Exactly optimal sampling designs for processes with a product covariance structure

The author considers the problem of finding exactly optimal sampling designs for estimating a second‐order, centered random process on the basis of finitely many observations. The value of the process at an unsampled point is estimated by the best linear unbiased estimator. A weighted integrated mean squared error or the maximum mean squared error is used to measure the performance of the estimator. The author presents a set of necessary and sufficient conditions for a design to be exactly optimal for processes with a product covariance structure. Expansions of these conditions lead to conditions for asymptotic optimality.     

Caliper pair-matching on a continuous variable in case-control studies

The use of matched pairs has been criticized as being less efficient than estimators based on random samples. This paper compares the mean square error of an analysis of covariance estimator based on random samples with two estimators based on caliper matched pairs. The first of these is a simple mean difference estimator and the second a regression estimator suggested by Rubin (1973b). Under conditions which commonly occur in epidemiologic case-control studies, both of the matched pair estimators can have smaller mean square errors than analysis o f covariance estimator. When there is a weak relationship between the matching and response variate, the mean difference estimator has a lower mean square error than the regression estimator.     

Estimating multivariate random effects without replication

This paper considers the estimation of multivariate random effects that are measured with error, but for which there are no replications. Using structural simplification of the correlation of the data, separate estimates are generated for the covariance of the random effects and the covariance of the error. An estimator of the random effects based on a truncated eigen structure is defined, and matrix mean squared error and its trace (risk) are analyzed, with comparison to the maximum likelihood estimator (m.l.e) and also to the Stein-like estimator of Efron and Morris (1972). It is shown that the estimator has risk which is smaller than the risk of the maximum likelihood estimator and the Efron-Morris estimator in most cases.     

An Efficient Estimation for Switching Regression Models: A Monte Carlo Study

This article investigates an efficient estimation method for a class of switching regressions based on the characteristic function (CF). We show that with the exponential weighting function, the CF-based estimator can be achieved from minimizing a closed form distance measure. Due to the availability of the analytical structure of the asymptotic covariance, an iterative estimation procedure is developed involving the minimization of a precision measure of the asymptotic covariance matrix. Numerical examples are illustrated via a set of Monte Carlo experiments examining the implementation, finite sample property and the efficiency of the proposed estimator.     

GMM estimation of spatial autoregressive models in a system of simultaneous equations with heteroskedasticity

This paper proposes a GMM estimation framework for the SAR model in a system of simultaneous equations with heteroskedastic disturbances. Besides linear moment conditions, the proposed GMM estimator also utilizes quadratic moment conditions based on the covariance structure of model disturbances within and across equations. Compared with the QML approach, the GMM estimator is easier to implement and robust under heteroskedasticity of unknown form. We derive the heteroskedasticity-robust standard error for the GMM estimator. Monte Carlo experiments show that the proposed GMM estimator performs well in finite samples.     

Mean-Squared errors of small area estimators under a multivariate linear model for repeated measures data

In this paper, we discuss the derivation of the first and second moments for the proposed small area estimators under a multivariate linear model for repeated measures data. The aim is to use these moments to estimate the mean-squared errors (MSE) for the predicted small area means as a measure of precision. At the first stage, we derive the MSE when the covariance matrices are known. At the second stage, a method based on parametric bootstrap is proposed for bias correction and for prediction error that reflects the uncertainty when the unknown covariance is replaced by its suitable estimator.     

Estimation of Fibre Length Distributions from Fibre Endpoints

Estimating the fibre length distribution in composite materials is of practical relevance in materials science. We propose an estimator for the fibre length distribution using the point process of fibre endpoints as input. Assuming that this point process is a realization of a Neyman–Scott process, we use results for the reduced second moment measure to derive a consistent and unbiased estimator for the fibre length distribution. We introduce various versions of the estimator taking anisotropy or errors in the observation into account. The estimator is evaluated using a heuristic for its mean squared error as well as a simulation study. Finally, the estimator is applied to the fibre endpoint process extracted from a tomographic image of a glass fibre composite.     

