Estimators of the Small Success Probabilities in Bernoulli Trials and Their d-Risk

We investigate the d-risk value of an success probability estimator in Bernoulli trials with a priori information about the extreme smallness of this probability. A special type 1–0 loss function is introduced that takes into consideration the relative error in the probability estimation. Also, the prior beta distribution with the appropriate selection of its parameters is considered. The d-risk function of Bayesian estimators from Bernoulli trial observations are calculated for a fixed sample size and in the case of inverse binomial sampling. The problem of d-guarantee of the estimators is discussed for the example of estimating the probability of a misfire when firing a shot in the case of practically acceptable sample sizes from a produced batch of cartridges.