Sankhya: The Indian Journal of Statistics

1993, Volume 55, Series A, Pt. 2, 244--258

EDGEWORTH EXPANSIONS OF A FUNCTION OF SAMPLE MEANS UNDER MINIMAL MOMENT CONDITIONS AND PARTIAL CRAMÉR'S CONDITION

By

GUTTI JOGESH BABU, *The Pennsylvania State University*

And

Z. D. BAI, *Temple University*

SUMMARY. A wide range of statistics can be expressed as smooth functions of sample means of random vectors. Edgeworth expansions of such statistics are generally obtained under Cramer's condition. In many practical situations, like the case of ratio statistics, only one of the components of the random vector satisfies the Cramer's condition, while the rest do not. Edgeworth expansions are established under partial Cramer's condition. Further the conditions on the moments are relaxed to the minimum needed to define the expansion.

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*AMS (1980) subject classification.* 60F05,62E20, 62F20

*Key words and phrases*. Asymptotic expansions, Cramer's condition, Edgeworth expansion, Student's $t$-statistic