AISM 52, 197-214

## Quantitative approximation to the ordered Dirichlet distribution under varying basic probability spaces

### Tomoya Yamada^{1} and Tadashi Matsunawa^{2}

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

^{2}The 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.

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