Reduced‐bias kernel estimators of a positive extreme value index

In this paper, we deal with the semi‐parametric estimation of the extreme value index, an important parameter in extreme value analysis. It is well known that many classic estimators, such as the Hill estimator, reveal a strong bias. This problem motivated the study of two classes of kernel estimators. Those classes generalize the classical Hill estimator and have a tuning parameter that enables us to modify the asymptotic mean squared error and eventually to improve their efficiency. Since the improvement in efficiency is not very expressive, we also study new reduced bias estimators based on the two classes of kernel statistics. Under suitable conditions, we prove their asymptotic normality. Moreover, an asymptotic comparison, at optimal levels, shows that the new classes of reduced bias estimators are more efficient than other reduced bias estimator from the literature. An illustration of the finite sample behaviour of the kernel reduced‐bias estimators is also provided through the analysis of a data set in the field of insurance.     

Cross-validated bagged learning

Many applications aim to learn a high dimensional parameter of a data generating distribution based on a sample of independent and identically distributed observations. For example, the goal might be to estimate the conditional mean of an outcome given a list of input variables. In this prediction context, bootstrap aggregating (bagging) has been introduced as a method to reduce the variance of a given estimator at little cost to bias. Bagging involves applying an estimator to multiple bootstrap samples and averaging the result across bootstrap samples. In order to address the curse of dimensionality, a common practice has been to apply bagging to estimators which themselves use cross-validation, thereby using cross-validation within a bootstrap sample to select fine-tuning parameters trading off bias and variance of the bootstrap sample-specific candidate estimators. In this article we point out that in order to achieve the correct bias variance trade-off for the parameter of interest, one should apply the cross-validation selector externally to candidate bagged estimators indexed by these fine-tuning parameters. We use three simulations to compare the new cross-validated bagging method with bagging of cross-validated estimators and bagging of non-cross-validated estimators.     

Second-order risk comparison of SLSE with GLSE and MLE in a regression with serial correlation

First, the second-order bias of the estimator of the autoregressive parameter based on the ordinary least squares residuals in a linear model with serial correlation is given. Second, the second-order expansion of the risk matrix of a generalized least squares estimator with the above estimated parameter is obtained. This expansion is the same as that based on a suitable estimator of the autoregressive parameter independent of the sample. Third, it is shown that the risk matrix of the generalized least squares estimator is asymptotically equivalent to that of the maximum likelihood estimator up to the second order. Last, a sufficient condition is given for the term due to the estimation of the autoregressive parameter in this expansion to vanish under Grenander's condition for the explanatory variates.     

Improved estimation of a covariance matrix in an elliptically contoured matrix distribution

In this paper, the problem of estimating the covariance matrix of the elliptically contoured distribution (ECD) is considered. A new class of estimators which shrink the eigenvalues towards their arithmetic mean is proposed. It is shown that this new estimator dominates the unbiased estimator under the squared error loss function. Two special classes of ECD, namely, the multivariate-elliptical t distribution and the ε-contaminated normal distribution are considered. A simulation study is carried out and indicates that this new shrinkage estimator provides a substantial improvement in risk under most situations.     

Estimation of the eigenvalues of noncentrality parameter matrix in noncentral Wishart distribution

We consider the problem of estimating the eigenvalues of noncentrality parameter matrix in noncentral Wishart distribution when the scale parameter is known. A decision theoretic approach is taken with squared error as the loss function. We propose two new estimators and show their superior performance to an usual estimator theoretically and numerically.     

Wavelet estimation of conditional density with truncated, censored and dependent data

In this paper we define a new nonlinear wavelet-based estimator of conditional density function for a random left truncation and right censoring model. We provide an asymptotic expression for the mean integrated squared error (MISE) of the estimator. It is assumed that the lifetime observations form a stationary α-mixing sequence. Unlike for kernel estimators, the MISE expression of the wavelet-based estimators is not affected by the presence of discontinuities in the curves. Also, asymptotic normality of the estimator is established.     

Maximum integrated likelihood estimator of the interest parameter when the nuisance parameter is location or scale

The problem of estimation of an interest parameter in the presence of a nuisance parameter, which is either location or scale, is studied. Two estimators are considered: the usual maximum likelihood estimator and the estimator based on maximization of the integrated likelihood function. The estimators are compared, asymptotically, with respect to the bias and with respect to the mean squared error. The examples are given.     

