Unbiasedness of two-stage estimation and prediction procedures for mixed linear models

The traditional method for estimating or predicting linear combinations of the fixed effects and realized values of the random effects in mixed linear models is first to estimate the variance components and then to proceed as if the estimated values of the variance components were the true values. This two-stage procedure gives unbiased estimators or predictors of the linear combinations provided the data vector is symmetrically distributed about its expected value and provided the variance component estimators are translation-invariant and are even functions of the data vector. The standard procedures for estimating the variance components yield even, translation-invariant estimators.     

A resampling approach to estimate variance components of multilevel models

In a multilevel model for complex survey data, the weight‐inflated estimators of variance components can be biased. We propose a resampling method to correct this bias. The performance of the bias corrected estimators is studied through simulations using populations generated from a simple random effects model. The simulations show that, without lowering the precision, the proposed procedure can reduce the bias of the estimators, especially for designs that are both informative and have small cluster sizes. Application of these resampling procedures to data from an artificial workplace survey provides further evidence for the empirical value of this method. The Canadian Journal of Statistics 40: 150–171; 2012 © 2012 Statistical Society of Canada     

The Superiorities of Empirical Bayes Estimation of Variance Components in Random Effects Model

In this article, the Bayes estimators of variance components are derived and the parametric empirical Bayes estimators (PEBE) for the balanced one-way classification random effects model are constructed. The superiorities of the PEBE over the analysis of variance (ANOVA) estimators are investigated under the mean square error (MSE) criterion, some simulation results for the PEBE are obtained. Finally, a remark for the main results is given.     

Robust-efficient fitting of mixed linear models: Methodology and theory

In many areas of application mixed linear models serve as a popular tool for analyzing highly complex data sets. For inference about fixed effects and variance components, likelihood-based methods such as (restricted) maximum likelihood estimators, (RE)ML, are commonly pursued. However, it is well-known that these fully efficient estimators are extremely sensitive to small deviations from hypothesized normality of random components as well as to other violations of distributional assumptions. In this article, we propose a new class of robust-efficient estimators for inference in mixed linear models. The new three-step estimation procedure provides truncated generalized least squares and variance components' estimators with hard-rejection weights adaptively computed from the data. More specifically, our data re-weighting mechanism first detects and removes within-subject outliers, then identifies and discards between-subject outliers, and finally it employs maximum likelihood procedures on the “clean” data. Theoretical efficiency and robustness properties of this approach are established.     

Estimating variance components by using survey data

Summary. Inflation-type weighted estimators for variance components can be badly biased. Modified weighted estimators suggested in the literature are also badly biased for certain sampling designs. We propose new estimators for variance components, some of which are approximately unbiased regardless of the sampling design. These estimators require knowledge of the joint inclusion probabilities of the observations. The small sample properties of the estimators are studied via simulation for the simple one-way random-effects model. An application is given by using data from the US Hispanic Health and Nutrition Examination Survey.     

A statistical model for orientation mechanism

A variance components model with response variable depending on both fixed effects of explanatory variables and random components is specified to model longitudinal circular data, in order to study the directional behaviour of small animals, as insects, crustaceans, amphipods, etc. Unknown parameter estimators are obtained using a simulated maximum likelihood approach. Issues concerning log-likelihood variability and the related problems in the optimization algorithm are also addressed. The procedure is applied to the analysis of directional choices under full natural conditions ofTalitrus saltator from Castiglione della Pescaia (Italy) beaches.     

On generalized variance of product of powered components and multiple stable Tweedie models

A flexible family of multivariate models, named multiple stable Tweedie (MST) models, is introduced and produces generalized variance functions which are products of powered components of the mean. These MST models are built from a fixed univariate stable Tweedie variable having a positive value domain, and the remaining random variables given the fixed one are also real independent Tweedie variables, with the same dispersion parameter equal to the fixed component. In this huge family of MST models, generalized variance estimators are explicitly pointed out by maximum likelihood method and, moreover, computably presented for the uniform minimum variance and unbiased approach. The second estimator is brought from modified Lévy measures of MST which lead to some solutions of particular Monge–Ampère equations.     

