Bounded risk estimation of a linear combination of location parameters in negative exponential distributions via three-stage sampling

The paper deals with the problem of bounded risk point estimation for a linear combination of location parameters of two negative exponential distributions. Isogai and Futschik considered the situation when the location and scale parameters are all unknown. They proposed purely sequential procedures and gave second order expansions of the average sample sizes and risks. In this paper we propose three-stage procedures and derive second order expansions of the average sample sizes and risks. Further, we compare the results with those from previous work.     

Locating libraries on a street

Hotelling (1929) studied two competing firms choosing their locations on a street. We consider instead a planner who builds two identical public facilities (e.g., libraries, parks, bridges, etc). We ask a normative question: Where should the planner build these facilities? We prove an axiomatic characterization of the efficient social choice rules that satisfy what is called the replacement-domination, which is a formulation of the idea of “solidarity” among the agents. Received: 26 November 1997/Accepted: 28 February 2000     

Power transformation of the F distribution and a power normal family

To transform the F distribution to a normal distribution, two types of formula for power transformation of the F variable are introduced. One formula is an extension of the Wilson-Hilferty transformation for the chi 2 variable, and the other type is based on the median of the F distribution. Combining those two formulas, a simple formula for the median of the F distribution is derived, and its numerical accuracy is evaluated. Simplification of the formula of the Wilson-Hilferty transformation, through the median formula, leads us to construct a power normal family from the generalized F distribution. Unlike the Box-Cox power normal family, our family has a property that the covariance structure of the maximum-likelihood estimates of the parameters is invariant under a scale transformation of the response variable. Numerical examples are given to show the diff erence between two power normal families.     

Sequential estimation of a linear function of location parameters of two negative exponential distributions

The paper considers the problem of bounded risk point estimation for a linear function of location parameters of two negative exponential distributions, including the difference in a special case, when two scale parameters are unknown. Purely sequential procedures are proposed and second order expansions of the average sample sizes and risk are given. Furthermore some simulation results are provided.     

Nonparametric recursive estimation in stationary markov processes

This paper deals with a class of recursive kernel estimators of the transition probability density function t(y|x) of a stationary Markov process. A sufficient condition for such estimators to be weakly and strongly 2 consistent for almost all (x,y)∈R² is given. Further an L, convergence result is obtained. No continuity conditions are imposed on t(y|x).     

Applications of a new power normal family

The main purpose of this paper is to give an algorithm to attain joint normality of non-normal multivariate observations through a new power normal family introduced by the author (Isogai, 1999). The algorithm tries to transform each marginal variable simultaneously to joint normality, but due to a large number of parameters it repeats a maximization process with respect to the conditional normal density of one transformed variable given the other transformed variables. A non-normal data set is used to examine performance of the algorithm, and the degree of achievement of joint normality is evaluated by measures of multivariate skewness and kurtosis. Besides the above topic, making use of properties of our power normal family, we discuss not only a normal approximation formula of non-central F distributions in the frame of regression analysis but also some decomposition formulas of a power parameter, which appear in a Wilson-Hilferty power transformation setting.     

Higher order approximations by a two-stage procedure for a negative exponential distribution

This paper deals with the problem of fixed-width confidence interval estimation of the location μ of a negative exponential distribution with unknown scale σ. Suppose we have information on the scale parameter σ such that σ>σ_L where σ_L(>0) is known to the experimenter from past experiences. We propose a two-stage procedure and provide higher order asymptotic expansions of the expected sample size and the coverage probability.