Binomial probability estimates with restrictions on their $d$-risks

The standard problem of assessing the proportion of substandard products was considered in the framework of the Bayesian paradigm in the sense of the problem of optimal estimation. This problem was reduced to assessing the probability of success in a binomial scheme with a quadratic loss function for which a prior beta distribution applies. Unlike the classical approach to parameter estimation, we used the $d$-posterior approach to construct statistical guarantee solutions. Estimates with the uniformly minimal $d$-risk and the Bayesian estimate are constructed. The last one is necessary for designing a $d$-guaranteed sequential “first crossing” procedure. The sequential procedure leads to significant reduction of the inspection volume of products batch. In this regard, the task of planning the volume of tests that guarantees a given restriction on $d$-risk was solved.