On Bayesian estimators in multistage binomial designs

A new class of Bayesian estimators for a proportion in multistage binomial designs is considered. Priors belong to the beta-J distribution family, which is derived from the Fisher information associated with the design. The transposition of the beta parameters of the Haldane and the uniform priors in fixed binomial experiments into the beta-J distribution yields bias-corrected versions of these priors in multistage designs. We show that the estimator of the posterior mean based on the corrected Haldane prior and the estimator of the posterior mode based on the corrected uniform prior have good frequentist properties. An easy-to-use approximation of the estimator of the posterior mode is provided. The new Bayesian estimators are compared to Whitehead's and the uniformly minimum variance estimators through several multistage designs. Last, the bias of the estimator of the posterior mode is derived for a particular case.     

On empirical Bayes estimation of variance components in random effects model

In this paper, Bayes estimators of variance components are derived for the one-way random effects model, and empirical Bayes (EB) estimators are constructed by the kernel estimation method of a multivariate density and its mixed partial derivatives. It is shown that the EB estimators are asymptotically optimal and convergence rates are established. Finally, an example concerning the main results is given.     

Quadratic estimators of covariance components in a multivariate mixed linear model

It is known that the Henderson Method III (Biometrics 9:226–252, 1953) is of special interest for the mixed linear models because the estimators of the variance components are unaffected by the parameters of the fixed factor (or factors). This article deals with generalizations and minor extensions of the results obtained for the univariate linear models. A MANOVA mixed model is presented in a convenient form and the covariance components estimators are given on finite dimensional linear spaces. The results use both the usual parametric representations and the coordinate-free approach of Kruskal (Ann Math Statist 39:70–75, 1968) and Eaton (Ann Math Statist 41:528–538, 1970). The normal equations are generalized and it is given a necessary and sufficient condition for the existence of quadratic unbiased estimators for covariance components in the considered model.     

Maximum likelihood estimation of variance components of heteroscedastic random anova model

the estimation of variance components of heteroscedastic random model is discussed in this paper. Maximum Likelihood (ML) is described for one-way heteroscedastic random models. The proportionality condition that cell variance is proportional to the cell sample size, is used to eliminate the efffect of heteroscedasticity. The algebraic expressions of the estimators are obtained for the model. It is seen that the algebraic expressions of the estimators depend mainly on the inverse of the variance-covariance matrix of the observation vector. So, the variance-covariance matrix is obtained and the formulae for the inversions are given. A Monte Carlo study is conducted. Five different variance patterns with different numbers of cells are considered in this study. For each variance pattern, 1000 Monte Carlo samples are drawn. Then the Monte Carlo biases and Monte Carlo MSE’s of the estimators of variance components are calculated. In respect of both bias and MSE, the Maximum Likelihood (ML) estimators of variance components are found to be sufficiently good.     

C239. The continued fraction for lnr( z): a stieljes conjecture

Estimation in logistic-normal models for correlated and overdispersed binomial data is complicated by the numerical evaluation of often intractable likelihood functions. Penalized quasilikelihood (PQL) estimators of fixed effects and variance components are known to be seriously biased for binary data. A simple correction procedure has been proposed to improve the performance of the PQL estimators. The proposed method is illustrated by analyzing infectious disease data. Its performance is compared, by means of simulations, with that of the Bayes approach using the Gibbs sampler.     

Minimum Variance Quadratic Unbiased Estimation for the Variance Components in Simple Linear Regression with Onefold Nested Error

The explicit forms of the minimum variance quadratic unbiased estimators (MIVQUEs) of the variance components are given for simple linear regression with onefold nested error. The resulting estimators are more efficient as the ratio of the initial variance components estimates increases and are asymptotically efficient as the ratio tends to infinity.     

