Sul modo di vincolare la funzione di verosimiglianza quando si è in presenza di informazioni a priori approssimative

Abstract

The penalty function method is a mathematical approach generally used in non linear programming and is based on the maximization of a (penalty) function obtained combining linearly through some coefficients called penalty parameters an objective function and the constraints expressed in quadratic form. If the objective function satisfies certain conditions about the continuity, the maximization of the penalty function produces a constrained solution only increasing indefinitely the penalty parameters while for finite values assigned to these parameters we obtain a solution placed between the maximum point and the constrained one. From a statistical viewpoint this approach can be used to constraint a likelihood function when we have prior informations on the parameters of approssimative nature. In this case the particular form (quadratic) of the restraints can be seen as a way to catch all the uncertainty included in this type of prior knowledges and the assigned values to the penalty parameters as a way to quantify this uncertainty. Statistically the maximization of the likelihood through the penalty function approach emphasizes two problems: the analysis of the property of the estimator produced by this method and the interpretation of the penalty parameter whose value determines the estimates. In the first case if the usual regularity conditions on the likelihood function are satisfied, the estimator is consistent and asintotically normal whatever is the value of the penalty parameter. In the second case the penalty parameter may be interpreted both as a cost we must pay to depart from the prior informations and as a variance of a stochastic constraint or of a prior distribution. In such a case the penalty function method produces exactly some well known bayesian solutions.