This book is a reconstruction of Masafumi Akahira’s works on statistical estimation and its related fields, especially higher order asymptotics and non-regular cases with consideration based on information amounts. There have been few books on higher order asymptotics and non-regular cases, but the book helps the reader to understand the meaning and implications of the hierarchical structure of higher order asymptotics and gives some insights into the structure of non-regular estimation. After the results on the second and third order asymptotic efficiency in the volume entitled “Joint Statistical Papers of Akahira and Takeuchi” are summarized, the positive resolution of the conjecture “third order efficiency implies fourth order efficiency” of J. K. Ghosh is described, from which the fourth order asymptotic efficiency of the bias-adjusted maximum likelihood estimator and the bias-adjusted generalized Bayes estimator is shown. In non-regular situations, the (maximum) order of consistency and the (second order) asymptotic sufficiency are discussed including the view of loss of information, and in regular cases, the asymptotic deficiency of asymptotically efficient estimators is stated. For a truncated family of distributions, the influence of a nuisance parameter on the estimation of the interest parameter is investigated through maximum likelihood estimators. Also discussed is the higher order sequential estimation with the Bhattacharyya type bound, including the presence of a nuisance parameter. In interval estimation, a systematic method of the construction of a confidence interval for the difference between means is discussed including the Behrens–Fisher type problem, and also ordinary, Bayesian, likelihood ratio and combined Bayesian–frequentist type confidence intervals for a positive parameter are provided and compared. Further, the higher order approximations to percentage points of the non-central t-distribution and the distribution of a non-central t-statistic without the normality assumptions are given. Finally, the large deviation efficiency and large deviation approximations are discussed up to the higher order. Masafumi Akahira is Professor Emeritus at the University of Tsukuba. He has served as Vice President of the University of Tsukuba and President of the Japan Statistical Society.
