Hajime Yamato

Relations between limiting bayes estimates and the u-statistics for estimable parameters of degrees 2 and 3

The Bayes estimates of estimable parameters of degrees 2 and 3 are obtained against a Dirichlet prior process and the squared error loss. The relations between the limits of the Bayes estimates and the U-statistics are shown. Some examples are given.

Ordered and unordered random partitions of an integer and the GEM distribution

The joint distribution of ordered (or unordered) frequencies of a sample from random proportions is the Donnelly-Tavare-Griffiths formula (or the Ewens sampling formula) if and only if the size-biased permutation of the random proportions has the GEM distribution.

A pólya urn model with a continuum of colors

For a Pólya urn model with a continuum of colors introduced by Blackwell and MacQueen ((1973),Ann. Statist.,2, 1152–1174), we show the joint distribution of colors aftern draws from which several properties of the urn model are derived. The similar results hold for the case where the initial distribution of colors is invariant under a finite group of transformations.

ORDERED SAMPLE FROM TWO-PARAMETER GEM DISTRIBUTION

Pitmans two-parameter GEM distribution is characterized by the product moments. Based on that fact, sampling distributions of some statistics of a random sample from the two-parameter GEM distribution are derived. The main statistics are the frequencies of categories in a sample in the order of appearance, the intervals between new categories, and number of distinct categories.

Nonparametric bayes estimates of estimable parameters with a dirichlet invarant process and inveriant u-statistics

Tne Bayes estimates of estimable parameters of arbitrary degree in the one sample case are obtained against a Dirichlet invariant. process prior and the squared error loss. We also oive the limits of Bayes estimates, which are related to the in- a- variant U-statistics. For a fixed distribution, the limits of Bayes estimates have the asymptotic normal distribution under certain conditjons.

Bayes estimates of estimable parameters with a direchlet invariant process

The Bayes estimates of estimable parameters of degree 2 are obtained against a Dirichlet invariant process prior and the squared error loss. Some examples are given with the limits of the Bayes estimates.

Invariant U-statistics

For a class of distributions which are invariant under a group of transformations, we propose an estimator ot an estimable parameter. The estimator, which we call the invariant U-statistic, is the uniformly minimum variance unbiased estimator of the corresponding estimable parameter for the class of all continuous distributions which are invariant under the group of transformations.

Characterization of some random partitions

Four types of celebrated random partitions are characterized. The types are ordered and unordered partitions of a finite set and those of a number. The random partitions are closely related and have been studied in population genetics under the names Hoppe’s urn, Donnelly-Tavaré-Griffiths formula and Ewens’ sampling formula. Characterizations of Ewens’ sampling formula in this paper improve those by Kingman (1978) and Donnelly (1986).

ON THE DONNELLY-TAVAR ´ E-GRIFFITHS FORMULA ASSOCIATED WITH THE COALESCENT

We evaluate the moments of the Donnelly-Tavare-Griffiths formula appearing in the n-coalescent with mutation, which characterizes this formula. The formula is also characterized by using Waring distribution and Yule distribution. The asymptotic distributions of the related statistics are obtained as n tends to infinity.

Jackknifing a Convex Combination of One-Sample U-Statistics

As an estimator of an estimable parameter, we consider a convex combination of U-statistics (Toda and Yamato, 2001) Yn which includes U-statistic, V-statistic, S-statistic, and limit of Bayes estimate. We introduce the bias-reduced jackknifed estimator based on this convex combination. This statistic is written as a linear combination of U-statistics. From this, we know that jackknifed version of the V-statistic is not equal to the U-statistic for the degree more than or equal to three. To see the effects of jackknifing the statistic Yn, we show the difference between Yn and by the second-order deficiency. By second-order deficiency, we can not see the difference between Un and and between different two s. Therefore, we show the differences between them by the fourth-order deficiency. We also give some examples using the V-statistic, limit of Bayes estimate, and S-statistic.