Asymptotic equivalence of the jackknife and infinitesimal jackknife variance estimators for some smooth statistics

The jackknife variance estimator and the infinitesimal jackknife variance estimator are shown to be asymptotically equivalent if the functional of interest is a smooth function of the mean or a trimmed L-statistic with Hölder continuous weight function.     

One-step jackknife for M-estimators computed using Newton's method

To estimate the dispersion of an M-estimator computed using Newton's iterative method, the jackknife method usually requires to repeat the iterative process n times, where n is the sample size. To simplify the computation, one-step jackknife estimators, which require no iteration, are proposed in this paper. Asymptotic properties of the one-step jackknife estimators are obtained under some regularity conditions in the i.i.d. case and in a linear or nonlinear model. All the one-step jackknife estimators are shown to be asymptotically equivalent and they are also asymptotically equivalent to the original jackknife estimator. Hence one may use a dispersion estimator whose computation is the simplest. Finite sample properties of several one-step jackknife estimators are examined in a simulation study.The research was supported by Natural Sciences and Engineering Research Council of Canada.     

Jackknifing in partially linear regression models with serially correlated errors

In this paper jackknifing technique is examined for functions of the parametric component in a partially linear regression model with serially correlated errors. By deleting partial residuals a jackknife-type estimator is proposed. It is shown that the jackknife-type estimator and the usual semiparametric least-squares estimator (SLSE) are asymptotically equivalent. However, simulation shows that the former has smaller biases than the latter when the sample size is small or moderate. Moreover, since the errors are correlated, both the Tukey type and the delta type jackknife asymptotic variance estimators are not consistent. By introducing cross-product terms, a consistent estimator of the jackknife asymptotic variance is constructed and shown to be robust against heterogeneity of the error variances. In addition, simulation results show that confidence interval estimation based on the proposed jackknife estimator has better coverage probability than that based on the SLSE, even though the latter uses the information of the error structure, while the former does not.     

Jackknifing in generalized linear models

In a generalized linear model, the jackknife estimator of the asymptotic covariance matrix of the maximum likelihood estimator is shown to be consistent. The corresponding jackknife studentized statistic is asymptotically normal. In addition, these results remain true even if there exist unequal dispersion parameters in the model. On the other hand, the variance estimator and the studentized statistic based on the standard method (substitution and linearization) do not enjoy this robustness property against the presence of unequal dispersion parameters.This research was supported by an Operating Grant from the Natural Science and Engineering Research Council of Canada.     

Delete-group Jackknife Estimate in Partially Linear Regression Models with Heteroscedasticity

Abstract Consider a partially linear regression model with an unknown vector parameter β,an unknownfunction g(.),and unknown heteroscedastic error variances.Chen,You proposed a semiparametric generalizedleast squares estimator(SGLSE)for β,which takes the heteroscedasticity into account to increase efficiency.Forinference based on this SGLSE,it is necessary to construct a consistent estimator for its asymptotic covariancematrix.However,when there exists within-group correlation, the traditional delta method and the delete-1jackknife estimation fail to offer such a consistent estimator.In this paper, by deleting grouped partial residualsa delete-group jackknife method is examined.It is shown that the delete-group jackknife method indeed canprovide a consistent estimator for the asymptotic covariance matrix in the presence of within-group correlations.This result is an extension of that in[21].     

Asymptotic Comparisons of Several Variance Estimators and their Effects for Studentizations

In this paper we obtain asymptotic representations of several variance estimators of U-statistics and study their effects for studentizations via Edgeworth expansions. Jackknife, unbiased and Sen's variance estimators are investigated up to the order o^p(n^-1). Substituting these estimators to studentized U-statistics, the Edgeworth expansions with remainder term o(n^-1) are established and inverting the expansions, the effects on confidence intervals are discussed theoretically. We also show that Hinkley's corrected jackknife variance estimator is asymptotically equivalent to the unbiased variance estimator up to the order o^p(n^-1).     

