QUANTITATIVE APPROXIMATION TO THE ORDERED DIRICHLET DISTRIBUTION UNDER VARYING BASIC PROBABILITY SPACES

TOMOYA YAMADA^{1} AND TADASHI MATSUNAWA^{2}

1The Graduate Universities for Advanced Studies, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106-8569, Japan

2The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106-8569, Japan

(Received May 1, 1998; revised April 8, 1999)

Abstract. An approximate expansion of a sequence of ordered Dirichlet densities is given under the set-up with varying dimensions of the relating basic probability spaces. The problem is handled as the approximation to the joint distribution of an increasing number of selected order statistics based on the random sample drawn from the uniform distribution U(0, 1). Some inverse factorial series to the expansion of logarithmic function enable us to give quantitative error evaluations to our problem. With the help of them the relating modified K-L information number, which is defined on an approximate main domain and different from the usual ones, is accurately evaluated. Further, the proof of the approximate joint normality of the selected order statistics is more systematically presented than those given in existing works. Concerning the approximate normality the modified affinity and the half variation distance are also evaluated.

Key words and phrases: Ordered Dirichlet distribution, approximate distribution, sample quantiles, modified K-L information number, modified affinity, half variation distance, approximate main domain, approximate joint normality.