Small sample behavior of a robust heteroskedasticity consistent covariance matrix estimator

In heteroskedastic regression models, the least squares (OLS) covariance matrix estimator is inconsistent and inference is not reliable. To deal with inconsistency one can estimate the regression coefficients by OLS, and then implement a heteroskedasticity consistent covariance matrix (HCCM) estimator. Unfortunately the HCCM estimator is biased. The bias is reduced by implementing a robust regression, and by using the robust residuals to compute the HCCM estimator (RHCCM). A Monte-Carlo study analyzes the behavior of RHCCM and of other HCCM estimators, in the presence of systematic and random heteroskedasticity, and of outliers in the explanatory variables.     

A polar coordinate transformation for estimating bivariate survival functions with randomly censored and truncated data

This paper proposes a new estimator for bivariate distribution functions under random truncation and random censoring. The new method is based on a polar coordinate transformation, which enables us to transform a bivariate survival function to a univariate survival function. A consistent estimator for the transformed univariate function is proposed. Then the univariate estimator is transformed back to a bivariate estimator. The estimator converges weakly to a zero-mean Gaussian process with an easily estimated covariance function. Consistent truncation probability estimate is also provided. Numerical studies show that the distribution estimator and truncation probability estimator perform remarkably well.     

The asymptotic moment profile and maximum likelihood: Applications to gamma and gamma ratio densities

In previous papers ( Bowman and Shenton, 1998, 1999a ) we have given expressions for the asymptotic skewness and kurtosis for maximum likelihood estimators in the case of several parameters. Skewness is measured by the third standardized central moment, and kurtosis by the fourth standardized central moment. Moments of the basic structure are assumed to exist. The overarching entity is the covariance matrix ( Hessian form ), and elements of its inverse. These entities involve Stieltjes integrals relating to sums of products of multiple derivatives linked to the basic structure. The first paper dealt with skewness and gives a simple expression read¬ily computerized. The second paper is devoted to the forth standardized central moment and although a certain simplification is discovered, the resulting formula is still somewhat complicated, ft is surprising to find that the asymptotic kurtosis in general requires the evaluation of several hundred components. The present paper studies cases involving estimator for two parameter gamma and one, and three pa¬rameter gamma ratio densities, and mentions strategies aimed at avoiding algebraic and numerical errors.     

ON ESTIMATION OF THE POPULATION SPECTRAL DISTRIBUTION FROM A HIGH‐DIMENSIONAL SAMPLE COVARIANCE MATRIX

Sample covariance matrices play a central role in numerous popular statistical methodologies, for example principal components analysis, Kalman filtering and independent component analysis. However, modern random matrix theory indicates that, when the dimension of a random vector is not negligible with respect to the sample size, the sample covariance matrix demonstrates significant deviations from the underlying population covariance matrix. There is an urgent need to develop new estimation tools in such cases with high‐dimensional data to recover the characteristics of the population covariance matrix from the observed sample covariance matrix. We propose a novel solution to this problem based on the method of moments. When the parametric dimension of the population spectrum is finite and known, we prove that the proposed estimator is strongly consistent and asymptotically Gaussian. Otherwise, we combine the first estimation method with a cross‐validation procedure to select the unknown model dimension. Simulation experiments demonstrate the consistency of the proposed procedure. We also indicate possible extensions of the proposed estimator to the case where the population spectrum has a density.     

Asymptotic normality of kernel type regression estimators for random fields

The asymptotic normality of the Nadaraya–Watson regression estimator is studied for α-mixing$\alpha \text{-}\mathrm{mixing}$ random fields. The infill-increasing setting is considered, that is when the locations of observations become dense in an increasing sequence of domains. This setting fills the gap between continuous and discrete models. In the infill-increasing case the asymptotic normality of the Nadaraya–Watson estimator holds, but with an unusual asymptotic covariance structure. It turns out that this covariance structure is a combination of the covariance structures that we observe in the discrete and in the continuous case.     

Reduction of n-th moment measures and the special case of the third moment measure of stationary and isotropic planar point processes

Formulae for the reduction of n-th moment measures of stationary point processes and an unbiased estimator for the reduced n-th moment measure are given. In particular, the special case of the third moment measure of planar point processes is considered and the application of the quantities studied is illustrated by an example.     