Asymptotic expansions associated with the jackknife functional. II

This paper is a sequel to part I [Ukr. Mat. Zh.,47, No. 4, 443–452 (1995)]. By using the results of the first part, we obtain the initial terms of the asymptotic expansions of the bias and variance for the jackknife estimator of the variance of the error of observations in a nonlinear regressive model.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 6, pp. 731–736, June, 1995.The present work was financially supported by the Ukrainian State Committee on Science and Technology.     

Jackknife variance estimators of the location estimator in the one-way random-effects model

In a one-way random-effects model, we frequently estimate the variance components by the analysis-of-variance method and then, assuming the estimated values are true values of the variance components, we estimate the population mean. The conventional variance estimator for the estimate of the mean has a bias. This bias can become severe in contaminated data. We can reduce the bias by using the delta method. However, it still suffers from a large bias. We develop a jackknife variance estimator which is robust with respect to data contamination.This research was supported by the Korea Science and Engineering Foundation.     

The best EBLUP in the Fay�CHerriot model

We show that in the case of Fay?CHerriot model for small area estimation, there is an estimator of the variance of the random effects so that the resulting EBLUP is the best in the sense that it minimizes the leading term in the asymptotic expansion of the mean squared error (MSE) of the EBLUP. In particular, in the balanced case, i.e., when the sampling variances are equal, this best EBLUP has the minimal MSE in the exact sense. We also propose a modified Prasad?CRao MSE estimator which is second-order unbiased and show that it is less biased than the jackknife MSE estimator in a suitable sense in the balanced case. A real data example is discussed.     

Imputation of mean of ratios for missing data and its application to PPSWR sampling

In practical survey sampling, nonresponse phenomenon is unavoidable. How to impute missing data is an important problem. There are several imputation methods in the literature. In this paper, the imputation method of the mean of ratios for missing data under uniform response is applied to the estimation of a finite population mean when the PPSWR sampling is used. The imputed estimator is valid under the corresponding response mechanism regardless of the model as well as under the ratio model regardless of the response mechanism. The approximately unbiased jackknife variance estimator is also presented. All of these results are extended to the case of non-uniform response. Simulation studies show the good performance of the proposed estimators.     

Sample rotation theory with missing data

This paper studies how the sample rotation method is applied to the case where item non-response occurs in surveys. The two cases where the response to the first occasion is complete or incomplete are considered. Using ratio imputation method, the estimators of the current population mean are proposed, which are valid under uniform response regardless of the model and under the ratio model regardless of the response mechanism. Under uniform response, the variances of the proposed estimators are derived. Interestingly, although their expressions are similar, the estimator for the case of incomplete response on the first occasion can have smaller variance than the one for the case of complete response on the first occasion under uniform response. The linearized jackknife variance estimators are also given. These variance estimators prove to be approximately design-unbiased under uniform response. It should be noted that similar property on variance estimators has not been discussed in literature.     

样本函数条件极值中减低偏差的方法

对样本函数条件极值中偏差项的阶进行了分析,探讨了减低偏差项的方法,分析表明古典折刀法、减-d折刀法均不能减低偏差项;在此基础上,提出了减低偏差项的自助法,并论证了在均方误差意义下,θnab是一种较优的估计.     

Kullback-Leibler Information Consistent Estimation for Censored Data

This paper is intended as an investigation of parametric estimation for the randomly right censored data. In parametric estimation, the Kullback-Leibler information is used as a measure of the divergence of a true distribution generating a data relative to a distribution in an assumed parametric model M. When the data is uncensored, maximum likelihood estimator (MLE) is a consistent estimator of minimizing the Kullback-Leibler information, even if the assumed model M does not contain the true distribution. We call this property minimum Kullback-Leibler information consistency (MKLI-consistency). However, the MLE obtained by maximizing the likelihood function based on the censored data is not MKLI-consistent. As an alternative to the MLE, Oakes (1986, Biometrics, 42, 177–182) proposed an estimator termed approximate maximum likelihood estimator (AMLE) due to its computational advantage and potential for robustness. We show MKLI-consistency and asymptotic normality of the AMLE under the misspecification of the parametric model. In a simulation study, we investigate mean square errors of these two estimators and an estimator which is obtained by treating a jackknife corrected Kaplan-Meier integral as the log-likelihood. On the basis of the simulation results and the asymptotic results, we discuss comparison among these estimators. We also derive information criteria for the MLE and the AMLE under censorship, and which can be used not only for selecting models but also for selecting estimation procedures.     

