CoInvestigator(Kenkyūbuntansha) 
AKI Shigeo The Institute of Statistical Mathematics, Assistant Professor, 統計基礎研究系, 助手 (90132696)
KONISHI Sadanori The Institute of Statistical Mathematics, Associate Professor, 統計基礎研究系, 助教授 (40090550)
HIRANO Katsuomi The Institute of Statistical Mathematics, Associate Professor, 統計教育情報センター, 助教授 (30000186)

Research Abstract 
In the theory of sampling distributions we often encounter with the situation who distribution of a statistic can not be obtained in a closed from, or even if it is obtain distribution is of no use because of its complexity. Therefore, studies on approximation distributions become important. Limiting forms of the sampling distributions, which are relatively simple, have been c wide class of statistics. Much has also been done about the expansion of the sampling distril its limiting distribution. However, from the practical, as well as theoretical, point of important problem in approximating a probability distribution by some sort of functions is accuracy of the approximation, i.e., to evaluate an upper bound of the error of the approxim In this research project we investigated asymptotic expansions of the density of the dis a random variable of the form Y=sigmaX, where epsilon(>0) and X are independent and X follows normal distribution N(0,1), assuming implicitly, that sigma is close 1 in the sense that both E E(sigma^<2> 1)^k are small. The expansion is given in terms of E(sigma^<+2> 1)^j,j = 0,1,...,k1,polynomials, and the difference between f and its approximation is absolutely integrable o space and is bounded by, roughly speaking, constant times the sum of E(sigma^2 1)^k and E( particular, the probability of the event YepsilonA is approximated, within this error, by the in approximating function over the set A. The result is expected to be applicable to the multivar Another result of our research includes determination of the limit distribution of some test
