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

## Author Info

• Luis Mendo

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## 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→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

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File URL: http://hdl.handle.net/10.1007/s11749-011-0267-x

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## Bibliographic Info

Article provided by Springer in its journal TEST.

Volume (Year): 21 (2012)
Issue (Month): 4 (December)
Pages: 656-675

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Handle: RePEc:spr:testjl:v:21:y:2012:i:4:p:656-675

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## Related research

Keywords: Sequential estimation; Point estimator; Inverse binomial sampling; Asymmetric loss function; 62L12; 62F12;

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• 62L - - - - - -
• 62F - - - - - -

## References

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1. J. A. Adell & P. Jodrá, 2005. "The median of the poisson distribution," Metrika, Springer, vol. 61(3), pages 337-346, 06.
2. Fikri Akdeniz, 2004. "New biased estimators under the LINEX loss function," Statistical Papers, Springer, vol. 45(2), pages 175-190, April.
3. Christoffersen & Diebold, . "Optimal Prediction Under Asymmetric Loss," Home Pages 167, 1996., University of Pennsylvania.
4. Jerzy Baran & Ryszard Magiera, 2010. "Optimal sequential estimation procedures of a function of a probability of success under LINEX loss," Statistical Papers, Springer, vol. 51(3), pages 511-529, September.
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