Estimation of a probability in inverse binomial sampling under normalized linear-linear and inverse-linear loss

TL;DR

This paper develops estimators for the success probability in inverse binomial sampling that minimize risk under asymmetric normalized linear-linear and inverse-linear loss functions, ensuring controlled risk levels across all success probabilities.

Contribution

It introduces new estimators with asymptotic and approximate minimax properties under asymmetric loss functions in inverse binomial sampling.

Findings

Estimators achieve asymptotic risk as p approaches 0.

Risk is bounded below the asymptotic value for all p in (0,1).

Estimators are approximately minimax when loss asymmetry is small.

Abstract

Sequential estimation of the success probability $p$ in inverse binomial sampling is considered in this paper. For any estimator $\hat p$, its quality is measured by the risk associated with normalized loss functions of linear-linear or inverse-linear form. These functions are possibly asymmetric, with arbitrary slope parameters $a$ and $b$ for $\hat p<p$ and $\hat p>p$ respectively. Interest in these functions is motivated by their significance and potential uses, which are briefly discussed. Estimators are given for which the risk has an asymptotic value as $p$ tends to $0$, and which guarantee that, for any $p$ in $(0,1)$, the risk is lower than its asymptotic value. This allows selecting the required number of successes, $r$, to meet a prescribed quality irrespective of the unknown $p$. In addition, the proposed estimators are shown to be approximately minimax when $a/b$ does not deviate too much from $1$, and asymptotically minimax as $r$ tends to infinity when $a=b$.