Sequential d-guaranteed estimate of the normal mean with bounded relative error

R.F. Salimov ^∗, I.N. Volodin^∗∗, N.F. Nasibullina ^∗∗∗∗

Kazan Federal University, Kazan, 420008 Russia

E-mail: ^∗rustem.salimov@gmail.com, ^∗∗igorvolodin@gmail.com, ^∗∗∗nasibusha@bk.ru

Received December 6, 2017

Full text PDF

DOI: 10.26907/2541-7746.2019.1.145-151

For citation: 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.

Abstract

In this paper, we continue our research on evaluation of the mean value of the normal distribution with prior information that this parameter is positive and very small. These data are obtained by using a prior exponential distribution with a large intensity parameter. The estimation problem with guaranteed relative error is considered. This issue is more important when small fractions are estimated. In addition to restrictions on the relative error, the procedure must have a given level of d-risk. We suggest a sequential procedure based on the first achievement by posterior probability of estimate reliability of a given level 1–в. The procedure is adapted to the problem of estimating harmful impurities in food products.

Keywords: first crossing procedure, normal mean estimation, d-posterior approach, sequential estimation

Acknowledgements. This work was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities (project no. 1.7629.2017/8.9)

References

1. Salimov R.F., Su-Fen Yang, Turilova E.A., Volodin I.N. 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.

 2. Volodin I.N. Guaranteed statistical inference procedure (determination of the sample size). J. Sov. Math., 1989, vol. 44, no. 5, pp. 568–600. doi: 10.1007/BF01095166.

 3. Salimov R.F. A sequential d-guaranteed test for distinguishing two interval hypotheses. Lobachevskii J. Math., 2016, vol. 37, no. 4, pp. 500–503. doi: 10.1134/S1995080216040156.

 4. Roughani G., Mahmoudi E. Exact risk evaluation of the two-stage estimation of the gamma scale parameter under bounded risk constraint. Sequential Anal., 2015, vol. 34, no. 3, pp. 387–405. doi: 10.1080/07474946.2015.1030303.

 5. Mukhopadhyay N. Improved sequential estimation of means of exponential distributions. Ann. Inst. Stat. Math., 1994, vol. 46, no. 3, pp. 509–519. doi: 10.1007/BF00773514.

 6. Mukhopadhyay N., Datta S. On fine-tuned bounded risk sequential point estimation of the mean of an exponential distribution. S. Afr. Stat. J., 1995, vol. 29, no. 1, pp. 9–27.

 7. Mukhopadhyay N., Pepe W. Exact bounded risk estimation when the terminal sample size and estimator are dependent: The exponential case. Sequential Anal., 2006, vol. 25, no. 1, pp. 85–101. doi: 10.1080/07474940500452254.

 8. Zacks S., Mukhopadhyay N. Bounded risk estimation of the exponential parameter in a two-stage sampling. Sequential Anal., 2006, vol. 25, no. 4, pp. 437–452. doi: 10.1080/07474940600934896.

 9. Zacks S., Mukhopadhyay N. Exact risks of sequential point estimators of the exponential parameter. Sequential Anal., 2006, vol. 25, no. 2, pp. 203–220. doi: 10.1080/07474940600596703.

 10.  Volodin I.N. Optimum sample size in statistical inference procedures. Izv. Vyssh. Uchebn. Zaved. Mat., 1978, no. 12, pp. 33–45. (In Russian)

 11. Simushkin S.V., Volodin I.N. Statistical inference with a minimal d-risk. Lect. Notes Math., 1983, vol. 1021, pp. 629–636.

The content is available under the license Creative Commons Attribution 4.0 License.