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Minimax Admissible Estimation of a Multivariate Normal Mean and Improvement Upon the James-Stein Estimator

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Minimax Admissible Estimation of a Multivariate Normal Mean and Improvement Upon the James-Stein Estimator Minimax Admissible Estimation of a Multivariate Normal Mean and Improvement upon the James-Stein Estimator By Yuzo Maruyama Faculty of Mathematics, Kyushu University A dissertation submitted to the Graduate School of Economics of University of Tokyo in partial fulfillment of the requirements for the degree of Doctor of Economics June, 2000 (Revised December, 2000) Abstract This thesis mainly focuses on the estimation of a multivariate normal mean, from the decision-theoretic point of view. We concentrate our energies on investigating a class of estimators satisfying two optimalities: minimaxity and admissibility. We show that all classes of minimax admissible generalized Bayes estimators in the hitherto researches are divided in two, on the basis of whether a certain function of the prior distribution is bounded or not. Furthermore by using Stein’s idea, we propose a new class of minimax admissible estimators. The relations among minimaxity, a prior distribution and a shrinkage factor of the genaralized Bayes estimator, are also investigated. For the problem of finding admissible estimators which dominate the inadmissible James-Stein estimator, we have some results. Moreover, in non-normal case, we consider the problem of proposing admissible minimax estimators. For a certain class of spherically symmetric distributions which includes at least multivariate-t distributions, admissible minimax estimators are de- rived. ii Acknowledgements I am indebted to my adviser, Professor Tatsuya Kubokawa for his guidance and encour- agement during the research for and preparation of this thesis. I wish to take opportunity to express my deep gratitude to Professors of Gradu- ate School of Economics, University of Tokyo: Akimichi Takemura, Naoto Kunitomo, Yoshihiro Yajima, Kazumitsu Nawata and Nozomu Matsubara, and to Professors of Faculty of Mathematics, Kyushu University: Takashi Yanagawa, Sadanori Konishi, Hiroto Hyakutake, Yoshihiko Maesono, Masayuki Uchida and Kaoru Fueda, for their encouragement and suggestions. Finally, special thanks to my wife, Hiromi, for her support, encouragement, tender- ness and patience. iii Contents Abstract ii Acknowledgements iii 1 Introduction 1 2 Basic notions of statistical decision theory 6 3 Estimation of a multivariate normal mean 14 3.1 Inadmissibility of the usual estimator . 14 3.2 Admissibility and permissibility . 20 3.2.1 Preparations . 20 3.2.2 The necessary condition for the admissibility . 22 3.2.3 The sufficient condition for admissibility . 23 3.2.4 Exterior boundary problem and recurrent diffusions . 30 3.2.5 Permissibility . 32 3.3 The class of admissible minimax estimators . 34 3.3.1 Generalized Bayes estimators and their admissibility . 34 3.3.2 The construction of minimax Bayes estimator . 38 3.3.3 Alam-Maruyama’s type generalized Bayes estimator . 44 3.3.4 Generalization of Stein’s Bayes estimator . 50 3.4 Improvement upon the James-Stein estimator . 59 iv 4 Estimation of a mean vector of scale mixtures of multivariate normal distributions 66 4.1 Introduction and summary . 66 4.2 The construction of the generalized Bayes estimators . 67 4.3 Admissible Minimax estimators of the mean vector . 74 5 Estimation problems on a multiple regression model 78 5.1 Minimax estimators of a normal variance . 78 5.1.1 Introduction . 78 5.1.2 Main results . 79 5.2 Estimation of a multivariate normal mean with unknown variance . 84 5.2.1 Introduction . 84 5.2.2 Main results . 85 6 A New Positive Estimator of Loss Function 89 6.1 Introduction . 89 6.2 A positive estimator improving on Johnstone’s estimator . 90 7 Conclusion 97 A Technical results 99 v Chapter 1 Introduction Statistical decision theory, which was made by Wald(1950), characterizes the mode of modern statistical thinking. The contrast between the simplicity of its basic notions and the broad area of applicability is astonishing. The minimax principle and least favorable distributions are central notions of statistical decision theory and are close parallel to the corresponding concepts of the game theory. Admissibility is another basic concept of statistical decision theory and is related to the concept of Pareto optimality in economics. During the last thirty years, the study of statistical decision theory, in particular of admissibility, led to unexpected, amazing connections to many branches of mathematics, for example, partial differential equations of mathematical physics, differential inequalities and recurrence properties of stochastic process. For the good accounts, see Brown(1971), Eaton(1992) and Rukhin(1995). It is noted that these connections were discovered and have developed in the process of investigating the estimation problem of a multivariate normal mean, a typical topic of what is called the Stein phenomenon, which is introduced as follows. Many statisticians would like to believe that a natural estimator as maximum like- lihood estimator(MLE) is admissible