The General Growth Curve Model Of Optimal Prediction

The general growth curve model is a more generalized multivariate linear model that is widely applied in biology, technology substitutions and economic forecast, ect. The prediction problem is to predict future observation values with the past known ones. There are mainly two questions to be solve for a predictable variable. One is to derive the optimal prediction and the other is to find its necessary and sufficient conditions. There is, however, a more design matrix in this model than is in multivariate linear model, which has caused difficulties such as solving a exceptional unlinear matrix equation groups especially when deriving the optimal prediction. As a result, we are not only unable to obtain the optimal prediction in general cases but incapable of finding the necessary and sufficient conditions as well.Considering linear and O-linear predictable variables, the author investigates optimal prediction problems by the trace of matrix. A few necessary conditions are derived and accordingly optimal linear and optimal O-linear unbiased predictors, which are unique with probabitity one, are obtained respectively by the author. In the meanwhile, seeing that the parameters are restricted for many practical problems, the author also studies the problems of optimal conditional prediction in the model with respect to two classes of restriction of linear parameter equations. What is more, the optimal conditional linear and optimal conditional ^-linear unbiased predictors are also obtained respectively, which extends the results given by the predecessors and enrich the theory of optimal prediction. Finally, some illustrative examples are given in this paper as a proof that optimal linear unbiased predictor is a better predictor than others.