Estimation of the Mean Value for the Normal Distribution with Constraints on d-Risk

Abstract

We consider the problem of an estimation of the mean value of the normal distribution with a prior information that this parameter is positive and very small. The prior information is implemented in terms of the exponential prior distribution. The estimation procedures are constructed for two cases: fixed sample size and sequential estimation that guarantee the given constraints on the precision and the d-risk of the estimator. An analytical review of the comprehensive literature for the problems of guaranteed statistical inference (d-risk and pFDR) is provided. For the practical applications of the proposed estimators with the unknown value of the prior distribution parameter, we solve the problem of choosing this parameter in the framework of empirical (parametric) Bayesian approach or in the framework of existing State Standards on the precision and output quality of the estimated parameter. As an implementation of the proposed statistical procedures, the problem of estimation of the chemical element of arsenic (As) in a food product is considered. The model parameters are chosen according to the State Standards for carrying out a laboratory tests for As detection. For the chosen values of the parameters, the probability of stopping for the experiment is estimated for each step by the method of statistical simulations. The histogram of the Bayesian estimate for the As content is presented.