Simulation Analysis Of Regression Estimators Based On Coefficients Of Uncertainty

Let us consider the ordinary linear model Y = X*beta +Z, where Y is a vector of observations of the dependent variable, X is a matrix of the observations of explanatory variables, (beta) is a vector of unknown regression coefficients and Z is a vector of random disturbances (all quantities of appropriate dimensions). Let us assume E(Y ) = X*beta and Cov(Y ) = Sigma. The paper is devoted to the problem of incorporating prior information in the estimation of the model coefficients. The usual least squares estimator is known to have several deficiencies - e.g. it is inadmissible (under the quadratic loss) if k >= 3, it is numerically unstable when the matrix X(transposed)X is nearly singular and it does not incorporate often ex- isting prior information about the regression equation. In choosing alternatives to the least-squares estimator one is presented with a wide range of estimators. Some of the estimators were developed to deal with the problem of the regression estimation in the presence of prior knowledge about the parameter beta and/or the covariance matrix Sigma. Sometimes the prior information about beta is expressed in terms of the probability distribution of the parameter. If the distribution is specified precisely it leads to Bayes formulation of the problem, if not then usually the distribution is assumed to belong to a given class of distributions. Such description of the prior information (and its uncertainty) often leads to robust Bayes or minimax estimators, see e.g. [1, 2]. Other estimators have been obtained when the prior information about beta had the form of the restricted parameter space, see e.g. [4]. So, the theory provides us with many tools to incorporate prior information in regression estimation. However, in real terms it is very difficult to decide what description of our prior knowledge would be the most suitable - the knowledge may have different nature and various origins. In the same time we are aware that the decision in uence the choice of estimators; different estimators perform best depending on the description of the prior information. In the paper we propose some intuitive method of incorporating informa- tion about the parameter beta in regression estimation. We assume that the prior information is derived from previous regression analysis performed for some phe- nomenon which is described, according to our beliefs, by the same regression equation as the one we investigate. To describe the uncertainty connected with such information we introduce some coefficients which are functions of uncer- tainty indices - improved versions of indices of utility of the prior information proposed in [3]. On the base of the coefficients of uncertainty we define some nonlinear estimators of regression parameters. The usefulness of the uncertainty indices as well as the performance of the introduced estimators are examined via computer simulations. During the simulations we generate the prior informa- tion as well as the observations for regression analysis (changing at random all characteristics of examined models). Consequently we study the performance of considered estimators for thousands data sets.