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Convex Analysisの二,三の進展について

1. Introduction and Overview. 2. Detecting Influential Observations and Outliers. 3. Detecting and Assessing Collinearity. 4. Applications and Remedies. 5. Research Issues and Directions for Extensions. Bibliography. Author Index. Subject Index.

Regression Diagnostics: Identifying Influential Data and Sources of Collinearity

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Table Of Integrals Series And Products

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Methods of information geometry

SUMMARY Second-order asymptotic theory of predictive distributions is investigated. It is shown that estimative distributions with asymptotically efficient estimators can be improved to predictive distributions that do not belong to the model. The model is assumed to be a multidimensional curved exponential family. Predictive distributions can be constructed by shifting estimative distributions in a direction orthogonal to the model. The average Kullback-Leibler divergence from the true distribution to a predictive distribution is represented as the sum of two components. One of them depends on the choice of the estimative distribution and the other depends on the shift orthogonal to the model. The optimal shift orthogonal to the model is obtained. A quantity that can be regarded as the mixture mean curvature of the model in the space of all probability distributions is introduced. The difference in average Kullback-Leibler divergence between an estimative distribution and the optimal predictive distribution obtained by shifting it can be regarded as the mixture mean curvature of the model. An asymptotic expression for Bayesian predictive distributions is also obtained.

On asymptotic properties of predictive distributions

SUMMARY We investigate shrinkage methods for constructing predictive distributions. We consider the multivariate Normal model with a known covariance matrix and show that there exists a shrinkage predictive distribution dominating the Bayesian predictive distribution based on the vague prior when the dimension is not less than three. Kullback-Leibler divergence from the true distribution to a predictive distribution is adopted as a loss function. Somiie key w,ords: Invariance; James-Stein estimator; Kullback-Leibler divergence; Stein's prior; Vague prior.

A shrinkage predictive distribution for multivariate Normal observables

We investigate shrinkage priors for constructing Bayesian predictive distributions. It is shown that there exist shrinkage predictive distributions asymptotically dominating Bayesian predictive distributions based on the Jeffreys prior or other vague priors if the model manifold satisfies some differential geometric conditions. Kullback--Leibler divergence from the true distribution to a predictive distribution is adopted as a loss function. Conformal transformations of model manifolds corresponding to vague priors are introduced. We show several examples where shrinkage predictive distributions dominate Bayesian predictive distributions based on vague priors.

Shrinkage priors for Bayesian prediction

We investigate shrinkage priors for constructing Bayesian predictive distributions. It is shown that there exist shrinkage predictive distributions asymptotically dominating Bayesian predictive distributions based on the Jeffreys prior or other vague priors if the model manifold satisfies some differential geometric conditions. Kullback-Leibler divergence from the true distribution to a predictive distribution is adopted as a loss function. Conformal transformations of model manifolds corresponding to vague priors are introduced. We show several examples where shrinkage predictive distributions dominate Bayesian predictive distributions based on vague priors.

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Simultaneous predictive distributions for independent Poisson observables are investigated. A class of improper prior distributions for Poisson means is introduced. The Bayesian predictive distributions based on priors from the introduced class are shown to be admissible under the Kullback-Leibler loss. A Bayesian predictive distribution based on a prior in this class dominates the Bayesian predictive distribution based on the Jeffreys prior.

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Fumiyasu Komaki

Papers

On asymptotic properties of predictive distributions

A shrinkage predictive distribution for multivariate Normal observables

Shrinkage priors for Bayesian prediction

Shrinkage priors for Bayesian prediction

Simultaneous prediction of independent Poisson observables