On boundary correction in kernel density estimation

It is well known now that kernel density estimators are not consistent when estimating a density near the finite end points of the support of the density to be estimated. This is due to boundary effects that occur in nonparametric curve estimation problems. A number of proposals have been made in the kernel density estimation context with some success. As of yet there appears to be no single dominating solution that corrects the boundary problem for all shapes of densities. In this paper, we propose a new general method of boundary correction for univariate kernel density estimation. The proposed method generates a class of boundary corrected estimators. They all possess desirable properties such as local adaptivity and non-negativity. In simulation, it is observed that the proposed method perform quite well when compared with other existing methods available in the literature for most shapes of densities, showing a very important robustness property of the method. The theory behind the new approach and the bias and variance of the proposed estimators are given. Results of a data analysis are also given.     

The mixed model for survey regression estimation

Estimation of the population mean under the regression model with random components is considered. Conditions under which the random components regression estimator is design consistent are given. It is shown that consistency holds when incorrect values are used for the variance components. The regression estimator constructed with model parameters that differ considerably from the true parameters performed well in a Monte Carlo study. Variance estimators for the regression predictor are suggested. A variance estimator appropriate for estimators constructed with a biased estimator for the between-group variance component performed well in the Monte Carlo study.     

The corrected VIF (CVIF)

In this paper, we propose a new corrected variance inflation factor (VIF) measure to evaluate the impact of the correlation among the explanatory variables in the variance of the ordinary least squares estimators. We show that the real impact on variance can be overestimated by the traditional VIF when the explanatory variables contain no redundant information about the dependent variable and a corrected version of this multicollinearity indicator becomes necessary.     

Variance components estimation for the balanced random effects model under mixed prior distributions

For the balanced random effects models, when the variance components are correlated either naturally or through common prior structures, by assuming a mixed prior distribution for the variance components, we propose some new Bayesian estimators. To contrast and compare the new estimators with the minimum variance unbiased (MVUE) and restricted maximum likelihood estimators (RMLE), some simulation studies are also carried out. It turns out that the proposed estimators have smaller mean squared errors than the MVUE and RMLE.     

Random effects selection in generalized linear mixed models via shrinkage penalty function

In this paper, we discuss the selection of random effects within the framework of generalized linear mixed models (GLMMs). Based on a reparametrization of the covariance matrix of random effects in terms of modified Cholesky decomposition, we propose to add a shrinkage penalty term to the penalized quasi-likelihood (PQL) function of the variance components for selecting effective random effects. The shrinkage penalty term is taken as a function of the variance of random effects, initiated by the fact that if the variance is zero then the corresponding variable is no longer random (with probability one). The proposed method takes the advantage of a convenient computation for the PQL estimation and appealing properties for certain shrinkage penalty functions such as LASSO and SCAD. We propose to use a backfitting algorithm to estimate the fixed effects and variance components in GLMMs, which also selects effective random effects simultaneously. Simulation studies show that the proposed approach performs quite well in selecting effective random effects in GLMMs. Real data analysis is made using the proposed approach, too.     

A two-stage estimation for panel data models with grouped fixed effects

ABSTRACT

This paper considers panel data models with fixed effects which have grouped patterns with unknown group membership. A two-stage estimation (TSE) procedure is developed to improve the properties of the GFE estimators of common parameters when the time span is small. Firstly, the common parameters are estimated. Subsequently, the optimal group assignment and the estimators of group effects are obtained by the K-means algorithm. Monte Carlo results reveal that the TSE estimator has a much smaller bias than the GFE estimator when the values of difference between effects are moderately small or at high variance of the idiosyncratic error.     

Robustness of Parameter Estimation Procedures in Multilevel Models When Random Effects are MEP Distributed

In this paper we examine maximum likelihood estimation procedures in multilevel models for two level nesting structures. Usually, for fixed effects and variance components estimation, level-one error terms and random effects are assumed to be normally distributed. Nevertheless, in some circumstances this assumption might not be realistic, especially as concerns random effects. Thus we assume for random effects the family of multivariate exponential power distributions (MEP); subsequently, by means of Monte Carlo simulation procedures, we study robustness of maximum likelihood estimators under normal assumption when, actually, random effects are MEP distributed.