Asymptotically optimum estimation of a probability in inverse binomial sampling
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The optimum quality that can be asymptotically achieved in the estimation of a probability $p$ using inverse binomial sampling is considered in this paper. A general definition of quality is used, in terms of the risk associated with a loss function that satisfies certain assumptions. It is shown that the limit superior of the risk for $p$ asymptotically small has a minimum over all (possibly randomized) estimators. This minimum is achieved by certain non-randomized estimators. The model includes commonly used quality criteria as particular cases. Applications to the non-asymptotic regime for specific loss functions are discussed.