Pioneering Works on Distribution Theory: In Honor of Masaaki Sibuya (SpringerBriefs in Statistics)

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. Preface Contents 1 Gibbs Base Random Partitions 1.1 Introduction 1.1.1 Partitions of a Number 1.1.2 Majorization Order of Partitions 1.1.3 Adjacency 1.2 Gibbs Base Random Partitions 1.2.1 The Ewens-Pitman Sampling Formula 1.2.2 Probability Measure on mathcalPn,k 1.2.3 Gibbs Base Random Partitions 1.3 Testing Statistical Hypotheses on α 1.3.1 Properties of GBRP 1.3.2 Testing Statistical Hypotheses 1.3.3 Problems in Computation References 2 Asymptotic and Approximate Discrete Distributions for the Length of the Ewens Sampling Formula 2.1 Introduction 2.2 Convergence in Distribution of Kn (I) 2.3 Convergence in Distribution of Kn (II) 2.4 Approximate Distributions for mathcalL(Kn) 2.4.1 Poisson Approximation 2.4.2 Binomial Approximation 2.5 Applications 2.5.1 An Approximation to the p.f. of the MLE of θ 2.5.2 An Estimation of the p.f. of Kn When θ Is Unknown References 3 Error Bounds for the Normal Approximation to the Length of a Ewens Partition 3.1 Introduction 3.1.1 Assumptions and Asymptotic Regimes 3.2 Main Results 3.2.1 An Upper Error Bound 3.2.2 Evaluation of the Decay Rate 3.3 Preliminary Results 3.3.1 A Representation of K Using a Bernoulli Sequence 3.3.2 Evaluations of Moments 3.4 Proofs of the Results in Sect.3.2 3.4.1 Proof of the Results in Sect.3.2.1 3.4.2 Proof of the Results in Sect.3.2.2 3.5 Conclusion References 4 Distribution of Number of Levels in an [s]-Specified Random Permutation 4.1 Introduction 4.2 Distribution of Number of Levels 4.3 Discussion References 5 Properties of General Systems of Orthogonal Polynomials with a Symmetric Matrix Argument 5.1 Introduction 5.2 Zonal Polynomials and General Systems of Orthogonal Polynomials with Matrix Arguments and Examples 5.2.1 Zonal Polynomials 5.2.2 General Systems of Orthogonal Polynomials 5.2.3 Examples 5.2.4 The Extended Orthogonal Polynomials with Multiple Matrix Arguments 5.3 Three-Term Recurrence Relations for Pλ(X) 5.3.1 One-Dimensional Case 5.3.2 Extension to { Pλ(X) } 5.3.3 (2k+1)-Term Recurrence Relations 5.4 Christoffel–Darboux Formula for Pλ(X) 5.4.1 Christoffel–Darboux Formulas 5.4.2 Extending the Christoffel–Darboux Formulas 5.5 Other Properties 5.5.1 Linearization Problem 5.5.2 Representation of the Hλ(X) as Moments References 6 A Characterization of Jeffreys' Prior with Its Implications to Likelihood Inference 6.1 Introduction 6.2 Preliminaries 6.2.1 Jeffreys' Prior in Conjugate Analysis 6.2.2 Kullback–Leibler Divergence 6.2.3 Optimal Predictors 6.3 A Characterization of Jeffreys' Prior 6.4 Implications to Likelihood Inferential Procedures 6.5 Optimality Property and Possible Improvements 6.5.1 Optimality Property 6.5.2 Improvements on the MLE 6.5.3 Improvements on the Posterior Mean Under Jeffreys' Prior 6.6 Discussion and Recommendations References