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.
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Ikeda, S. and Matsunawa, T. (1972). On the uniform asymptotic joint normality of sample quantiles, Ann. Inst. Statist. Math., 24, 33–52.
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions, Vol. 1, 2nd ed., Wiley, New York.
Matsunawa, T. (1976). Some inequalities based on inverse factorial series, Ann. Inst. Statist. Math., 28, 291–305.
Matsunawa, T. (1982). Uniform ø — equivalence of probability distributions based on information and related measures of discrepancy, Ann. Inst. Statist. Math., 34, 1–17.
Matsunawa, T. (1986). Modified information criteria for a uniform approximate equivalence of probability distributions, Ann. Inst. Statist. Math., 38, 205–222.
Matsunawa, T. (1995). Development of distributions — The Legendre transformation and canonical information criteria —, Pro. Inst. Statist. Math., 43, 293–311 (in Japanese).
Mosteller, F. (1946). On some useful “inefficient” statistics, Ann. Math. Statist., 17, 377–408.
Reiss, R. D. (1974). The asymptotic normality and asymptotic expansions for the joint distribution of several order statistics, Colloq. Math. Soc. János Bolyai, 297–340.
Walker, A. M. (1968). A note on the asymptotic distribution of sample quantiles, J. Roy. Statist. Soc. Ser. B, 30, 570–575.
Weiss, L. (1969). The asymptotic joint distribution of an increasing number of sample quantiles, Ann. Inst. Statist. Math., 21, 257–263.
Yamada, T. and Matsunawa, T. (1998). Uniform approximations to probability distributions based on K-L information defined on approximate main domains — With applications to quantitative evaluations of fluctuations in multivariate general exponential families —, Proc. Inst. Statist. Math., 46, 461–476 (in Japanese).
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Yamada, T., Matsunawa, T. Quantitative Approximation to the Ordered Dirichlet Distribution under Varying Basic Probability Spaces. Annals of the Institute of Statistical Mathematics 52, 197–214 (2000). https://doi.org/10.1023/A:1004153419736
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DOI: https://doi.org/10.1023/A:1004153419736