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地获得与本项目标题相关的研究成果。主要研究成果如下：(1)针对预设均值积分平方误差界的非参数概率密度函数估计问题，提出了序列密度估计器，并证明了序列密度估计器的渐近有效性。(2)考虑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定宽置信区间估计。构造了具有预分配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一致性。(3)考虑相对误差平方加线性代价下的非零均值序列点估计作为损失函数。我们提出了一种序贯法，并证明了该序贯法对某一类分布的风险小于现有序贯法。(4)考虑了在一定损失函数下负指数分布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量，并给出了与它们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.