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About this book

This book highlights the forefront of research on statistical distribution theory, with a focus on unconventional random quantities, and on phenomena such as random partitioning. The respective papers reflect the continuing appeal of distribution theory and the lively interest in this classic field, which owes much of its expansion since the 1960s to Professor Masaaki Sibuya, to whom this book is dedicated.
The topics addressed include a test procedure for discriminating the (multivariate) Ewens distribution from the Pitman Sampling Formula, approximation to the length of the Ewens distribution by discrete distributions and the normal distribution, and the distribution of the number of levels in [s]-specified random permutations. Also included are distributions associated with orthogonal polynomials with a symmetric matrix argument and the characterization of the Jeffreys prior.

Table of Contents

Frontmatter

Chapter 1. Gibbs Base Random Partitions

Abstract
As a typical family of random partitions on \(\mathcal {P}_{n,k}\), the set of partitions of n into k parts, the conditional distribution of Pitman’s random partition, termed as the Gibbs base random partition, GBRP \((\alpha )\), is investigated. The set \(\mathcal {P}_{n,k}\) is a lattice with respect to majorization partial order with unique minimum and maximum, and GBRP \((\alpha )\) has TP2 with respect to this order. The main purpose of this paper is to study such a family of random partitions and the inference on its parameter.
Masaaki Sibuya

Chapter 2. Asymptotic and Approximate Discrete Distributions for the Length of the Ewens Sampling Formula

Abstract
The Ewens sampling formula is well known as the probability for a partition of a positive integer. Here, we discuss the asymptotic and approximate discrete distributions of the length of the formula. We give a sufficient condition for the length to converge in distribution to the shifted Poisson distribution. This condition is proved using two methods: One is based on the sum of independent Bernoulli random variables, and the other is based on an expression of the length that is not the sum of independent random variables. As discrete approximations of the length, we give those based on the Poisson distribution and the binomial distribution. The results show that the first two moments of the approximation based on the binomial distribution are almost equal to those of the length. Two applications of this approximation are given.
Hajime Yamato

Chapter 3. Error Bounds for the Normal Approximation to the Length of a Ewens Partition

Abstract
Let \(K(=K_{n,\theta })\) be a positive integer-valued random variable with a distribution given by \(\mathrm{P}(K = x) = \bar{s}(n,x) \theta ^x/(\theta )_n\) \((x=1,\ldots ,n) \), where \(\theta \) is a positive value, n is a positive integer, \((\theta )_n=\theta (\theta +1)\cdots (\theta +n-1)\), and \(\bar{s}(n,x)\) is the coefficient of \(\theta ^x\) in \((\theta )_n\) for \(x=1,\ldots ,n\). This formula describes the distribution of the length of a Ewens partition, which is a standard model of random partitions. As n tends to infinity, K asymptotically follows a normal distribution. Moreover, as n and \(\theta \) simultaneously tend to infinity, if \(n^2/\theta \rightarrow \infty \), K also asymptotically follows a normal distribution. This study provides error bounds for the normal approximation. The results show that the decay rate of the error varies with asymptotic regimes.
Koji Tsukuda

Chapter 4. Distribution of Number of Levels in an -Specified Random Permutation

Abstract
Successions, Eulerian and Simon Newcomb numbers, and levels are the best-known patterns associated with \([\varvec{s}]\)-specified random permutations. The distribution of the number of rises was first studied in 1755 by Euler. However, the distribution of the number of levels in an \([\varvec{s}]\)-specified random permutation remained unknown. In this study, our main goal is to identify the distribution of the number of levels, which we achieve using the finite Markov chain imbedding technique and insertion procedure. An example is given to illustrate the theoretical result.
James C. Fu

Chapter 5. Properties of General Systems of Orthogonal Polynomials with a Symmetric Matrix Argument

Abstract
There exists a large literature of the orthogonal polynomials (OPs) with a single variable associated with a univariate distribution. The theory of these OPs is well established and many properties of them are developed. Then, some authors have discussed the OPs with matrix arguments in the past. However, there are many unsolved properties, owing to the complex structures of the OPs with a matrix argument. In this paper, we extend some properties, which are well known for the OPs with a single variable, to those with a matrix argument. We give a brief discussion on the zonal polynomials and the general system of OPs with a symmetric matrix argument, with examples, the Hermite, the Laguerre, and the Jacobi polynomials. We derive the so-called three-term recurrence relations, and then, the Christoffel–Darboux formulas satisfied by the OPs with a symmetric matrix argument as a consequence of the three-term recurrence relations. Also, we present the “\((2k+1)\)-term recurrence relations”, an extension of the three-term recurrence relations, and then an extension of the Christoffel–Darboux formulas as its consequence. Finally we give a brief discussion on the linearization problem and the representation of Hermite polynomials as moments. For the derivations of those results, the theory of zonal and invariant polynomials with matrix arguments is useful.
Yasuko Chikuse

Chapter 6. A Characterization of Jeffreys’ Prior with Its Implications to Likelihood Inference

Abstract
A characterization of Jeffreys’ prior for a parameter of a distribution in the exponential family is given by the asymptotic equivalence of the posterior mean of the canonical parameter to the maximum likelihood estimator. A promising role of the posterior mean is discussed because of its optimality property. Further, methods for improving estimators are explored, when neither the posterior mean nor the maximum likelihood estimator performs favorably. The possible advantages of conjugate analysis based on a suitably chosen prior are examined.
Takemi Yanagimoto, Toshio Ohnishi
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