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

Sequential estimation of the success probability p in inverse binomial sampling is considered in this paper. For any estimator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\hat{p}$</EquationSource> </InlineEquation>, 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\hat{p}> p$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\hat{p}> p$</EquationSource> </InlineEquation>, 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→0, and which guarantee that, for any p∈(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→∞ when a=b. Copyright Sociedad de Estadística e Investigación Operativa 2012