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

TL;DR

This paper investigates the asymptotic optimality of probability estimation using inverse binomial sampling under general loss functions, identifying estimators that minimize the worst-case risk as the probability approaches zero.

Contribution

It establishes the existence of non-randomized estimators that asymptotically achieve the minimum possible risk for probability estimation under broad loss functions.

Findings

Minimum asymptotic risk is achieved by certain non-randomized estimators.

The model encompasses common quality criteria as special cases.

Minimax estimators are derived for specific loss functions in the non-asymptotic regime.

Abstract

The optimum quality that can be asymptotically achieved in the estimation of a probability p using inverse binomial sampling is addressed. 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 are discussed considering specific loss functions, for which minimax estimators are derived.