Uniformly minimum variance nonnegative quadratic unbiased estimation in a generalized growth curve model

Consider the generalized growth curve model subject to R(X_m)⊆?⊆R(X₁), where B_i are the matrices of unknown regression coefficients, and E=(ε₁,…,ε_s)^′ and are independent and identically distributed with the same first four moments as a random vector normally distributed with mean zero and covariance matrix Σ. We derive the necessary and sufficient conditions under which the uniformly minimum variance nonnegative quadratic unbiased estimator (UMVNNQUE) of the parametric function with C≥0 exists. The necessary and sufficient conditions for a nonnegative quadratic unbiased estimator with of to be the UMVNNQUE are obtained as well.     

Direct variable selection for discrimination among several groups

We propose a criterion for variable selection in discriminant analysis. This criterion permits to arrange the variables in decreasing order of adequacy for discrimination, so that the variable selection problem reduces to that of the estimation of suitable permutation and dimensionality. Then, estimators for these parameters are proposed and the resulting method for selecting variables is shown to be consistent. In a simulation study, we compute proportions of correct classification after variable selection in order to gain understanding of the performance of our proposal and to compare it to existing methods.     

Item response theory for longitudinal data: population parameter estimation

In this work we propose IRT models to estimate ability distribution parameters of a population of individuals submitted to different tests along the time, having or not common items. The item parameters are considered known and several covariance structures are proposed to accommodate the possible dependence among the abilities of the same individual, measured at different instants. Maximum likelihood equations and some simulation results are presented.     

Minimax estimation for singular linear multivariate models with mixed uncertainty

The problem of minimax estimation is examined for the linear multivariate statistically indeterminate observation model with mixed uncertainty. The a priori information on the distributions of model parameters is formulated in terms of second-order moment characteristics. It is shown that in the regular case the minimax estimate is defined explicitly via the solution of the dual optimization problem. For singular models, the method of dual optimization is developed by means of using the Tikhonov regularization techniques. Several particular cases which are widely used in practice are also considered.     

Asymptotics of Bayesian median loss estimation

We establish the consistency, asymptotic normality, and efficiency for estimators derived by minimizing the median of a loss function in a Bayesian context. We contrast this procedure with the behavior of two Frequentist procedures, the least median of squares (LMS) and the least trimmed squares (LTS) estimators, in regression problems. The LMS estimator is the Frequentist version of our estimator, and the LTS estimator approaches a median-based estimator as the trimming approaches 50% on each side. We argue that the Bayesian median-based method is a good tradeoff between the two Frequentist estimators.     

On the conic section fitting problem

Adjusted least squares (ALS) estimators for the conic section problem are considered. Consistency of the translation invariant version of ALS estimator is proved. The similarity invariance of the ALS estimator with estimated noise variance is shown. The conditions for consistency of the ALS estimator are relaxed compared with the ones of the paper Kukush et al. [Consistent estimation in an implicit quadratic measurement error model, Comput. Statist. Data Anal. 47(1) (2004) 123-147].     

Restricted estimation in multivariate measurement error regression model

We study a multivariate ultrastructural measurement error (MUME) model with more than one response variable. This model is a synthesis of multivariate functional and structural models. Three consistent estimators of regression coefficients, satisfying the exact linear restrictions have been proposed. Their asymptotic distributions are derived under the assumption of a non-normal measurement error and random error components. A simulation study is carried out to investigate the small sample properties of the estimators. The effect of departure from normality of the measurement errors on the estimators is assessed.     

Nonparametric variance function estimation with missing data

In this paper, a fixed design regression model where the errors follow a strictly stationary process is considered. In this model the conditional mean function and the conditional variance function are unknown curves. Correlated errors when observations are missing in the response variable are assumed. Four nonparametric estimators of the conditional variance function based on local polynomial fitting are proposed. Expressions of the asymptotic bias and variance of these estimators are obtained. A simulation study illustrates the behavior of the proposed estimators.     

Nonparametric estimation of competing risks models with covariates

In competing risks model, several failure times arise potentially. The smallest failure time and its index only are observed. Without specific assumptions, the joint or even the marginal distribution functions of the underlying failure times are not identifiable (A. Tsiatis, Proc. Natl. Acad. Sci. USA 72 (1975) 20). Nonetheless, if each individual is characterized by a “sufficiently informative” set of covariates, these distributions are identifiable under some conditions of regularity (J.J. Heckman and B. Honoré, Biometrika 76 (1989) 325). In this paper, nonparametric kernel estimators of the joint distribution function of failure times conditional on the covariates are proposed. Their weak and strong consistency are discussed.     

Unbiased invariant minimum norm estimation in generalized growth curve model

This paper considers the generalized growth curve model subject to R(X_m)⊆R(X_m-1)⊆?⊆R(X₁), where B_i are the matrices of unknown regression coefficients, X_i,Z_i and U are known covariate matrices, i=1,2,…,m, and E splits into a number of independently and identically distributed subvectors with mean zero and unknown covariance matrix Σ. An unbiased invariant minimum norm quadratic estimator (MINQE(U,I)) of tr(CΣ) is derived and the conditions for its optimality under the minimum variance criterion are investigated. The necessary and sufficient conditions for MINQE(U,I) of tr(CΣ) to be a uniformly minimum variance invariant quadratic unbiased estimator (UMVIQUE) are obtained. An unbiased invariant minimum norm quadratic plus linear estimator (MINQLE(U,I)) of is also given. To compare with the existing maximum likelihood estimator (MLE) of tr(CΣ), we conduct some simulation studies which show that our proposed estimator performs very well.     

