Researches on the higher order asymptotic theory of statistical inference

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We studied the interval estimation in the theory of higher order asymptotics in statistical inference and an application to a percentage point of the distribution of sample correlation coefficient, and could get useful results in both cases. For discrete distributions it was usually impossible to obtain a non-randomized test or confidence interval with given size, and an actual size was often quite different from the prescribed level. But randomized procedures, which was quite nice in theory, were not easily acceptable to practitioners. So, in this research we constructed a randomized confidence interval from an optimal randomized test, i.e.a uniformly most powerful unbiased test and discussed its approximation using the Edgeworth expansion of the distribution of sufficient statistics. We also investigated that the approximation was accurate in the case of Poisson and binomial distributions.Next, an inference on the correlation coefficient rho is one of the important topics, and until … More now a percentage point of the sample correlation coefficient R has been approximated up to the higher order using the Cornish-Fisher expansion for Fisher's Z-transformation of R.But, unfortunately the approximation way was very complex and a computational treatment must be used. So, in the research we derived a new approximation formula of a percentage point of the distribution of R in a similar way to Akahira (1995) who introduced the approximation formula of a percentage point of the non-central t-distribution. Indeed, we derived the approximation formula using the Cornish-Fisher expansion for the statistic based on a linear combination of a normal random variable and chi-random variables. In numerical calculations, the approximation formula was seen to be that it dominated the normal approximation, the approximation by Fisher's Z-approximation, etc.and gives almost precise values in various cases of alpha and rho even for size 10 of sample.The research was carried out according to plan and the above results were obtained. They were also widely applied to practical problems. Discussions with researchers in related fields were very useful. Less