On Efron's Parameterization

This paper discusses a curved exponential family of distributions which is defined by a differential equation with respect to the expectation parameters in the two–dimensional exponential family. The differential equation considered here is the same as the one given by Efron (1975) for the trinomial distribution. This equation is extended here to a general exponential family, and called Efron's parameterization in the two–dimensional exponential family. The solution of Efron's parameterization is obtained explicitly in an exponential family, although Kumagai & Inagaki (1996) showed that there exists no proper solution of Efron's equation for the trinomial distribution, in line with the counterexample given by Efron (1975 p. 1206). The paper gives some characterizations of Efron's parameterization with special reference to Fisher's circle model. The implications of these characterizations are the two–dimensional normal distribution and a spiral curve in the plane.     

Iterative weighted least-squares estimates in a heteroscedastic linear regression model

The aim of this study is to improve the efficiency of weighted least-squares estimates for a regression parameter. An iterative procedure, starting with an unbiased estimate other than the unweighted least-squares estimate, yields estimates which are asymptotically more efficient than the feasible generalized least-squares estimate when errors are spherically distributed. The result has an application in the improvement of the Graybill–Deal estimate of the common mean of several normal populations.     

Distributions of simple patterns in some kinds of exchangeable sequences

In this article, we consider the distributions of simple patterns in some types of sequences of infinite exchangeable multi-state trials. The distributions on exchangeable multi-state trials are considered in terms of an extension of de Finetti's theorem. As an application of partially exchangeable sequences, distributions on a Markov exchangeable sequence are studied. Furthermore, we propose a new type of partially exchangeable sequence and examine its properties. In addition, we discuss the distribution theory in the case of the finite exchangeable sequences. The results presented here provide a wide framework for developing the exact distribution theory of simple patterns. Finally, some examples are given in order to illustrate our theoretical results.     

Methods for Studying Generalized Birthday and Coupon Collection Problems

In this paper, we consider generalizations of two classical probability problems: the birthday problem and the coupon collector's problem. These problems are discussed in terms of urn models and captured through generating functions. Some methods for the study of the problems are presented. Furthermore, we also formulate the generalized birthday and coupon collector's problems as the waiting time problems. In each case, numerical examples are given in order to illustrate the feasibility of our methods.     

On discrete distributions generated from drawing balls with bivariate labels

In this article, we consider urn models under three types of sampling schemes in terms of the probability-generating functions. The tools are developed for the evaluation of the distributions arising from the urn models along with some examples. Furthermore, the distributions are investigated by making use of the Bell polynomials. The results presented here provide a wide framework for developing the theory of urn models. As examples, we propose new class of probability models, which are called multiple-player problems and examine their properties. Finally, we treat the parameter estimation problem in the waiting time distributions with a numerical example.     

Distributions of Runs and Scans on Higher-Order Markov Trees

In this article, we consider the distributions of the number of success runs of specified length and scans on a higher-order Markov tree under three different enumeration schemes (the “non overlapping”, the “at least”, and the “overlapping” scheme). Recursive formulae for the evaluation of their probability generating functions are established. We provide a proper framework for extending the exact distribution theory of runs and scans from based on sequences to based on directed trees. Some numerical results for the run and scan statistics are given in order to illustrate the computational aspects and the feasibility of our theoretical results. Finally, two special reliability systems are considered, which are closely related to our general results.