ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

12. Probability and Statistics

1991 MS Classification: 62E10 Hwang, Tea-Yuan, National Tsing-Hua University; Hu, Chin-Yuan, National Changhua University of Education, Taiwan, ROC On some characterizations of popultaion distributions Earlier works of the present authors and a method due to Anosov for solving certain intego-functional equations are combined to show that the independence of the sample mean X̄_{n} and the Z_{n}-statistic characterizes the normal population, when the random samples are iid from a population having a continuous density function on R, and the sample size n ≥ 3; obiously, the sample standard devitation is an Z_{n}statistic. An important subclass of Z_{n}-statistic with the form of a linear combination ∑_{i=1}^{n} a_{i}X_{(i)} of order statistics has been found, where a_{1} ≤ ⋯ ≤ a_{n}, not all equal and ∑_{i=1}^{n} a_{i} = 0; the sample range and Gini’s mean difference are included in this subclass. Similar approaches can be applied to prove that the independence of X̄_{n} and Z_{n}/X̄_{n} characterizes the gamma distribution. References: (1) D. V. Anosov, on an integral equation arising in statistics, Vestnik Leningrad Univ., 7(1964), 151-154, (2) T. Y. Hwang, and C. Y. Hu, On the joint distribution of studentized order statistics. Ann. Inst. Statist. Math., 46, (1994), 165-177.