Researches on the higher order asymptotic theory of statistical inference

我们研究了统计推断中高阶渐近理论中的区间估计，并将其应用于样本相关系数的一个百分点的分布，在这两种情况下都能得到有用的结果。对于离散分布，通常不可能获得给定大小的非随机化检验或可信区间，并且实际大小往往与规定的水平大不相同。但随机程序，这在理论上是相当好的，并不容易被从业者接受。因此，在本研究中，我们从最优随机化检验，即一致最强无偏检验出发，构造了一个随机化的可信区间，并利用充分统计量分布的Edgeworth展开式讨论了它的逼近。我们还考察了在泊松分布和二项分布的情况下近似是准确的。其次，对相关系数Rho的推断是重要的课题之一，直到…此外，对于R的Fisher Z变换，现在已经使用Corish-Fisher展开将样本相关系数R的一个百分点逼近到更高的阶数，但遗憾的是，近似方式非常复杂，必须使用计算处理。因此，在研究中，我们用类似于Akahira(1995)提出的非中心t-分布的一个百分点的近似公式，推导出了一个新的关于R分布一个百分点的近似公式。事实上，我们利用Corish-Fisher展开推导了基于正态随机变量和卡方随机变量的线性组合的统计量的近似公式。在数值计算中发现，近似公式主要有正态近似、Fisher Z近似等，在α和Rho的各种情况下，甚至在样本大小为10的情况下，都给出了几乎精确的数值。它们也被广泛应用于实际问题。与相关领域的研究人员进行的讨论非常有用。较少

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