution of the mle of a transition probability in the countable state space case is also considered.