A sequential analytic approach to the methods of optiaml statistical decisions

主要研究者和每位研究者直接或间接获得了与本项目标题相关的研究结果。主要研究结果如下：（1）对于具有预先给定的均方误差界的非参数概率密度函数的估计问题，我们提出了序贯密度估计，并证明了它们的渐近有效性。(2)本文研究了均值和方差均未知的正态分布均值和标准差线性组合的序贯固定宽度置信区间估计。我们构造了具有预先指定的固定宽度和覆盖概率的序贯置信区间，并讨论了它们的渐近相合性。(3)考虑了非零均值在平方相对误差加线性代价下的序贯点估计。我们提出了一个顺序的程序，并表明这种程序的风险小于现有的程序的某一类分布。(4)研究了负指数分布在一定损失函数下尺度参数的有界风险点估计问题。我们提出了两个序贯估计，并给出了与之相关的风险的渐近展开式。

Head investigator and each of the investigators obtained the research results concerning the title of this project directly or indirectly. The main results by head investigator are as follows :(1) For the problem of estimating nonparametric probability density function with preassigned bound of mean integrated squared error we proposed sequential density estimators, and showed their asymptotic efficiency.(2) We considered sequential fixed-width confidence interval estimation for a linear combination of mean and standard deviation of a normal distribution with mean and variance both unknown. We constructed sequential confidence intervals with preassigned fixed-width and coverage probability, and their asymptotic consistency.(3) We considered the sequential point estimation of the nonzero mean under squared relative error plus linear cost as a loss function. We proposed a sequential procedure and showed this procedure has a risk less than that of the existing procedure for a certain clas of distributions.(4) We considered the problem of bounded risk point estimation for the scale parameter of a nagative exponential distribution under a certain loss function. We proposed two sequential estimators, and gave the asymptotic expansions of the risk associated with them.