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Estimation on Restricted Parameter Spaces

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Estimation on Restricted Parameter Spaces ESTIMATION ON RESTRICTED PARAMETER SPACES by William Franklin Watson, Jr. Institute of Statistics Mimeograph Series No. 1026 Raleigh, N. C. ABSTRACT WATSON, WILLIAM FRANKLIN~ JR. Esti.mation on Restricted Parameter Spaces. (Under the direction of HUBERTUS ROBERT VAN DER VAART and BENEE FRANK SWINDEL.) The problem of finding point estimates of parameters when the feasible parameter space is a proper and convex subset of Euclidean m~space waS studied. The algorithms of maximum likelihood estimation for the parameters of linear models, restricted in such a manner, were reviewed for the case in which the elements of the error vector have a normal distribution. These estimators were shown to be biased, to possess a type ,of consistency, and, in the univariate case, to have a mean square error no larger than the unrestricted maximum likelihood estimator. Also, these estimators were shown to map all unrestricted estimates which are not in the feasible parameter space to the boundary of the feasible parameter space. It would be difficult to believe that the parameter is on the boundary so often. The Bayesian estimators, the median and mean of the posterior distribution, were shown to have different unpleasant properties when the parameter space is a proper, convex subset in Euclidean m-space, The median of the posterior distribution W3.S found to take on points on the boundary of the feasible parameter space only if a supporting hyperplane of the posterior contained at least half of the probability mass of the posterior distribution. Si.mi.larly, the mean of the posterior distributi.on would never take on Some of the points in the feasible parameter space as estimates unless the posterior distribution tended to a dengenerate distribution at these points for Some point in the sample space. However. the mean of the univariate and a bivariate truncated normal posterior distribution, were, shown to take on every point in the support of the posterior for some value of the random variable. Assuming the prior density to be proportional to either a uniform, exponential, or truncated normal density over the feasible space, zero elsewhere, lead to a truncated normal posterior when the random variable was distributed normally. A detailed examination was made of the estimators for the mean parameter of a univariate normal distribution for the situation in which the parameter; was known to be contained in a half-line. Neither the mean of appropriate truncated normal posteriors using any of the priors mentioned above nor the restricted maximum likeli­ hood estimators had uniformly smaller mean square error over the feasible parameter space. The regret function was then introduced and was defined to be the difference in the mean square error of an estimator at a point in parameter space and the smallest mean square error of the candidate estimators of that point. The strategy chosen waS to find an estimator which would minimize, among the candidate estimators, the maximum regret over the sample space. Joined estimation procedures were proposed, in which the mean of a posterior (exponential prior) was used over a portion of the sample space and maximum likelihood procedures were used over the remainder of the sample space. An optimal joined estimator was found to give an 18% reduction in maximum regret over the best of the classical estimators. To extend the technique) opti.mal Bayesian estimators of this type were found for se.veral subsets of the sample space. The resulting estimator gave a 48% reduction in the maximum regret over what was found for the best of the classical estimators. Similar results were found for a bivariate example. ESTIMATION ON RESTRICTED PARAMETER SPACES by WILLIAM FRANKLIN WATSON JR. A thesis submitted to the Graduate Faculty of North Carolina State University at Raleigh in partial fulfillment of the requirements for the Degree of Doctor of Philosophy DEPARTMENTS OF FORESTRY AND STATISTICS RALEIGH 1 9 7 4 APPROVED BY: Co-Chairman of Advisory Committee Co-Chairman of Advisory Committee BIOGRAPHY William F. Watson Jr. was born September 18, 194.5, i.n Tifton, Georgia, and was raised in the farming community of Eldorado which is near Tifton. He received his elementary and secondary education in the Tift County, Georgia, school system and was graduated from Tift County High School in 1963. He attended Abraham Baldwin Agricultural College, Auburn University. and the University of Georgia. From the latter, he received a Bachelor of Science degree in forestry in 1967 and the Master of Science degree in 1969. In 1969 he was inducted into the U.S. Army where he served as a th rd computer programmer and systems analyst for the ~ and 2~ Infantry Divisions in the Republi.c of Viet Nam. Upon his release from active duty in 1971, he entered North Carolina State University to pursue the Doctor of Philosophy degree. In 1974, he assumed a research