The saddlepoint approximations introduced by Daniels (1954) has been used as a simple and useful way for inverting the characteristic function to corresponding p.d.f. However, nowadays the ability of the computing environment enables us to calculate the integration such as Lévy’s inversion formula that includes complex numbers. We shall compare the saddlepoint method with the formula by through some numerical examples. Although the accuracy of the former is inferior to the latter, saddlepoint itself has an inversion formula to get exact p.d.f. Moreover, it naturally explains the performance of the asymptotic normality of a statistics (Takeuchi (2013, 2014)). This is demonstrated when we evaluate the asymptotic normality of the kernel-type density estimator at each point.

This article consists of two parts. In Part I, we briefly summarize some results in three kinds of optimization problems, which have been obtained through our many years' study. More specifically, these are concerned with the problems on maximum likelihood estimation of some parameters, on optimization of correlation with a quadratic constraint, and on optimum stratification. Especially for the stabilities of those optimal solutions, some additional examinations are investigated in this study. In Part II, miscellaneous problems of statistical education in Japan today are pointed out, and some discussion on the direction to be developed is given. We also show some action plans and some proposals.