The $d $-risk of Bayesian estimation for the probability of success in Bernoulli trials

This article considers the problem of estimating the probability $p$ of success in Bernoulli trials when it is a priori the smallest. Using the $d$-posterior approach to the problem of guaranteed statistical inference, a Bayesian estimation of $p$ was performed for a special loss function of type $1$-$0$ with the relative error restriction and the beta prior distribution of the estimated parameter. The $d$-risk of the Bayesian estimation was calculated, and the impossibility to design a $d$-guaranteed estimation procedure for a fixed amount of tests was revealed.