position with the Forestry Department at Mississippi'State University. The author is married to the former Linda Diane Hamlin and they have one son, Hank. iii ACKNOWLEDGMENTS Any expression of gratitude would be insufficient for the Co­ Chairmen of thi.s author's Advisory Committee, Professors' H. R. van der Vaart and B. F. Swindel. These gentlemen were extremely generous with their time, and provided the counsel which inspired many of the significant portions of this study. A special word of thanks is due Professor W. 1. Hafley who served as guide through the admin.istrative hurdles while serving as Co-Chairman of the Advisory Committee. The author wishes to also thank the other members of the Graduate Faculty who served on his committee, T. M. Gerig, T. E. Maki, and T. O. Perry; all of whom made significant contributions to his educational experience. Professor J. M. Danby's suggestions dealing with the problems of numerical integration encountered in this study were also appreciated. The author was supported during his graduate study by the North Carolina Agricultural Experiment Station. Sincere gratitude is extended to the author's wife and Son for their sacrifices, and especially to his wife for her efforts in the completion of this paper. iv TABLE OF CONTENTS Page 1. INTRODUCTION •••• 1 1.1 The Problem • 1 1.2 Terminology 3 1.3 Review of Literature 3 1.4 Scope, Objectives, and Organization of This Paper 4 2. MAXIMUM LIKELIHOOD ESTIMATION ON RESTRICTED PARAMETER SPACES 6 2.1 General Discussion 6 2.2 Quadratic Programming 7 2.3 Isotonic Regression •• 14 2.4 Properties of the Restricted Maximum Likelihood Estima. tes . 0 0 • • 9 •••••••••• 18 3. BAYES IAN ESTIMATION ON RESTRICTED PARAMETER SPACES 23 3.1 Properties and Problems ••••• 23 3.2 Alternative Bayesian Procedures • 26 4. BAYESIAN ESTIMATORS DERIVED FROM TRUNCATED NORMAL POSTERIOR DISTRIBUTIONS ••••••••••••• 30 4.1 Mean of a Truncated Normal Distribution ••••• 30 4.2 Priors Producing a Truncated Normal Posterior Distribution for the Problem of Isotonic Regression .••.. •••.•.•• •• 32 4.3 Construction of Several Bayesian Estimators and Comparison With the Restricted Maximum Likelihood Estimators ••• ••••••• 36 4.4 Comparison of Mean Square Errors of Restricted Maximum Likelihood Estimators and Bayesian Estimators •••• •••••• 49 5. IMPROVED ESTIMATORS 58 5.1 Joining Estimators 58 5.2 The Criterion of Regret •• •• 60 5.3 The Application of Minimax Regret to the Construction of a Joined Estimator 61 5.4 Other Joined Estimators ••• ••• 69 5.5 Extending the Technique •••••••• 77 5.6 Estimating Two Ordered Parameters ••• 84 5.7 Estimators for m Ordered Parameters • 94 v ~BLE OF CONTENTS (Continued) Page 6. SUMMARY 98 7. LIST OF REFERENCES 103 8. APPENDIX 106 8.1 Theorems and Proofs •••• 107 8.2 Values of the Function f(x)/F(x) •••• 112 8.3 The Mean of a Truncated Multivariate Normal Posterior Distribution •••••• us 8.4 Truncated Normal Posteriors Arising From Unequal Samples From Several Populat.ions ••••••• 133 1. INTRODUCTION L 1 The Problem The statistician is often confronted with the problem of estimating the parameters for the linear model y=X~+~. (l.1.1) In this problem, y is an n element vector of responses, X is an n X m design matrix, ~, is an m element vector containing the un~ known parameters, and ~ is an n element vector of the random components. There are many situations where the true value of the m parameter vector ~ is known to lie in a proper subset of R (Euclidean m-space). Often such information can be written as linear inequality constraints in terms of the parameter vector ~. An example of such restrictions is (l.1.2) where C is a matrix of order k X m ,and d is a vector of k elements. Not all restrictions are of this simple form: Hudson (1969) cites a case of polynomial regression where the derivative of the polynomial must be positive over an interval. Modelers of growth in biological population can often rule out m subsets of the parameter space, R , because values of the parameters in these sets would violate known biological laws. An example of such a violation would be a model for the amount of wood fiber accumulated in the bole of a tree at various ages with parameters which give de­ creasing predictions of fiber accumulation over age. 2 It would be desirable if statisti.cians could prescribe a uniform set of rules for the modeler who has prior information that the true value of the parameters are certain to be found in a proper subset of m R • Unfortunately, such a set of rules has not been forthcoming. Searle (1971) has listed the alternatives that have been proposed to resolve the problem of negati.ve variance components. Many of these alternatives are applicable to the problem of estimation when the true values of the parameters of a linear model are known to be in a subset m of R • Some statisticians view estimates which violate known constraints as being indicative of a failure of the model to represent the true situation, and investigate alternative formulations of the model.
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