Binomial probability estimates with restrictions on their d-risks

R.F. Salimov^∗ , I.A. Kareev^∗∗

Kazan Federal University, Kazan, 420008 Russia

E-mail: ^∗rustem.salimov@kpfu.ru, ^∗∗kareevia@gmail.com

Received December 24, 2019

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DOI: 10.26907/2541-7746.2020.1.91-97

For citation: Salimov R.F., Kareev I.A. Binomial probability estimates with restrictions on their d-risks. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2020, vol. 162, no. 1, pp. 91–97. doi: 10.26907/2541-7746.2020.1.91-97. (In Russian)

Abstract

The standard problem of assessing the proportion of substandard products was considered in the framework of the Bayesian paradigm in the sense of the problem of optimal estimation. This problem was reduced to assessing the probability of success in a binomial scheme with a quadratic loss function for which a prior beta distribution applies. Unlike the classical approach to parameter estimation, we used the d-posterior approach to construct statistical guarantee solutions. Estimates with the uniformly minimal d-risk and the Bayesian estimate are constructed. The last one is necessary for designing a d-guaranteed sequential “first crossing” procedure. The sequential procedure leads to significant reduction of the inspection volume of products batch. In this regard, the task of planning the volume of tests that guarantees a given restriction on d-risk was solved.

Keywords: binomial probability estimate, Bayesian estimate, estimate with uniformly minimal d-risk, quadratic loss function, prior Beta distribution, necessary sample size

Acknowledgments. The study was supported by the Russian Foundation for Basic Research (project no. 18-31-00094).

References

1. Salimov R.F., Turilova E.A., Volodin I.N. Sequential procedures for assessing the percentage of harmful impurietes with the given limitations on the accuracy and reliability of statistical inference. Proc. 16th Int. Multidiscip. Sci. GeoConf. SGEM 2016, SGEM Vienna GREEN Ext. Sci. Sess., 2016, book 1, vol. 4, pp. 175–180. doi: 10.5593/SGEM2016/HB14/S01.023.
2. Salimov R.F., Yang S.-F., Turilova E.A., Volodin I.N. Estimation of the mean value for the normal distribution with constrains on d-risk. Lobachevskii J. Math., 2018, vol. 39, no. 3, pp. 377–387. doi: 10.1134/S1995080218030174.
3. 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.
4. Lehmann E.L. Theory of Point Estimation. John Wiley & Sons, 1983. 506 p.
5. Simushkin S.V., Volodin I.N. Statistical inference with a minimal d -risk. Lect. Notes Math., 1983, vol. 1021, pp. 629–636. doi: 10.1007/BFb0072958.
6. Simushkin S.V., Volodin I.N. Statistical inference with minimum d -risk. J. Sov. Math, 1988, vol. 42, no. 1, pp. 1464–1472. doi: 10.1007/BF01098858.
7. Zaikin A.A. Estimates with asymptotically uniformly minimal d -risk. Theory Probab. Its Appl., 2019, vol. 63, no. 3, pp. 500–505. doi: 10.1137/S0040585X97T989192.
8. 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.
9. Gupta A.K., Nadarajah S. Handbook of Beta Distribution and Applications. New York, Marcel Dekker, 2004. 600 p.
10. Belyayev Yu.K. Veroyatnostnyye metody vyborochnogo kontrolya [Probabilistic Methods in Sample Control]. Moscow, Nauka, 1975. 408 p. (In Russian)
11. Volodin I.N. Optimum sample size in statistical inference procedures. Izv. Vyssh. Uchebn. Zaved., Mat., 1978, no. 21, pp. 33–45. (In Russian)
12. Lutsenko M.M. A game theoretic method for estimation of a parameter of the binomial law. Theory Probab. Its Appl., 1990, vol. 35, no. 3, pp. 467–477. doi: 10.1137/1135066.

References

1. Salimov R.F., Turilova E.A., Volodin I.N. Sequential procedures for assessing the percentage of harmful impurietes with the given limitations on the accuracy and reliability of statistical inference. Proc. 16th Int. Multidiscip. Sci. GeoConf. SGEM 2016, SGEM Vienna GREEN Ext. Sci. Sess., 2016, book 1, vol. 4, pp. 175–180. doi: 10.5593/SGEM2016/HB14/S01.023.
2. Salimov R.F., Yang S.-F., Turilova E.A., Volodin I.N. Estimation of the mean value for the normal distribution with constrains on d -risk. Lobachevskii J. Math., 2018, vol. 39, no. 3, pp. 377–387. doi: 10.1134/S1995080218030174.
3. 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.
4. Lehmann E.L. Theory of Point Estimation. John Wiley & Sons, 1983. 506 p.
5. Simushkin S.V., Volodin I.N. Statistical inference with a minimal d -risk. Lect. Notes Math., 1983, vol. 1021, pp. 629–636. doi: 10.1007/BFb0072958.
6. Simushkin S.V., Volodin I.N. Statistical inference with minimum d -risk. J. Sov. Math, 1988, vol. 42, no. 1, pp. 1464–1472. doi: 10.1007/BF01098858.
7. Zaikin A.A. Estimates with asymptotically uniformly minimal d -risk. Theory Probab. Its Appl., 2019, vol. 63, no. 3, pp. 500–505. doi: 10.1137/S0040585X97T989192.
8. 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.
9. Gupta A.K., Nadarajah S. Handbook of Beta Distribution and Applications. New York, Marcel Dekker, 2004. 600 p.
10. Belyayev Yu.K. Veroyatnostnyye metody vyborochnogo kontrolya [Probabilistic Methods in Sample Control]. Moscow, Nauka, 1975. 408 p. (In Russian)
11. Volodin I.N. Optimum sample size in statistical inference procedures. Izv. Vyssh. Uchebn. Zaved., Mat., 1978, no. 21, pp. 33–45. (In Russian)
12. Lutsenko M.M. A game theoretic method for estimation of a parameter of the binomial law. Theory Probab. Its Appl., 1990, vol. 35, no. 3, pp. 467–477. doi: 10.1137/1135066.

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