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).     

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.     

Iterative Estimation of the Extreme Value Index

Let {X_n, n ≥ 1} be a sequence of independent random variables with common continuous distribution function F having finite and unknown upper endpoint. A new iterative estimation procedure for the extreme value index γ is proposed and one implemented iterative estimator is investigated in detail, which is asymptotically as good as the uniform minimum varianced unbiased estimator in an ideal model. Moreover, the superiority of the iterative estimator over its non iterated counterpart in the non asymptotic case is shown in a simulation study.AMS 2000 Subject Classification: 62G32Supported by Swiss National Science foundation.     

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.     

Statistical estimation in partial linear models with covariate data missing at random

In this paper, we consider the partial linear model with the covariables missing at random. A model calibration approach and a weighting approach are developed to define the estimators of the parametric and nonparametric parts in the partial linear model, respectively. It is shown that the estimators for the parametric part are asymptotically normal and the estimators of g(·) converge to g(·) with an optimal convergent rate. Also, a comparison between the proposed estimators and the complete case estimator is made. A simulation study is conducted to compare the finite sample behaviors of these estimators based on bias and standard error.     

Confidence intervals for quantile estimation using Jackknife techniques

We consider the inference on quantiles, Q _y (β), with jackknife techniques, in finite populations of a variable, Y, using the quantile information on an auxiliary variable, X. Jackknife techniques are applied to estimate quantiles and the behaviour of these estimators is analyzed. Their properties are studied for simple random sampling. We also examine the confidence intervals obtained with jackknife variances.     

Second order asymptotic comparison of estimators of a common parameter in the double exponential case

Summary The problem to estimate a common parameter for the pooled sample from the double exponential distributions is discussed in the presence of nuisance parameters. The maximum likelihood estimator, a weighted median, a weighted mean and others are asymptotically compared up to the second order, i.e. the ordern ^−1/2 with the asymptotic expansions of their distributions. University of Electro-communications     

Two-stage local M-estimation of additive models

This paper studies local M-estimation of the nonparametric components of additive models.A two-stage local M-estimation procedure is proposed for estimating the additive components and their derivatives.Under very mild conditions,the proposed estimators of each additive component and its derivative are jointly asymptotically normal and share the same asymptotic distributions as they would be if the other components were known.The established asymptotic results also hold for two particular local M-estimations:the local least squares and least absolute deviation estimations.However,for general two-stage local M-estimation with continuous and nonlinear ψ-functions,its implementation is time-consuming.To reduce the computational burden,one-step approximations to the two-stage local M-estimators are developed.The one-step estimators are shown to achieve the same effciency as the fully iterative two-stage local M-estimators,which makes the two-stage local M-estimation more feasible in practice.The proposed estimators inherit the advantages and at the same time overcome the disadvantages of the local least-squares based smoothers.In addition,the practical implementation of the proposed estimation is considered in details.Simulations demonstrate the merits of the two-stage local M-estimation,and a real example illustrates the performance of the methodology.     

<Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-nearest neighbor estimators of entropy

For estimating the entropy of an absolutely continuous multivariate distribution, we propose nonparametric estimators based on the Euclidean distances between the n sample points and their k _n -nearest neighbors, where {k _n : n = 1, 2, …} is a sequence of positive integers varying with n. The proposed estimators are shown to be asymptotically unbiased and consistent.      

交替迭代最大似然估计量的偏倚校正

In this paper, we give a definition of the alternating iterative maximum likelihood estimator (AIMLE) which is a biased estimator. Furthermore we adjust the AIMLE to result in asymptotically unbiased and consistent estimators by using a bootstrap iterative bias correction method as in Kuk (1995). Two examples and simulation results reported illustrate the performance of the bias correction for AIMLE.     

Nonparametric density estimation for linear processes with infinite variance

We consider nonparametric estimation of marginal density functions of linear processes by using kernel density estimators. We assume that the innovation processes are i.i.d. and have infinite-variance. We present the asymptotic distributions of the kernel density estimators with the order of bandwidths fixed as h =  cn ^−1/5, where n is the sample size. The asymptotic distributions depend on both the coefficients of linear processes and the tail behavior of the innovations. In some cases, the kernel estimators have the same asymptotic distributions as for i.i.d. observations. In other cases, the normalized kernel density estimators converge in distribution to stable distributions. A simulation study is also carried out to examine small sample properties.     

Weighted L quantile distance estimators for randomly censored data

The asymptotic properties of a family of minimum quantile distance estimators for randomly censored data sets are considered. These procedures produce an estimator of the parameter vector that minimizes a weighted L² distance measure between the Kaplan-Meier quantile function and an assumed parametric family of quantile functions. Regularity conditions are provided which insure that these estimators are consistent and asymptotically normal. An optimal weight function is derived for single parameter families, which, for location/scale families, results in censored sample analogs of estimators such as those suggested by Parzen.     

On the estimation of entropy

Motivated by recent work of Joe (1989,Ann. Inst. Statist. Math.,41, 683–697), we introduce estimators of entropy and describe their properties. We study the effects of tail behaviour, distribution smoothness and dimensionality on convergence properties. In particular, we argue that root-n consistency of entropy estimation requires appropriate assumptions about each of these three features. Our estimators are different from Joe's, and may be computed without numerical integration, but it can be shown that the same interaction of tail behaviour, smoothness and dimensionality also determines the convergence rate of Joe's estimator. We study both histogram and kernel estimators of entropy, and in each case suggest empirical methods for choosing the smoothing parameter.     

