Takemi Yanagimoto

Families of positively dependent random variables

SummaryPossible definitions of positive dependence of random variables are systematically and exhaustively examined and previous results on this notion are improved. Three approaches to the definitions are proposed. They are similar to those given by the author and Sibuya in defining “stochastically larger component of a random vector.” Unbiasedness of rank tests of independence is treated.

Smoothing Serial Count Data Through a State-Space Model

A method is proposed for smoothing serial Poisson count data through state-space modeling. Recursive formulas for evaluating the exact likelihood are given. The exact likelihood yields the likelihood ratio test for homogeneity of the Poisson means. The method is versatile enough to permit various extensions, including that for serial binomial data. The proposed method is applied to three sets of weekly disease incidence data.

Combining moment estimates of a parameter common through strata

Abstract A method for combining moment estimates of a parameter common through strata is developed, which derives new and known examples including the Mantel-Haenszel estimator. The method is based on the theory of an unbiased estimating equation developed by Godambe and his coworkers. Other interpretations of the proposed method, such as an extension of the quasi- likelihood function, are also given.

Stochastically larger component of a random vector

SummaryDefinitions of different strengths are given to the notion of ‘a stochastically larger component of a two-dimensional random vector.’ Some of them reduce to the known definitions of stochastic order relationship when the components are stochastically independent. The definitions and the approach are related to nonparametric problems.

Isotonic tests for spread and tail

SummaryOne-sample test problem for ‘stochastically more (or less) spread’ is defined and a family of tests with isotonic power is given. The problem is closely related to that for ‘longer (or shorter) tail’ in the reliability theory and the correspondence between them is shown.To characterize the tests three spread preorders inRn and corre-sponding tail preorders inR+n are introduced. Functions which are ‘monotone’ in these orders, and subsets which are ‘centrifugal’ or ‘centripetal’ with respect to these orders are studied. These notions generalize the Schur convexity.

The Kullback-Leibler risk of the Stein estimator and the conditional MLE

The decomposition of the Kullback-Leibler risk of the maximum likelihood estimator (MLE) is discussed in relation to the Stein estimator and the conditional MLE. A notable correspondence between the decomposition in terms of the Stein estimator and that in terms of the conditional MLE is observed. This decomposition reflects that of the expected log-likelihood ratio. Accordingly, it is concluded that these modified estimators reduce the risk by reducing the expected log-likelihood ratio. The empirical Bayes method is discussed from this point of view.

A notion of an obstructive residual likelihood

SummaryA new notion of an obstructive residual likelihood is proposed and explored. Examples where the conditional maximum likelihood estimator is preferable to the unconditional maximum likelihood estimator are discussed. In these examples the residual likelihood can be obstructive in deriving a preferable estimator, when the maximum likelihood criterion is applied. This notion is different from a similar notion ancillarity, which simply emphasizes that a residual likelihood is un-informative.

Estimation for the negative binomial distribution based on the conditional likelihood

For estimating the dispersion parameter of the negative binomial distribution, the conditional maximum likelihood estimator is compared with the maximum likelihood and moment estimators. The biases, mean squared errors and the Kullback-Leibler risks of the estimators are examined by simulation studies for a single population and multiple ones with a common parameter. We observe that the conditional maximum likelihood estimator is superior to the others in frequency for a single population as a whole, and it is more definite for multiple populations.

Possible superiority of the conditional MLE over the unconditional MLE

The possibility that the conditional maximum likelihood estimator (CMLE) is superior to the unconditional maximum likelihood estimator (UMLE) is discussed in examples where the residual likelihood is obstructive. We observe relatively smaller risks of the CMLE for a finite sample size. The models in the study include the normal, inverse Gauss, gamma, two-parameter exponential, logit, negative binomial and two-parameter geometric ones.

Extensions of the conjugate prior through the Kullback-Leibler separators

The 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 will be 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.