Asymptotic properties of conditional maximum likelihood estimator in a certain exponential model

The conditional maximum likelihood estimator is suggested as an alternative to the maximum likelihood estimator and is favorable for an estimator of a dispersion parameter in the normal distribution, the inverse-Gaussian distribution, and so on. However, it is not clear whether the conditional maximum likelihood estimator is asymptotically efficient in general. Consider the case where it is asymptotically efficient and its asymptotic covariance depends only on an objective parameter in an exponential model. This remand implies that the exponential model possesses a certain parallel foliation. In this situation, this paper investigates asymptotic properties of the conditional maximum likelihood estimator and compares the conditional maximum likelihood estimator with the maximum likelihood estimator. We see that the bias of the former is more robust than that of the latter and that two estimators are very close, especially in the sense of bias-corrected version. The mean Pythagorean relation is also discussed.     

IMPROVED ESTIMATES OF THE COVARIANCE MATRIX IN GENERAL LINEAR MIXED MODELS

In this article, the problem of estimating the covariance matrix in general linear mixed models is considered. Two new classes of estimators obtained by shrinking the eigenvalues towards the origin and the arithmetic mean, respectively, are proposed. It is shown that these new estimators dominate the unbiased estimator under the squared error loss function. Finally, some simulation results to compare the performance of the proposed estimators with that of the unbiased estimator are reported. The simulation results indicate that these new shrinkage estimators provide a substantial improvement in risk under most situations.     

Estimation of the Scale Matrix and its Eigenvalues in the Wishart and the Multivariate F Distributions

In this paper, the problem of estimating the scale matrix and their eigenvalues in a Wishart distribution and in a multivariate F distribution (which arise naturally from a two-sample setting) are considered. A new class of estimators which shrink the eigenvalues towards their arithmetic mean are proposed. It is shown that the new estimator which dominates the usual unbiased estimator under the squared error loss function. A simulation study was carried out to study the performance of these estimators.     

Properties of nonparametric estimators of autocovariance for stationary random fields

Summary We introduce nonparametric estimators of the autocovariance of a stationary random field. One of our estimators has the property that it is itself an autocovatiance. This feature enables the estimator to be used as the basis of simulation studies such as those which are necessary when constructing bootstrap confidence intervals for unknown parameters. Unlike estimators proposed recently by other authors, our own do not require assumptions such as isotropy or monotonicity. Indeed, like nonparametric function estimators considered more widely in the context of curve estimation, our approach demands only smoothness and tail conditions on the underlying curve or surface (here, the autocovariance), and moment and mixing conditions on the random field. We show that by imposing the condition that the estimator be a covariance function we actually reduce the numerical value of integrated squared error.     

Methods for improvement in estimation of a normal mean matrix

This paper is concerned with the problem of estimating a matrix of means in multivariate normal distributions with an unknown covariance matrix under invariant quadratic loss. It is first shown that the modified Efron-Morris estimator is characterized as a certain empirical Bayes estimator. This estimator modifies the crude Efron-Morris estimator by adding a scalar shrinkage term. It is next shown that the idea of this modification provides a general method for improvement of estimators, which results in the further improvement on several minimax estimators. As a new method for improvement, an adaptive combination of the modified Stein and the James-Stein estimators is also proposed and is shown to be minimax. Through Monte Carlo studies of the risk behaviors, it is numerically shown that the proposed, combined estimator inherits the nice risk properties of both individual estimators and thus it has a very favorable risk behavior in a small sample case. Finally, the application to a two-way layout MANOVA model with interactions is discussed.     

Pre-averaged kernel estimators for the drift function of a diffusion process in the presence of microstructure noise

We consider estimation of the drift function of a stationary diffusion process when we observe high-frequency data with microstructure noise over a long time interval. We propose to estimate the drift function at a point by a Nadaraya–Watson estimator that uses observations that have been pre-averaged to reduce the noise. We give conditions under which our estimator is consistent and asympotically normal. Its rate and asymptotic bias and variance are the same as those without microstructure noise. To use our method in data analysis, we propose a data-based cross-validation method to determine the bandwidth in the Nadaraya–Watson estimator. Via simulation, we study several methods of bandwidth choices, and compare our estimator to several existing estimators. In terms of mean squared error, our new estimator outperforms existing estimators.     

