@misc{oai:ir.soken.ac.jp:00000759,
 author = {大西, 俊郎 and オオニシ, トシオ and OONISHI, Toshio},
 month = {2016-02-17},
 note = {Simultaneous estimation of a high-dimensional parameter is and will be a very important subject of research.  The recent rapid progress in computational environment has made it easy to collect a complex dataset.  The role of a statistical model containing a high-dimensional parameter, which is suited for such a complex dataset, is getting more and more significant in accordance with this progress.<br />　　The aim of this dissertation is investigate simultaneous estimation of a high-dimensional parameter from a view point of a Pythagorean relationship both theoretically and practically.  Theoretical researches on estimation in such complex models are needed from the viewpoint of application.  Also actual estimation procedures with nice properties are in high demand.  The James-Stein estimator (James and Stein, 1961) is a breakthrough estimator in this area.  A number of works have been devoted to elucidating the reason why the James-Stein estimator or its modifiers perform well.  There seems to be the following two approaches to casting light to the reason.  One is a Pythagorean relationship holding among the maximum likelihood estimator (MLE), the James-Stein estimator and the true parameter, which was pointed out first by Stein (1981).  The other approach is the interpretation of the James-Stein estimator as an empirical Bayes estimator, as proposed in Efion and Morris (1973).<br />　　The theoretical aspect of the motivation of this dissertation is as follows.  A Pythagorean relationship is one of the most natural and fundamental notions by which we can improve upon something or show a certain inequality.  For example, the Pythagorean relationship pointed out by Stein (1981) makes it clear how the James-Stein estimator dominates the MLE.  It is expected that such a Pythagorean relationship will lead to better estimation in a unified way.  Yanagimoto (1994, 2000) discussed Stein-type estimation from this point of view.  We show in Chapter 3 that a Pythagorean relationship holds in the field of estimating functions, although the original Pythagorean relationship by Stein (1981) does in the field of estimators.<br />　　Further, in Chapters 4 and 5 which investigate the Bayesian analysis, a Pythagorean relationship plays another important role.  The optimality of the Bayes estimator is understood quite easily through the Pythagorean relationship.  The three points, the Bayes estimator, an arbitrary estimator and the true parameter value, constitute on the average a modified right triangle.  The modified triangle makes it clear how the Bayes estimator is superior to an arbitrary estimator.<br />　　Here we state the practical aspect of the motivation of this dissertation.  As indicated by the fact that the James-Stein estimator is regarded as an empirical Bayes estimator (Efron and Morris, 1973), the Bayesian approach seems to be promising in the estimation of a high-dimensional parameter.  Although Chapter 3 investigates the problem from the frequentists' viewpoint, some of the obtained estimating functions can be interpreted from the Bayesian point of view.  The empirical Bayesian method provides us with practical inferential procedures for a vector parameter.  A difficulty in constructing an empirical Bayes estimator lies in that there are a restricted number of families of prior densities. This is why some useful families of prior densities are necessary.<br />　　Conjugate priors, originally introduced by Raiffa and Schlaifer (1961, p. 43-58), are of great use for their desirable properties and was assumed by Efion and Morris (1973) in deriving the empirical Bayes estimator.  To extend the notion of conjugate priors is of great significance in this respect.  A recent and extensive review of the conjugate priors is found in Gutierrez-Pena and Smith (1997).  Chapters 4 and 6 present two methods for eliciting prior densities.<br />　　The organization of this dissertation is as follows. In Chapter 2, some basic concepts and tools are presented, which lead to better understanding of the subsequent chapters.<br />　　In Chapter 3, which is based on Ohnishi and Yanagimoto (2003), a unified approach using a Pythagorean relationship reveals the mechanism through which the maximum likelihood estimation can be improved upon.  A Stein-type estimation of location vectors is discussed in terms of estimating functions.  We assess the superiority of an estimating equation by its mean squared norm.  The Coulomb potential function in electrostatics leads to a Pythagorean relationship with respect to this norm.  By making full use of the Pythagorean relationship, we improve upon the likelihood estimating function.  A further improvement is shown to be feasible under a certain condition.  We pursue possible strong relationships between the superiority over the likelihood estimating function and physical quantities appearing in the theory of electrostatics.<br />　　In Chapter 4, we enrich the notion of conjugate prior distributions in two directions and investigate the Bayesian analysis assuming the introduced prior densities.  A conjugate prior for the exponential family, referred to also as the natural conjugate prior, is represented in terms of the Kullback-Leibler separator.  This representation permits us to extend the conjugate prior to that for a general family of sampling distributions.  Further, by replacing the Kullback-Leibler separator with its dual form, we define another form of a prior, which is called the mean conjugate prior.  Various results on duality between the two conjugate priors are shown.  Implications of this approach include richer families of prior distributions induced by a sampling distribution and the empirical Bayes estimation of a high-dimensional mean parameter.  A Pythagorean relationship with respect to the Kullback-Leibler separator is used both to show the optimality of the Bayes estimator and to construct an empirical Bayes estimator.  This chapter is due to Yanagimoto and Ohnishi (2002, 2003).<br />　　In Chapter 5, we introduce specific location-dispersion models and discuss a conjugate analysis by assuming some prior density.  The models are called the 1-additive location-dispersion models, and we apply one of the prior elicitation procedures in Chapter 4.  The 1-additive location-dispersion model is generated by the density function whose logarithm satisfies a certain addition identity, an extension of the addition formula for the (hyperbolic) cosine function.  The addition identity can be interpreted in the light of statistical mechanics.  We show that 1-additive location-dispersion models consist of the familiar five models.  The assumed prior density is proved to be closed under sampling. We also calculate the Bayes estimator under a Kullback-Leibler loss function.  A unified approach proves that the posterior mode, which has an analytical form, is optimal.  Empirical Bayes estimators of location vectors are constructed explicitly in the five 1-additive location-dispersion models.  This chapter is based on Ohnishi and Yanagimoto (2002).<br />　　In Chapter 6, we propose a prior elicitation method other than the one in Chapter 5.  The key feature is the use of the likelihood of the distribution of the MLE.  The derived prior density is proved to be an extension of a conjugate prior density.  Three examples including the 1-additive location-dispersion model in Chapter 5 are presented in order to clarify our idea.  The applicability of our method is discussed with the use of the Barndorff-Nielsen's p*-formula　(1983)., 総研大乙第120号},
 title = {Simultaneous estimation of a high-dimensional  parameter through a Pythagorean relationship},
 year = {}
}