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


  • Luis Mendo



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

Suggested Citation

  • Luis Mendo, 2012. "Estimation of a probability in inverse binomial sampling under normalized linear-linear and inverse-linear loss," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(4), pages 656-675, December.
  • Handle: RePEc:spr:testjl:v:21:y:2012:i:4:p:656-675
    DOI: 10.1007/s11749-011-0267-x

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    References listed on IDEAS

    1. Fikri Akdeniz, 2004. "New biased estimators under the LINEX loss function," Statistical Papers, Springer, vol. 45(2), pages 175-190, April.
    2. Christoffersen, Peter F. & Diebold, Francis X., 1997. "Optimal Prediction Under Asymmetric Loss," Econometric Theory, Cambridge University Press, vol. 13(06), pages 808-817, December.
    3. 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.
    4. J. A. Adell & P. Jodrá, 2005. "The median of the poisson distribution," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 61(3), pages 337-346, June.
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