If is shown that in linear regression models we do not make a great mistake if we substitute some sufficiently precise approximations for the unknown covariance matrix and covariance vector in the expressions for computation of the best linear unbiased estimator and predictor.
The method of least wquares is usually used in a linear regression model for estimating unknown parameters . The case when is an autoregressive process of the first order and the matrix corresponds to a linear trend is studied and the Bayes approach is used for estimating the parameters . 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.
The least squres invariant quadratic estimator of an unknown covariance function of a stochastic process is defined and a sufficient condition for consistency of this estimator is derived. The mean value of the observed process is assumed to fulfil a linear regresion model. A sufficient condition for consistency of the least squares estimator of the regression parameters is derived, too.
The model of quadratic regression is studied by means of the projection pursuit method. This method leads to a decomposition of the matrix of quadratic regression, which can be used for an estimation of this matrix from the data observed.
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