because admissibility is a weak criterion for opti- 0 mality. For the estimation problem of a mean vector θ = (θ1; : : : ; θp) based on random 0 vector X = (X1;:::;Xp) having p-variate normal distribution N(θ; Ip) relative to a quadratic loss function, Stein(1956) showed that the MLE X, which is also minimax, 1 is inadmissible for p ¸ 3 while it is admissible for p = 1, 2. James and Stein(1961) succeeded in giving an explicit form of an estimator improving on X as µ ¶ p ¡ 2 ±JS = 1 ¡ X; kXk2 which is called the James-Stein estimator. This result surprises us because this means that a usual estimator is inadmissible in the framework of the simultaneous estimation of several parameters, although the components of the estimator are separately admissible to estimate the corresponding one-dimensional parameters, and we call it the Stein phenomenon. Generally if ± dominates a minimax estimator, then ± is also minimax and, thus, definitely preferred. It is, therefore, important from decision-theoretic point of view, to ascertain whether a proposed minimax estimator ± is admissible. Moreover if the estimator ± in the above is also inadmissible, it is interesting to derive admissible esti- mators improving upon ±. In this thesis, we mainly consider the estimation problem in Stein(1956)’s setting and concentrate our energies on investigating a class of estimators satisfying two optimalities: minimaxity and admissibility. We also consider the problem of deriving admissible estimators improving upon the James-Stein estimator which is known to be inadmissible. The outline of this thesis is as follows. In Chapter 2, we review basic notions of statistical decision theory, particular in esti- mation problem. The concepts of “admissibility”, “minimaxity”, “Bayes” and “permis- sibility” are introduced and results concerning close connections among these concepts are presented with proof. Chapter 3 contains a variety of results concerning an estimation of a multivariate normal mean. In Section 3.1, the minimaxity and the inadmissibility for p ¸ 3 of the natural estimator X are proved. In Section 3.2, results of Brown(1971,1988) and Srinivasan(1981) concerning admissibility and permissibility in this estimation problem are presented. In Section 3.3, we consider a generalized Bayes estimator ±h with respect 2 to scale mixtures of multivariate normal distribution whose density is proportional to Z µ ¶ µ ¶ 1 ¸ p=2 ¸ exp ¡ kθk2 ¸¡ah(¸)d¸; (1.1) 0 1 ¡ ¸ 2(1 ¡ ¸) where h(¸) is a measurable positive function on (0; 1). By Brown(1971)’s result, it is shown that ±h is admissible if and only if a · 2. Next we consider the minimaxity of ±h. As seen easily, all classes of minimax generalized Bayes estimators given in the hith- erto researches, that is, Strawderman(1971), Alam(1973), Berger(1976b), Faith(1978), Maruyama(1998) and Fourdrinier et al.(1998) are expressed as ±h although relations among classes above have not been clear. Hence we succeed in treating uniformly classes above. We point out that Fourdrinier et al.(1998)’s result is based on the boundedness of h(¸) although their result is powerful and includes results of Strawderman(1971), Berger(1976b) and Faith(1978). Indeed, the generalized Bayes estimators proposed by Alam(1973) and Maruyama(1998), are expressed as ±h in the case of unbounded h(¸). Thus we can see that the hitherto researches are divided in two, as Table 3.1 in Section 3.3.3, on the basis of whether h(¸) is bounded or not. Moreover we pay attention to the following Stein(1973)’s simple idea and provide it with the warrant for statistical decision theory. Stein(1973) suggested that the gener- alized Bayes estimator with respect to the prior distribution: the weighted sum of that determined by the density kθk2¡p, which corresponds to (1.1) with a = 2 and h ´ 1, and a measure concentrated at the origin, may dominate the James-Stein estimator. Since Kubokawa(1991) afterward showed that the generalized Bayes estimator with re- spect to the density kθk2¡p dominates the James-Stein estimator, Stein’s suggestion is to the point in the case where the ratio of a measure concentrated at the origin is very small. Efron and Morris(1976) expressed Stein(1973)’s estimator explicitly. As for the decision-theoretic properties of the estimator, admissibility is easily able to be checked by using Brown(1971)’s theorem. On the other hand, even minimaxity, to say noth- ing of dominance over the James-Stein estimator has not been proved yet. Extending Stein’s prior distribution, we consider a prior distribution: the weighted sum of a mea- sure concentrated at the origin and the prior distribution whose density is proportional 3 to (1.1). This prior distribution corresponds to one which has a density function (1.1) with h(¸) = ¯h(¸) + ±(¸ ¡ 1), where ¯ > 0 and ±(¢) is the Dirac delta function. We ¤ show that for h(¸) such that ±h is shown to be minimax, there exists a constant ¯ such that for ¯ ¸ ¯¤, the generalized Bayes estimator with respect to the above distribution is minimax. Furthermore the relations among minimaxity, the prior distribution, and the be- havior of Á(w), a shrinkage factor of the generalized Bayes estimator, are investigated.
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