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

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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.