Studies on.the structure of.methods, of.sequential estimation

首席研究者和每位研究者直接或间接地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)考虑了当损失函数为误差平方加线性代价时，均值和方差未知的正态分布标准差幂的点估计问题。当我们用最小的样本量来估计它们，使风险最小化时，渐近最优样本量包含未知参数。因此，我们提出了一个序贯估计量，并得到了期望样本量的渐近展开式和当每单位样本代价趋近于零时序贯估计量的风险。(2)考虑了正态分布尺度参数幂的点估计问题。我们希望通过使用最小的样本量来估计功率，这样当风险为均方误差时，风险小于或等于预分配的误差界。在这种情况下，渐近最优样本量包含未知参数。因此，我们定义了一个停止规则，并证明了风险小于或等于误差界。此外，我们还考虑了当损失函数为误差平方加线性代价时指数分布尺度参数的估计问题。(3)考虑指数分布尺度参数幂的有界风险点估计问题。我们希望通过使用最小的样本量来估计功率，这样当风险为均方误差时，风险小于或等于预分配的误差界。这个最小的样本量不能在实际中使用，因为它包含未知参数。因此，我们提出了一个停止规则，并证明了在足够小的误差范围内，风险的条件是满足的。

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)We consider the point estimation problem of the powers of a standard deviation of a normal distribution with unknown mean and variance when the loss function is squared error plus linear cost. When we estimate them by using the smallest sample size such that the risk is minimized, the asymptotic optimal sample size contains the unknown parameter. Therefore we propose a sequential estimator and obtain the asymptotic expansions of the expected sample size and the risk of the sequential estimator as the cost per unit sample approaches zero.(2)We consider the point estimation problem of the powers of scale parameter of a normal distribution. We want to estimate the powers by using the smallest sample size such that the risk is less than or equal to a preassigned error bound when the risk is mean squared error. In this case the asymptotic optimal sample size contains the unknown parameter. Therefore we define a stopping rule and show that the risk is less than or equal to the error bound. Also, we consider the problem of estimating a scale parameter of an exponential distribution when the loss function is squared error plus linear cost.(3)We consider the bounded risk point estimation problem of the powers of scale parameter of an exponential distribution. We want to estimate the powers by using the smallest sample size such that the risk is less than or equal to a preassigned error bound when the risk is mean squared error. This smallest sample size cannot be used in practice, because it contains the unknown parameter. Therefore we propose a stopping rule and show that the condition of the risk is satisfied for sufficiently small error bound.