Fixed and Random Covariance Models

This chapter reviews fixed and random mean regression modelsFixed and random mean regression models and then discusses their differences with respect to their expected test (prediction) errors based on the test data and the parameter estimator obtained from the training data. It is not surprising that the random setting model yields a larger test error than that of the fixed setting model. We then focus on the fixed and random covariance regression modelsFixed and random mean regression models. The theoretical properties of regression parameter estimators are established, which show that the random setting yields larger bias and variance in test errors. In addition, the estimators of the expected test errors under the Fixed-X and Random-X settings are obtained, which lead to their corresponding Mallows-type selection criteria C ^ p $$\widehat C_p$$ and R C p ^ $$\widehat {RC_p}$$ , respectively. These two criteria can be used to assess the performance of competing covariance regression modelsCovariance regression model. An empirical example is briefly discussed.