On an estimator in T3-class of linear estimators in sampling with varying probabilities from a finite population

Summary Horvitz and Thompson [4] introduced three classes of linear estimators for estimation of population characteristics on the basis of a sample drawn with varying probabilities and without replacement. TheirT ₃-class of estimators does not admit a best unbiased estimator. In this paper, the variance and an unbiased estimate of variance for an estimator in T₃-class, which is proved to have several good properties by Godambe [2], [3], are derived for sampling with varying probabilities with or without replacement.     

Smoothed Monte Carlo estimators for the time-in-the-red in risk processes

We consider a modified version of the de Finetti model in insurance risk theory in which, when surpluses become negative the company has the possibility of borrowing, and thus continue its operation. For this model we examine the problem of estimating the time-in-the red over a finite horizon via simulation. We propose a smoothed estimator based on a conditioning argument which is very simple to implement as well as particularly efficient, especially when the claim distribution is heavy tailed. We establish unbiasedness for this estimator and show that its variance is lower than the naïve estimator based on counts. Finally we present a number of simulation results showing that the smoothed estimator has variance which is often significantly lower than that of the naïve Monte-Carlo estimator.     

Sharp asymptotics for isotonic regression

 The asymptotic behavior of the isotonic estimator of a monotone regression function (that is the least-squares estimator under monotonicity restriction) is investigated. In particular it is proved that the ?₁-distance between the isotonic estimator and the true function is of magnitude n ^-1/3. Moreover, it is proved that a centered version of this ?₁-distance converges at the n ^1/2 rate to a Gaussian variable with fixed variance. Received: 20 September 1999 / Revised version: 10 May 2001 / Published online: 19 December 2001     

Estimating a Distribution Function for Censored Time Series Data

Consider a long term study, where a series of dependent and possibly censored failure times is observed. Suppose that the failure times have a common marginal distribution function, but they exhibit a mode of time series structure such as α-mixing. The inference on the marginal distribution function is of interest to us. The main results of this article show that, under some regularity conditions, the Kaplan–Meier estimator enjoys uniform consistency with rates, and a stochastic process generated by the Kaplan–Meier estimator converges weakly to a certain Gaussian process with a specified covariance structure. Finally, an estimator of the limiting variance of the Kaplan–Meier estimator is proposed and its consistency is established.     

Resampling time series using missing values techniques

Several techniques for resampling dependent data have already been proposed. In this paper we use missing values techniques to modify the moving blocks jackknife and bootstrap. More specifically, we consider the blocks of deleted observations in the blockwise jackknife as missing data which are recovered by missing values estimates incorporating the observation dependence structure. Thus, we estimate the variance of a statistic as a weighted sample variance of the statistic evaluated in a “complete” series. Consistency of the variance and the distribution estimators of the sample mean are established. Also, we apply the missing values approach to the blockwise bootstrap by including some missing observations among two consecutive blocks and we demonstrate the consistency of the variance and the distribution estimators of the sample mean. Finally, we present the results of an extensive Monte Carlo study to evaluate the performance of these methods for finite sample sizes, showing that our proposal provides variance estimates for several time series statistics with smaller mean squared error than previous procedures.     

Exact convergence rate of bootstrap quantile variance estimator

Summary It is shown that the relative error of the bootstrap quantile variance estimator is of precise order n ^-1/4, when n denotes sample size. Likewise, the error of the bootstrap sparsity function estimator is of precise order n ^-1/4. Therefore as point estimators these estimators converge more slowly than the Bloch-Gastwirth estimator and kernel estimators, which typically have smaller error of order at most n ^-2/5.