The d-risk of Bayesian estimation for the probability of success in Bernoulli trials

N.F. Bilalova

Kazan Federal University, Kazan, 420008 Russia

E-mail: bnfnal@gmail.com

Received August 10, 2022

ORIGINAL ARTICLE

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DOI: 10.26907/2541-7746.2022.4.271-284

For citation: Bilalova N.F. The d-risk of Bayesian estimation for the probability of success in Bernoulli trials. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2022, vol. 164, no. 4, pp. 271–284. doi: 10.26907/2541-7746.2022.4.271-284. (In Russian)

Abstract

This article considers the problem of estimating the probability p of success in Bernoulli trials when it is a priori the smallest. Using the d-posterior approach to the problem of guaranteed statistical inference, a Bayesian estimation of p was performed for a special loss function of type 1-0 with the relative error restriction and the beta prior distribution of the estimated parameter. The d-risk of the Bayesian estimation was calculated, and the impossibility to design a d-guaranteed estimation procedure for a fixed amount of tests was revealed.

Keywords: Bernoulli trials, Bayesian probability estimation, beta prior distribution, d-risk estimation

References

1. DeGroot M.H. Optimal Statistical Decisions. New York, McGraw-Hill, 1970. xvi, 489 p.
2. Gupta A.K., Nadaraiah S. Handbook of Beta Distribution and Its Applications. New York, Marcel Dekker, 2004. viii, 571 p.
3. Lehmann E.L. Testing Statistical Hypotheses. Berlin, Springer, 1997. 625 p.
4. Lehmann E.L., Casella G. Theory of Point Estimation. New York, Berlin, Hidelberg, Springer, 1998. 616 p.
5. Salimov R.F., Turilova E.A., Volodin I.N. Sequential procedures for assessing the percentage of harmful impurities with the given limitations on the accuracy and reliability of statistical inference. 16th Int. Multidiscip. Sci. GeoConf. SGEM 2016, SGEM Vienna GREEN Ext. Sci. Sess.: SGEM2016 Conf. Proc., 2016, book 1, vol. 4, pp. 175–180. doi: 10.5593/SGEM2016/HB14/S01.023.
6. Salimov R.F., Turilova E.A., Volodin I.N., Yang S.-F. Estimation of the mean value for the normal distribution with constraints on d-risk. Lobachevskii J. Math., 2018, vol. 39, no. 3, pp. 377–387. doi: 10.1134/S1995080218030174.
7. Salimov R.F., Volodin I.N. Nasibullina N.F. Sequential d-guaranteed estimate of the normal mean with bounded relative error. Uchenye Zapiski Kazanskogo Universiteta Seriya Fiziko-Matematicheskie Nauki, 2019, vol. 161, no. 1, pp. 145–151. doi: 10.26907/2541-7746.2019.1.145-151.
8. Salimov R.F., Volodin A.I., Volodin I.N., Yang S.-F. Estimation of mean value a normal distribution with constraints on the relative error and d-risk. J. Stat. Comput. Simul., 2020, vol. 90, no. 7, pp. 1286–1300. doi: 10.1080/00949655.2020.1724292.
9. Simushkin S.V. The empirical d-posterior approach to the problem of guaranteedness of statistical inference. Sov. Math. (Izv. Vyssh. Uchebn. Zaved.), 1983, vol. 27, no. 11, pp. 47–67.
10. Volodin I.N. Guaranteed procedures of statistical inference: Choosing the optimal sample size. Doct. Phys.-Math. Sci. Diss. Kazan, 1980. 239 p. (In Russian)
11. Volodin I.N. Guaranteed statistical inference procedures (determination of the optimal sample size). J. Sov. Math., 1989, vol. 44, no. 5, pp. 568–600. doi: 10.1007/Bf01095166.
12. Volodin I.N., Simushkin S.V. On the d-posterior approach to the problem of statistical inference. 3-ya Vil’nyusskaya mezhdunar. konf. po teorii veroyatnostei i matem. statistike: Tez. dokl. [Proc. 3rd Vilnius Int. Conf. on the Theory of Probability and Mathematical Statistics]. Vilnius, 1981, pp. 100–101. (In Russian)

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