Hazard function estimation with cause-of-death data missing at random

Hazard function estimation is an important part of survival analysis. Interest often centers on estimating the hazard function associated with a particular cause of death. We propose three nonparametric kernel estimators for the hazard function, all of which are appropriate when death times are subject to random censorship and censoring indicators can be missing at random. Specifically, we present a regression surrogate estimator, an imputation estimator, and an inverse probability weighted estimator. All three estimators are uniformly strongly consistent and asymptotically normal. We derive asymptotic representations of the mean squared error and the mean integrated squared error for these estimators and we discuss a data-driven bandwidth selection method. A simulation study, conducted to assess finite sample behavior, demonstrates that the proposed hazard estimators perform relatively well. We illustrate our methods with an analysis of some vascular disease data.     

An efficient shrinkage bootstrap bias estimator for smooth functions of sample means

Summary  Since it is not always possible to calculate bootstrap estimators, they are usually approximated by simulation. In this article, we propose a bootstrap bias estimator for smooth functions of sample means that has less mean squared error, due to the simulation process, than the ordinary bootstrap. The estimator is based on shrinking the bootstrap mean towards the original sample mean. It can easily be implemented while demanding almost no additional computational effort.     

Variance function estimation in multivariate nonparametric regression with fixed design

Variance function estimation in multivariate nonparametric regression is considered and the minimax rate of convergence is established in the iid Gaussian case. Our work uses the approach that generalizes the one used in [A. Munk, Bissantz, T. Wagner, G. Freitag, On difference based variance estimation in nonparametric regression when the covariate is high dimensional, J. R. Stat. Soc. B 67 (Part 1) (2005) 19-41] for the constant variance case. As is the case when the number of dimensions d=1, and very much contrary to standard thinking, it is often not desirable to base the estimator of the variance function on the residuals from an optimal estimator of the mean. Instead it is desirable to use estimators of the mean with minimal bias. Another important conclusion is that the first order difference based estimator that achieves minimax rate of convergence in the one-dimensional case does not do the same in the high dimensional case. Instead, the optimal order of differences depends on the number of dimensions.     

左截断右删失数据分位差估计及其渐近性质

我们研究了左截断右删失数据分位差,基于左截断右删失数据乘积限构造了分位差的经验估计,同时克服经验估计的非光滑性,提出了分位数差的核光滑估计.利用经验过程理论推导出这两个估计的渐近偏差和渐近方差,并且在左截断右删失数据下研究了这两个分位差的大样本性质,获得分位差估计的相合性和渐近正态性.同时给出计算模拟以验证光滑分位差估计的表现,在均方损失的意义下模拟结果表明光滑估计比经验估计具有更好的性质.     

Estimating the covariance matrix: a new approach

In this paper, we consider the problem of estimating the covariance matrix and the generalized variance when the observations follow a nonsingular multivariate normal distribution with unknown mean. A new method is presented to obtain a truncated estimator that utilizes the information available in the sample mean matrix and dominates the James-Stein minimax estimator. Several scale equivariant minimax estimators are also given. This method is then applied to obtain new truncated and improved estimators of the generalized variance; it also provides a new proof to the results of Shorrock and Zidek (Ann. Statist. 4 (1976) 629) and Sinha (J. Multivariate Anal. 6 (1976) 617).     

Second-order nonlinear least squares estimation

The ordinary least squares estimation is based on minimization of the squared distance of the response variable to its conditional mean given the predictor variable. We extend this method by including in the criterion function the distance of the squared response variable to its second conditional moment. It is shown that this “second-order” least squares estimator is asymptotically more efficient than the ordinary least squares estimator if the third moment of the random error is nonzero, and both estimators have the same asymptotic covariance matrix if the error distribution is symmetric. Simulation studies show that the variance reduction of the new estimator can be as high as 50% for sample sizes lower than 100. As a by-product, the joint asymptotic covariance matrix of the ordinary least squares estimators for the regression parameter and for the random error variance is also derived, which is only available in the literature for very special cases, e.g. that random error has a normal distribution. The results apply to both linear and nonlinear regression models, where the random error distributions are not necessarily known.     

Efficiency of a Liu-type estimator in semiparametric regression models

In this paper we consider the semiparametric regression model, y=Xβ+f+ε. Recently, Hu [11] proposed ridge regression estimator in a semiparametric regression model. We introduce a Liu-type (combined ridge-Stein) estimator (LTE) in a semiparametric regression model. Firstly, Liu-type estimators of both β and f are attained without a restrained design matrix. Secondly, the LTE estimator of β is compared with the two-step estimator in terms of the mean square error. We describe the almost unbiased Liu-type estimator in semiparametric regression models. The almost unbiased Liu-type estimator is compared with the Liu-type estimator in terms of the mean squared error matrix. A numerical example is provided to show the performance of the estimators.