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The optimum quality that can be asymptotically achieved in the estimation ofa probability p using inverse binomial sampling is addressed. A generaldefinition of quality is used in terms of the risk associated with a lossfunction that satisfies certain assumptions. It is shown that the limitsuperior of the risk for p asymptotically small has a minimum over all(possibly randomized) estimators. This minimum is achieved by certainnon-randomized estimators. The model includes commonly used quality criteria asparticular cases. Applications to the non-asymptotic regime are discussedconsidering specific loss functions, for which minimax estimators are derived.

Asymptotically optimum estimation of a probability in inverse binomial sampling under general loss functions