Some equalities for estimations of variance components in a general linear model and its restricted and transformed models

For the unknown positive parameter σ² in a general linear model , the two commonly used estimations are the simple estimator (SE) and the minimum norm quadratic unbiased estimator (MINQUE). In this paper, we derive necessary and sufficient conditions for the equivalence of the SEs and MINQUEs of the variance component σ² in the original model ?, the restricted model , the transformed model , and the misspecified model .     

Nonnegative quadratic estimation of the mean squared errors of minimax estimators in the linear regression model

In this paper, the problem of nonnegative quadratic estimation of the mean squared errors of minimax estimators of in the linear regression modelE(y)=X, VAR(y) = ² is discussed. An explicit formula for the admissible nonnegative minimum biased estimator is given. Some applications to one-way classification model are also considered.     

Empirical likelihood inference for censored median regression model via nonparametric kernel estimation

An alternative to the accelerated failure time model is to regress the median of the failure time on the covariates. In the recent years, censored median regression models have been shown to be useful for analyzing a variety of censored survival data with the robustness property. Based on missing information principle, a semiparametric inference procedure for regression parameter has been developed when censoring variable depends on continuous covariate. In order to improve the low coverage accuracy of such procedure, we apply an empirical likelihood ratio method (EL) to the model and derive the limiting distributions of the estimated and adjusted empirical likelihood ratios for the vector of regression parameter. Two kinds of EL confidence regions for the unknown vector of regression parameters are obtained accordingly. We conduct an extensive simulation study to compare the performance of the proposed methods with that normal approximation based method. The simulation results suggest that the EL methods outperform the normal approximation based method in terms of coverage probability. Finally, we make some discussions about our methods.     

An adjusted maximum likelihood method for solving small area estimation problems

For the well-known Fay-Herriot small area model, standard variance component estimation methods frequently produce zero estimates of the strictly positive model variance. As a consequence, an empirical best linear unbiased predictor of a small area mean, commonly used in small area estimation, could reduce to a simple regression estimator, which typically has an overshrinking problem. We propose an adjusted maximum likelihood estimator of the model variance that maximizes an adjusted likelihood defined as a product of the model variance and a standard likelihood (e.g., a profile or residual likelihood) function. The adjustment factor was suggested earlier by Carl Morris in the context of approximating a hierarchical Bayes solution where the hyperparameters, including the model variance, are assumed to follow a prior distribution. Interestingly, the proposed adjustment does not affect the mean squared error property of the model variance estimator or the corresponding empirical best linear unbiased predictors of the small area means in a higher order asymptotic sense. However, as demonstrated in our simulation study, the proposed adjustment has a considerable advantage in small sample inference, especially in estimating the shrinkage parameters and in constructing the parametric bootstrap prediction intervals of the small area means, which require the use of a strictly positive consistent model variance estimate.     

MOMENT ESTIMATION FOR MULTIVARIATE EXTREME VALUE DISTRIBUTION

Moment estimation for multivariate extreme value distribution is described in this paper. Asymptotic covariance matrix of the estimators is given. The relative efficiencies of moment estimators as compared with the maximum likelihood and the stepwise estimators are computed. We show that when there is strong dependence between the variates, the generalized variance of moment estimators is much lower than the stepwise estimators. It becomes more obvious when the dimension increases.     

REML estimation for binary data in GLMMs

The restricted maximum likelihood (REML) procedure is useful for inferences about variance components in mixed linear models. However, its extension to hierarchical generalized linear models (HGLMs) is often hampered by analytically intractable integrals. Numerical integration such as Gauss-Hermite quadrature (GHQ) is generally not recommended when the dimensionality of the integral is high. With binary data various extensions of the REML method have been suggested, but they have had unsatisfactory biases in estimation. In this paper we propose a statistically and computationally efficient REML procedure for the analysis of binary data, which is applicable over a wide class of models and design structures. We propose a bias-correction method for models such as binary matched pairs and discuss how the REML estimating equations for mixed linear models can be modified to implement more general models.     

On an additive decomposition of the BLUE in a multiple-partitioned linear model

Necessary and sufficient conditions are derived for the BLUE in a general multiple-partitioned linear model to be the sum of the BLUEs under the k small models , …, . Some consequences and further research topics are also given.     

Non-parametric kernel regression for multinomial data

This paper presents a kernel smoothing method for multinomial regression. A class of estimators of the regression functions is constructed by minimizing a localized power-divergence measure. These estimators include the bandwidth and a single parameter originating in the power-divergence measure as smoothing parameters. An asymptotic theory for the estimators is developed and the bias-adjusted estimators are obtained. A data-based algorithm for selecting the smoothing parameters is also proposed. Simulation results reveal that the proposed algorithm works efficiently.     

Fisher information for generalised linear mixed models

The Fisher information for the canonical link exponential family generalised linear mixed model is derived. The contribution from the fixed effects parameters is shown to have a particularly simple form.