In this doctoral thesis we consider topics related to linear estimation and prediction in the general Gauss—Markov model. The thesis consists of eleven articles and an introduction to concepts considered in the articles. The main contributions of the thesis concern the concepts of the best linear unbiased estimator, BLUE, the best linear unbiased predictor, BLUP, linear sufficiency, linear prediction sufficiency, the ordinary least squares estimator, OLSE, and the Watson efficiency.

In this thesis we consider linear sufficiency and linear completeness in the context of estimating the given estimable parametric function. Some new characterizations for linear sufficiency and linear completeness in a case of estimation of the parametric function are given, and also a predictive approach for obtaining the BLUE of the estimable parametric function is considered.

In the context of predicting the value of new observation under the general Gauss—Markov model, a new concept---linear prediction sufficiency---is introduced, and some basic properties of linear prediction sufficiency are given.

Furthermore, in this thesis the equality of the OLSE and BLUE of the given estimable parametric function is considered, and properties of the Watson efficiency are investigated particularly under the partitioned linear model.

This thesis contains also an article concerning the best linear unbiased estimation under the linear mixed effects model, and an article considering a particular matrix decomposition useful in the theory of linear models.