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

Mendo Tomás, Luis (2012). Estimation of a probability in inverse binomial sampling under normalized linear-linear and inverse-linear loss. "Test", v. 21 (n. 4); pp. 656-675. ISSN 1133-0686. https://doi.org/10.1007/s11749-011-0267-x.

## Description

Title: Estimation of a probability in inverse binomial sampling under normalized linear-linear and inverse-linear loss Mendo Tomás, Luis Article Test December 2012 1133-0686 21 Sequential estimation, Point estimator, Inverse binomial sampling, Asymmetric loss function. E.T.S.I. Telecomunicación (UPM) Señales, Sistemas y Radiocomunicaciones None

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## Abstract

Sequential estimation of the success probability $p$ in inverse binomial sampling is considered in this paper. For any estimator $\hatvap$, 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 $\hatvap < p$ and $\hatvap > p$ 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 $p \rightarrow 0$, 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, $\nnum$, 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 $\nnum \rightarrow \infty$ when $a=b$.

Item ID: 21857 http://oa.upm.es/21857/ oai:oa.upm.es:21857 10.1007/s11749-011-0267-x http://www.springer.com/statistics/journal/11749 Dr. Luis Mendo 05 Dec 2013 09:35 21 Apr 2016 12:39

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