Minimum f-divergence estimators and quasi-likelihood functions

Maximum quasi-likelihood estimators have several nice asymptotic properties. We show that, in many situations, a family of estimators, called the minimum f-divergence estimators, can be defined such that each estimator has the same asymptotic properties as the maximum quasi-likelihood estimator. The family of minimum f-divergence estimators include the maximum quasi-likelihood estimators as a special case. When a quasi-likelihood is the log likelihood from some exponential family, Amari's dual geometries can be used to study the maximum likelihood estimator. A dual geometric structure can also be defined for more general quasi-likelihood functions as well as for the larger family of minimum f-divergence estimators. The relationship between the f-divergence and the quasi-likelihood function and the relationship between the f-divergence and the power divergence is discussed.This work was supported by National Science Foundation grant DMS 88-03584.     

Third order efficiency implies fourth order efficiency: A resolution of the conjecture of J. K. Ghosh

Under suitable regularity conditions, it is shown that a third order asymptotically efficient estimator is fourth order asymptotically efficient in some class of estimators in the sense that the estimator has the most concentration probability in any symmetric interval around the true parameter up to the fourth order in the class. This is a resolution of the conjecture by Ghosh (1994, Higher Order Asymptotics, Institute of Mathematical Statistics, Hayward, California). It is also shown that the bias-adjusted maximum likelihood estimator is fourth order asymptotically efficient in the class.     

Estimation of intrinsic growth factors in a class of stochastic population model

This article discusses the problem of parameter estimation with nonlinear mean-reversion type stochastic differential equations (SDEs) driven by Brownian motion for population growth model. The estimator in the population model is the climate effects, population policy and environmental circumstances which affect the intrinsic rate of growth r. The consistency and asymptotic distribution of the estimator θ is studied in our general setting. In the calculation method, unlike previous study, since the nonlinear feature of the model, it is difficult to obtain an explicit formula for the estimator. To solve this, some criteria are used to derive an asymptotically consistent estimator. Furthermore Girsanov transformation is used to simplify the equations, which then gives rise to the corresponding convergence of the estimator being with respect to a family of probability measures indexed by the dispersion parameter, while in the literature the existing results have dealt with convergence with respect to a given probability measure.     

Efficient and fast spline-backfitted kernel smoothing of additive models

A great deal of effort has been devoted to the inference of additive model in the last decade. Among existing procedures, the kernel type are too costly to implement for high dimensions or large sample sizes, while the spline type provide no asymptotic distribution or uniform convergence. We propose a one step backfitting estimator of the component function in an additive regression model, using spline estimators in the first stage followed by kernel/local linear estimators. Under weak conditions, the proposed estimator’s pointwise distribution is asymptotically equivalent to an univariate kernel/local linear estimator, hence the dimension is effectively reduced to one at any point. This dimension reduction holds uniformly over an interval under assumptions of normal errors. Monte Carlo evidence supports the asymptotic results for dimensions ranging from low to very high, and sample sizes ranging from moderate to large. The proposed confidence band is applied to the Boston housing data for linearity diagnosis. Supported in part by NSF awards DMS 0405330, 0706518, BCS 0308420 and SES 0127722.     

Testing for serial correlation of unknown form in cointegrated time series models

Portmanteau test statistics are useful for checking the adequacy of many time series models. Here we generalized the omnibus procedure proposed by Duchesne and Roy (2004,Journal of Multivariate Analysis,89, 148–180) for multivariate stationary autoregressive models with exogenous variables (VARX) to the case of cointegrated (or partially nonstationary) VARX models. We show that for cointegrated VARX time series, the test statistic obtained by comparing the spectral density of the errors under the null hypothesis of non-correlation with a kernel-based spectral density estimator, is asymptotically standard normal. The parameters of the model can be estimated by conditional maximum likelihood or by asymptotically equivalent estimation procedures. The procedure relies on a truncation point or a smoothing parameter. We state conditions under which the asymptotic distribution of the test statistic is unaffected by a data-dependent method. The finite sample properties of the test statistics are studied via a small simulation study.     

Empirical Bayes estimation in a multiple linear regression model

Summary Estimation of the vector β of the regression coefficients in a multiple linear regressionY=Xβ+ε is considered when β has a completely unknown and unspecified distribution and the error-vector ε has a multivariate standard normal distribution. The optimal estimator for β, which minimizes the overall mean squared error, cannot be constructed for use in practice. UsingX, Y and the information contained in the observation-vectors obtained fromn independent past experiences of the problem, (empirical Bayes) estimators for β are exhibited. These estimators are compared with the optimal estimator and are shown to be asymptotically optimal. Estimators asymptotically optimal with rates nearO(n ⁻¹) are constructed. Supported in part by a Natural Sciences and Engineering Research Council of Canada grant.     

A Strong Representation of the Product-Limit Estimator for Left Truncated and Right Censored Data

In this paper we consider the TJW product-limit estimatorF_n(x) of an unknown distribution functionFwhen the data are subject to random left truncation and right censorship. An almost sure representation of PL-estimatorF_n(x) is derived with an improved error bound under some weaker assumptions. We obtain the strong approximation ofF_n(x)−F(x) by Gaussian processes and the functional law of the iterated logarithm is proved for maximal derivation of the product-limit estimator toF. A sharp rate of convergence theorem concerning the smoothed TJW product-limit estimator is obtained. Asymptotic properties of kernel estimators of density function based on TJW product-limit estimator is given.