Time-varying coefficient estimation in differential equation models with noisy time-varying covariates

We study the problem of estimating time-varying coefficients in ordinary differential equations. Current theory only applies to the case when the associated state variables are observed without measurement errors as presented in Chen and Wu (2008) [4] and [5]. The difficulty arises from the quadratic functional of observations that one needs to deal with instead of the linear functional that appears when state variables contain no measurement errors. We derive the asymptotic bias and variance for the previously proposed two-step estimators using quadratic regression functional theory.     

A subclass of bayes linear estimators that are minimax

In the linear regression model with ellipsoidal parameter constraints, the problem of estimating the unknown parameter vector is studied. A well-described subclass of Bayes linear estimators is proposed in the paper. It is shown that for each member of this subclass, a generalized quadratic risk function exists so that the estimator is minimax. Moreover, some of the proposed Bayes linear estimators are admissible with respect to all possible generalized quadratic risks. Also, a necessary and sufficient condition is given to ensure that the considered Bayes linear estimator improves the least squares estimator over the whole ellipsoid whatever generalized risk function is chosen.     

Confidence ellipsoids based on a general family of shrinkage estimators for a linear model with non-spherical disturbances

This paper considers a general family of Stein rule estimators for the coefficient vector of a linear regression model with nonspherical disturbances, and derives estimators for the Mean Squared Error (MSE) matrix, and risk under quadratic loss for this family of estimators. The confidence ellipsoids for the coefficient vector based on this family of estimators are proposed, and the performance of the confidence ellipsoids under the criterion of coverage probability and expected volumes is investigated. The results of a numerical simulation are presented to illustrate the theoretical findings, which could be applicable in the area of economic growth modeling.     

Conditional and unconditional methods for selecting variables in linear mixed models

In the problem of selecting the explanatory variables in the linear mixed model, we address the derivation of the (unconditional or marginal) Akaike information criterion (AIC) and the conditional AIC (cAIC). The covariance matrices of the random effects and the error terms include unknown parameters like variance components, and the selection procedures proposed in the literature are limited to the cases where the parameters are known or partly unknown. In this paper, AIC and cAIC are extended to the situation where the parameters are completely unknown and they are estimated by the general consistent estimators including the maximum likelihood (ML), the restricted maximum likelihood (REML) and other unbiased estimators. We derive, related to AIC and cAIC, the marginal and the conditional prediction error criteria which select superior models in light of minimizing the prediction errors relative to quadratic loss functions. Finally, numerical performances of the proposed selection procedures are investigated through simulation studies.     

On improved estimation of normal precision matrix and discriminant coefficients

The problem of estimating the precision matrix of a multivariate normal distribution model is considered with respect to a quadratic loss function. A number of covariance estimators originally intended for a variety of loss functions are adapted so as to obtain alternative estimators of the precision matrix. It is shown that the alternative estimators have analytically smaller risks than the unbiased estimator of the precision matrix. Through numerical studies of risk values, it is shown that the new estimators have substantial reduction in risk. In addition, we consider the problem of the estimation of discriminant coefficients, which arises in linear discriminant analysis when Fisher's linear discriminant function is viewed as the posterior log-odds under the assumption that two classes differ in mean but have a common covariance matrix. The above method is also adapted for this problem in order to obtain improved estimators of the discriminant coefficients under the quadratic loss function. Furthermore, a numerical study is undertaken to compare the properties of a collection of alternatives to the “unbiased” estimator of the discriminant coefficients.     

Estimation in mixed effects model with errors in variables

In this paper, we consider a linear mixed-effects model with measurement errors in both fixed and random effects and find the moment of estimators for the parameters of interest. The strong consistency and asymptotic normality of the estimators are obtained under regularity conditions. Moreover, we obtain the strong consistent estimators of the asymptotic covariance matrices involved in the limiting theory. Simulations are reported for illustration.     

Risk and Pitman closeness properties of feasible generalized double k-class estimators in linear regression models with non-spherical disturbances under balanced loss function

In this article, a family of feasible generalized double k-class estimator in a linear regression model with non-spherical disturbances is considered. The performance of this estimator is judged with feasible generalized least-squares and feasible generalized Stein-rule estimators under balanced loss function using the criteria of quadratic risk and general Pitman closeness. A Monte-Carlo study investigates the finite sample properties of several estimators arising from the family of feasible double k-class estimators.     

Analysis of variance and linear contrasts in experimental design with generalized secant hyperbolic distribution

We consider one-way classification model in experimental design when the errors have generalized secant hyperbolic distribution. We obtain efficient and robust estimators for block effects by using the modified maximum likelihood estimation (MML) methodology. A test statistic analogous to the normal-theory F statistic is defined to test block effects. We also define a test statistic for testing linear contrasts. It is shown that test statistics based on MML estimators are efficient and robust. The methodology readily extends to unbalanced designs.     

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.     

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.     