Robust estimation of mean and covariance for longitudinal data with dropouts

In this paper, we study estimation of linear models in the framework of longitudinal data with dropouts. Under the assumptions that random errors follow an elliptical distribution and all the subjects share the same within-subject covariance matrix which does not depend on covariates, we develop a robust method for simultaneous estimation of mean and covariance. The proposed method is robust against outliers, and does not require to model the covariance and missing data process. Theoretical properties of the proposed estimator are established and simulation studies show its good performance. In the end, the proposed method is applied to a real data analysis for illustration.     

Weighted quantile regression with missing covariates using empirical likelihood

This paper proposes an empirical likelihood-based weighted (ELW) quantile regression approach for estimating the conditional quantiles when some covariates are missing at random. The proposed ELW estimator is computationally simple and achieves semiparametric efficiency if the probability of missingness is correctly specified. The limiting covariance matrix of the ELW estimator can be estimated by a resampling technique, which does not involve nonparametric density estimation or numerical derivatives. Simulation results show that the ELW method works remarkably well in finite samples. A real data example is used to illustrate the proposed ELW method.     

Form and selection of covariance adjusted estimators in repeated measures models

The GMANOVA model is considered when one characteristic is measured at q time points. The covariance adjusted estimator is the OLS estimator adjusted using analysis of covariance. The covariates are obtained from the space vertical to that derived from the design matrix. Its form is obtained as a GLS estimator using a weight matrix of reduced rank unless all available covariates are used. The choice of the appropriate covariate combination is made by introducing a method based on cross validation (CV). A comparison is made with two other methods that appeared in statistical literature using simulation.     

Near Optimal Prediction from Relevant Components

Abstract. The random x regression model is approached through the group of rotations of the eigenvectors for the x ‐covariance matrix together with scale transformations for each of the corresponding regression coefficients. The partial least squares model can be constructed from the orbits of this group. A generalization of Pitman's Theorem says that the best equivariant estimator under a group is given by the Bayes estimator with the group's invariant measure as the prior. A straightforward application of this theorem turns out to be impossible since the relevant invariant prior leads to a non‐defined posterior. Nevertheless we can devise an approximate scale group with a proper invariant prior leading to a well‐defined posterior distribution with a finite mean. This Bayes estimator is explored using Markov chain Monte Carlo technique. The estimator seems to require heavy computations, but can be argued to have several nice properties. It is also a valid estimator when p>n.     

Selection Strategy for Covariance Structure of Random Effects in Linear Mixed‐effects Models

Linear mixed‐effects models are a powerful tool for modelling longitudinal data and are widely used in practice. For a given set of covariates in a linear mixed‐effects model, selecting the covariance structure of random effects is an important problem. In this paper, we develop a joint likelihood‐based selection criterion. Our criterion is the approximately unbiased estimator of the expected Kullback–Leibler information. This criterion is also asymptotically optimal in the sense that for large samples, estimates based on the covariance matrix selected by the criterion minimize the approximate Kullback–Leibler information. Finite sample performance of the proposed method is assessed by simulation experiments. As an illustration, the criterion is applied to a data set from an AIDS clinical trial.     

On the natural restrictions in the singular Gauss–Markov model

A Gauss–Markov model is said to be singular if the covariance matrix of the observable random vector in the model is singular. In such a case, there exist some natural restrictions associated with the observable random vector and the unknown parameter vector in the model. In this paper, we derive through the matrix rank method a necessary and sufficient condition for a vector of parametric functions to be estimable, and necessary and sufficient conditions for a linear estimator to be unbiased in the singular Gauss–Markov model. In addition, we give some necessary and sufficient conditions for the ordinary least-square estimator (OLSE) and the best linear unbiased estimator (BLUE) under the model to satisfy the natural restrictions.      

Monte Carlo integration with Markov chain

There are two conceptually distinct tasks in Markov chain Monte Carlo (MCMC): a sampler is designed for simulating a Markov chain and then an estimator is constructed on the Markov chain for computing integrals and expectations. In this article, we aim to address the second task by extending the likelihood approach of Kong et al. for Monte Carlo integration. We consider a general Markov chain scheme and use partial likelihood for estimation. Basically, the Markov chain scheme is treated as a random design and a stratified estimator is defined for the baseline measure. Further, we propose useful techniques including subsampling, regulation, and amplification for achieving overall computational efficiency. Finally, we introduce approximate variance estimators for the point estimators. The method can yield substantially improved accuracy compared with Chib's estimator and the crude Monte Carlo estimator, as illustrated with three examples.