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استخدام اسلوب بيز التجريبي في تقدير معلمات انموذج الانحدار الخطي == Using Empirical Bayes Approach For Estimating Parameters In A Linear Regression Model
Author name:
حازم منصور كوركيس عربو
Supervisor name:
اموري هادي كاظم الحسناوي
General topic:
Administration and Economics
Specific topic:
Statistics
Degree:
Doctorate
University:
University of Baghdad - Faculty Of Administration And Economics - Department Of Statistics
Language:
Arabic
University location:
Baghdad
First pages:
07T3528 - p.pdf
Abstract:
The primary purpose of our thesis is to assess the importance of using empirical Bayes approach in regression analysis. At the first some empirical Bayes techniques in point estimation are considered. It is well known that in point estimation with a squared error loss function the Bayes estimator is the posterior mean. In the empirical Bayes approach we must construct a consistent sequence of estimators for this posterior mean using past experience. This construction is done for three general families of distributions. New estimators for the parameters in the simple orthogonal linear regression model are presented using not only the usual random sample of observations but also past experience in the form of previous estimators of parameters in similar but independent situations. The regression parameters are considered to be random variables. Bayes estimators are given for a squared error loss function. Even though the prior density of the parameter is unknown the Bayes estimator can be written in terms of the marginal density of sufficient statistic. This marginal density can be estimated empirically , thus forming the empirical Bayes estimator. Empirical Bayes estimators for the parameters in the general linear regression model are presented. These estimators by pass exact knowledge of the prior distribution of parameters by means of supplementary informations from similar independent experiments. The case in which the error variance is unknown and may vary from one experiment to the next is included. Some basic concepts in shrinkage estimators are introduced. The definition of multicolinearity as the existence of near linear relationships among the independent variables is given. Effects of multicolinearity on estimated regression coefficients are explained. Sources of multicolinearity and methods of detecting multicolinerity are presented. The method of ridge regression is given as one of several methods that have been proposed to remedy multicolinearity problems by modifying the method of least squares to allow biased estimators of the regression coefficients. The technique of ridge regression first proposed by Hoerl and Kennard has become a popular tool for data analysts faced with a high degree of multicolinearity in their data. Ridge solution properties and methods for choosing the ridge parameter are presented. The first method is graphical applied by using graphical display called 'ridge trace' the second method is the iterative method proposed by Hoerl and Kennard. The equivalence of ridge regression estimator with Bayers estimator is proved. Bayesian methods are employed for choosing the ridge parameter. An empirical Bayes estimator of the ridge parameter is presented. An empirical Bayes estimator of the ridge parameter which result in minimax ridge regression estimator under strawderman's loss function (formula 2.81) is also presented. By minimax estimator we mean an estimator which is uniformly better than the least square estimator in terms of risk. In the practical part of the thesis we apply ridge regression analysis to the set of actual data suffer from multicolinearity. Three methods are employed to determine the value of the ridge parameter. Comparisons between the three methods are made on the basis of various statistics that might go into the choice of the ridge parameter. According to these comparisons we conclude that the value of the ridge parameter obtained by using empirical Bayes approach (formula 2.78) is better than the other two methods. Ridge analysis is repeated for another set of experimental data obtained by constraint simulation. The same conclusion is obtained when the dependent variable is any one of the variables Y2 ,..... Y6.
