A Fitting Characteristic Vector Based on the Mean Square Error with an Application

A vector measure is discussed which makes available the full information as to the fitness of a model contained implicitly in the mean square error. The vector measure is called the fitting characteristic vector (FCV) in the paper. Dimensionless quantities of three kinds are extracted from the mean square error. The FCV is defined by these quantities. It is shown that each of the quantities has its own intrinsic meaning concerning the fitness. The following concepts are introduced in connection with the FCV: 1) the latent goodness-of-fit, 2) the basic latent goodness-of-fit, 3) the alienation of the realized goodness-of-fit from the latent goodness-of-fit, and 4) the basic alienation. A new view of the fitness is given on the basis of the above concepts. It is shown that the FCV can be used as a criterion for both identification and ex post facto evaluation. The relation between the FCV and Theil's inequality proportions is given. The FCV is applied to a real problem to show its usage and its effectiveness in ex post facto evaluation.