Asymptotics of R-, MD- and LAD-estimators in linear regression models with long range dependent errors

Summary This paper establishes the uniform closeness of a weighted residual empirical process to its natural estimate in the linear regression setting when the errors are Gaussian, or a function of Gaussian random variables, that are strictly stationary and long range dependent. This result is used to yield the asymptotic uniform linearity of a class of rank statistics in linear regression models with long range dependent errors. The latter result, in turn, yields the asymptotic distribution of the Jaeckel (1972) rank estimators. The paper also studies the least absolute deviation and a class of certain minimum distance estimators of regression parameters and the kernel type density estimators of the marginal error density when the errors are long range dependent.Research of this author was partly supported by the NSF grant: DMS-9102041     

Minimum distance estimation in linear regression with unknown error distributions

This paper discusses minimum distance (m.d.) estimators of the paramter vector in the multiple linear regression model when the distributions of errors are unknown. These estimators are defined in terms of L₂-distances involving certain weighted empirical processes. Their finite sample properties and asymptotic behavior under heteroscedastic, symmetric and asymmetric errors are discussed. Some robustness properties of these estimators are also studied.     

Statistical inference in a panel data semiparametric regression model with serially correlated errors

We consider a panel data semiparametric partially linear regression model with an unknown vector β of regression coefficients, an unknown nonparametric function g(·) for nonlinear component, and unobservable serially correlated errors. The correlated errors are modeled by a vector autoregressive process which involves a constant intraclass correlation. Applying the pilot estimators of β and g(·), we construct estimators of the autoregressive coefficients, the intraclass correlation and the error variance, and investigate their asymptotic properties. Fitting the error structure results in a new semiparametric two-step estimator of β, which is shown to be asymptotically more efficient than the usual semiparametric least squares estimator in terms of asymptotic covariance matrix. Asymptotic normality of this new estimator is established, and a consistent estimator of its asymptotic covariance matrix is presented. Furthermore, a corresponding estimator of g(·) is also provided. These results can be used to make asymptotically efficient statistical inference. Some simulation studies are conducted to illustrate the finite sample performances of these proposed estimators.     

All admissible linear estimators of a regression coefficient under a balanced loss function

Admissibility of linear estimators of a regression coefficient in linear models with and without the assumption that the underlying distribution is normal is discussed under a balanced loss function. In the non-normal case, a necessary and sufficient condition is given for linear estimators to be admissible in the space of homogeneous linear estimators. In the normal case, a sufficient condition is provided for restricted linear estimators to be admissible in the space of all estimators having finite risks under the balanced loss function. Furthermore, the sufficient condition is proved to be necessary in the normal case if additional conditions are assumed.     

On the quadratic estimation of covariance matrices in multivariate linear models

This paper investigates the estimation of covariance matrices in multivariate mixed models. Some sufficient conditions are derived for a multivariate quadratic form and a linear combination of multivariate quadratic forms to be the BQUE (quadratic unbiased and severally minimum varianced) estimators of its expectations.     

Safe and tight linear estimators for global optimization

Global optimization problems are often approached by branch and bound algorithms which use linear relaxations of the nonlinear constraints computed from the current variable bounds. This paper studies how to derive safe linear relaxations to account for numerical errors arising when computing the linear coefficients. It first proposes two classes of safe linear estimators for univariate functions. Class-1 estimators generalize previously suggested estimators from quadratic to arbitrary functions, while class-2 estimators are novel. When they apply, class-2 estimators are shown to be tighter theoretically (in a certain sense) and almost always tighter numerically. The paper then generalizes these results to multivariate functions. It shows how to derive estimators for multivariate functions by combining univariate estimators derived for each variable independently. Moreover, the combination of tight class-1 safe univariate estimators is shown to be a tight class-1 safe multivariate estimator. Finally, multivariate class-2 estimators are shown to be theoretically tighter (in a certain sense) than multivariate class-1 estimators.     

Admissibility of linear estimators with respect to inequality constraints under matrix loss function

In this paper we investigate the admissibility of linear estimators in the multivariate linear model with respect to inequality constraints under matrix loss function. The necessary and sufficient conditions for a linear estimator to be admissible in the class of homogeneous linear estimators and the class of inhomogeneous linear estimators are obtained, respectively.     

Difference based ridge and Liu type estimators in semiparametric regression models

We consider a difference based ridge regression estimator and a Liu type estimator of the regression parameters in the partial linear semiparametric regression model, y=Xβ+f+ε. Both estimators are analyzed and compared in the sense of mean-squared error. We consider the case of independent errors with equal variance and give conditions under which the proposed estimators are superior to the unbiased difference based estimation technique. We extend the results to account for heteroscedasticity and autocovariance in the error terms. Finally, we illustrate the performance of these estimators with an application to the determinants of electricity consumption in Germany.     

Second-order accurate inference on eigenvalues of covariance and correlation matrices

Edgeworth expansions and saddlepoint approximations for the distributions of estimators of certain eigenfunctions of covariance and correlation matrices are developed. These expansions depend on second-, third-, and fourth-order moments of the sample covariance matrix. Expressions for and estimators of these moments are obtained. The expansions and moment expressions are used to construct second-order accurate confidence intervals for the eigenfunctions. The expansions are illustrated and the results of a small simulation study that evaluates the finite-sample performance of the confidence intervals are reported.