The paper concludes our investigations in looking for the locally best linear-quadratic estimators of mean value parameters and of the covariance matrix elements in a special structure of the linear model (2 variables case) where the dispersions of the observed quantities depend on the mean value parameters. Unfortunately there exists no linear-quadratic improvement of the linear estimator of mean value parameters in this model.

In the paper necessary and sufficient conditions for the existence and an explicit expression for the Bayes invariant quadratic unbiased estimate of the linear function of the variance components are presented for the mixed linear model $𝐭=𝐗\beta +ϵ$, $𝐄\left(𝐭\right)=𝐗\beta$, $\mathrm{𝐕𝐚𝐫}\left(𝐭\right)={\mathbf{0}}_{\mathbf{1}}{𝐔}_{\mathbf{1}}+{\mathbf{0}}_{\mathbf{2}}{𝐔}_{\mathbf{2}}+{\mathbf{0}}_{\mathbf{3}}{𝐔}_{\mathbf{3}}$, with three unknown variance components in the normal case. An application to some examples from the analysis of variance is given.

In the paper an explicit expression for the Bayes invariant quadratic unbiased estimate of the linear function of the variance components is presented for the mixed linear model $𝐭=𝐗\beta +ϵ$, $𝐄\left(𝐭\right)=𝐗\beta$, $𝐃\left(𝐭\right)={\mathbf{0}}_{\mathbf{1}}{𝐔}_{\mathbf{1}}+{\mathbf{0}}_{\mathbf{2}}{𝐔}_{\mathbf{2}}$ with the unknown variance componets in the normal case. The matrices ${𝐔}_{\mathbf{1}}$, ${𝐔}_{\mathbf{2}}$ may be singular. Applications to two examples of the analysis of variance are given.

The method of least wquares is usually used in a linear regression model $𝐘=𝐗\beta +ϵ$ for estimating unknown parameters $\beta$. The case when $ϵ$ is an autoregressive process of the first order and the matrix $𝐗$ corresponds to a linear trend is studied and the Bayes approach is used for estimating the parameters $\beta$. Unbiased Bayes estimators are derived for the case of a small number of observations. These estimators are compared with the locally best unbiased ones and with the usual least squares estimators.

This paper considers the problem of making statistical inferences about group judgements and group decisions using Qualitative Controlled Feedback, from the Bayesian point of view. The qualitative controlled feedback procedure was first introduced by Press (1978), for a single question of interest. The procedure in first reviewed here including the extension of the model to the multiple question case. We develop a model for responses of the panel on each stage. Many questions are treated simultaneously...

We consider the joint modelling of the mean and covariance structures for the general antedependence model, estimating their parameters and the innovation variances in a longitudinal data context. We propose a new and computationally efficient classic estimation method based on the Fisher scoring algorithm to obtain the maximum likelihood estimates of the parameters. In addition, we also propose a new and innovative Bayesian methodology based on the Gibbs sampling, properly adapted for longitudinal...

This paper proposes a bias reduction of the coefficients' estimator for linear regression models when observations are randomly censored and the error distribution is unknown. The proposed bias correction is applied to the weighted least squares estimator proposed by Stute [28] [W. Stute: Consistent estimation under random censorship when covariables are present. J. Multivariate Anal. 45 (1993), 89-103.], and it is based on model-based bootstrap resampling techniques that also allow us to work with...

We derive expressions for the asymptotic approximation of the bias of the least squares estimators in nonlinear regression models with parameters which are subject to nonlinear equality constraints. The approach suggested modifies the normal equations of the estimator, and approximates them up to $o_p\left(N^{-1}\right)$, where $N$ is the number of observations. The “bias equations” so obtained are solved under different assumptions on constraints and on the model. For functions of the parameters the invariance of the approximate...

General results giving approximate bias for nonlinear models with constrained parameters are applied to bilinear models in anova framework, called biadditive models. Known results on the information matrix and the asymptotic variance matrix of the parameters are summarized, and the Jacobians and Hessians of the response and of the constraints are derived. These intermediate results are the basis for any subsequent second order study of the model. Despite the large number of parameters involved,...

We study properties of Linear Genetic Programming (LGP) through several regression and classification benchmarks. In each problem, we decompose the results into bias and variance components, and explore the effect of varying certain key parameters on the overall error and its decomposed contributions. These parameters are the maximum program size, the initial population, and the function set used. We confirm and quantify several insights into the practical usage of GP, most notably that (a) the...