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.     

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 estimator for anm-dependent stationary process

The use of the jackknife method is successful in many situations. However, when the observations are from anm-dependent stationary process, the ordinary jackknife may provide an inconsistent variance estimator. It is shown in this note that this deficiency of the jackknife can be rectified and the jackknife variance estimator proposed is strongly consistent.     

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.     

Asymptotic expansions associated with the jackknife functional. I

We obtain an asymptotic expansion of the functional of the jackknife method, which is used for the estimation of the variance of observational errors in a nonlinear regression model.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 443–451, April, 1995.     

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.     

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.     

Smoothed jackknife empirical likelihood method for ROC curve

In this paper we propose a smoothed jackknife empirical likelihood method to construct confidence intervals for the receiver operating characteristic (ROC) curve. By applying the standard empirical likelihood method for a mean to the jackknife sample, the empirical likelihood ratio statistic can be calculated by simply solving a single equation. Therefore, this procedure is easy to implement. Wilks’ theorem for the empirical likelihood ratio statistic is proved and a simulation study is conducted to compare the performance of the proposed method with other methods.     

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).     

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.     

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].     

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.     

On the jackknife statistics for eigenvalues and eigenvectors of a correlation matrix

This paper deals with some problems of eigenvalues and eigenvectors of a sample correlation matrix and derives the limiting distributions of their jackknife statistics with some numerical examples.     

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.     

Balanced augmented jackknife empirical likelihood for two sample <Emphasis Type="Italic">U</Emphasis>-statistics

In this paper, we investigate the two sample U-statistics by jackknife empirical likelihood (JEL), a versatile nonparametric approach. More precisely, we propose the method of balanced augmented jackknife empirical likelihood (BAJEL) by adding two articial points to the original pseudo-value dataset, and we prove that the log likelihood ratio based on the expanded dataset tends to the χ² distribution.     

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.     

无穷方差厚尾过程中非参数函数的小波估计

研究无穷方差厚尾过程中含有变点的非参数函数的估计问题。通过小波方法给出变点位置的估计值并得到其收敛速度。在已知变点估计值的基础上,将截尾方法与小波压缩方法相结合得到非参数函数的估计值。模拟研究结果说明对于无穷方差厚尾过程中的函数估计问题小波方法是有效的。     

Estimating the intensity of a cyclic Poisson process in the presence of linear trend

We construct and investigate a consistent kernel-type nonparametric estimator of the intensity function of a cyclic Poisson process in the presence of linear trend. It is assumed that only a single realization of the Poisson process is observed in a bounded window. We prove that the proposed estimator is consistent when the size of the window indefinitely expands. The asymptotic bias, variance, and the mean-squared error of the proposed estimator are also computed. A simulation study shows that the first order asymptotic approximations to the bias and variance of the estimator are not accurate enough. Second order terms for bias and variance were derived in order to be able to predict the numerical results in the simulation. Bias reduction of our estimator is also proposed.     

Optimal variance estimation based on lagged second-order difference in nonparametric regression

Differenced estimators of variance bypass the estimation of regression function and thus are simple to calculate. However, there exist two problems: most differenced estimators do not achieve the asymptotic optimal rate for the mean square error; for finite samples the estimation bias is also important and not further considered. In this paper, we estimate the variance as the intercept in a linear regression with the lagged Gasser-type variance estimator as dependent variable. For the equidistant design, our estimator is not only \(n^{1/2}\)-consistent and asymptotically normal, but also achieves the optimal bound in terms of estimation variance with less asymptotic bias. Simulation studies show that our estimator has less mean square error than some existing differenced estimators, especially in the cases of immense oscillation of regression function and small-sized sample.