We compare two sequential d-guaranteed tests and an optimal d-guaranteed test based on a fixed number of observations with respect to the average number of observations within the most accepted practical applications of Bayesian probabilistic models. We consider the sequential “first skipping” test, the sequential locally efficient test based on the score statistic, and the test based on a fixed number of observations that minimizes the necessary sample size. We study various characteristics of these tests connected with the number of observations within three probabilistic models, namely, the normal (ϑ, σ 2) distribution of the observed random variable and the normal a priori distribution of ϑ with fixed σ 2; the exponential distribution with the intensity parameter ϑ and the a priori gamma distribution of ϑ; and the Bernoulli sampling with the success probability ϑ and the a priori beta distribution of ϑ. We discuss the connection of the d-posterior approach with the compound decision problem as applied to the analysis of data provided by microchips (when the false discovery rate, or FDR, for short, is treated as the d-risk of the first kind). We present the vast data on characteristics of the mentioned tests obtained by the method of mathematical modeling in several tables. We discuss unsolved problems of the d-guaranteed discrimination of hypotheses with the minimal number of observations and approaches to their solution.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Yu. K. Belyaev, Probabilistic Methods of Sample Control, Nauka, Moscow (1975).
Y. Benjiamini and Y. Hochberg, “Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing,” J. R. Statist. Soc., Ser. B. 57, No.1, 289–300 (1995).
S. N. Bernstein, “On the Fisher “Confidence” Probabilities,” Izv. AN SSSR, Ser. Matem., 5, 85–94 (1941).
F. S. Datta and R. Mukrjee, Probability Matching Priors: Higher Order Asymptotics, Springer–Verlag, New York (2004).
B. Efron, Large-Scale Inference. Empirical Bayes Methods for Estimation, Testing, and Prediction, Cambridge Univ. Press, London (2010).
B. K. Ghosh, Sequential Tests of Statistical Hypotheses, Addison–Wesley, Reading (1970).
J. Kruopis, “On comparison of many simple hypothesis,” in: Abstracts of papers at 2nd Vilnius Conf. on Probab. Theory and Math. Statist. Vilnius, 1 (1977), pp. 207–209.
J. Kruopis, “Minimization of objective systems in some quality control systems,” Optimization of Quality Control Systems, Inst. Math. Kibern. Akad. Nauk. Lit. SSR, Vilnius, (1981), pp. 12–27.
T. L. Lai, “Asymptotical optimality of invariant sequential probability ratio tests,” Ann. Statist., 9, No. 2, 318–333 (1981).
J. Neyman, “Two breakthroughs in the theory of statistical decision making,” Rev. Inst. Int. Stat., 30, No. 5, 11–27 (1962).
An. A. Novikov, “Asymptotic optimality of a sequential d-duaranteed test,” Theor. Probab. Appl., 32, No. 2, 387–391(1987).
An. A. Novikov and I. N. Volodin, “Asymptotics of necessary sample size in testing parametric hypotheses,” Math. Method. Stat., 7, No. 1, 111–121 (1998).
C. R. Rao, Advanced Statistical Methods in Biometric Research, John Wiley and Sons, New York (1952).
H. Robbins, “An empirical Bayes approach to statistics”, in: Proceedings Third Berkeley Symposium on Math. Statist. and Probab. I, Univ. California Press, Berkeley and Los Angeles, (1956), pp. 157–163.
E. D. Sherman, “Empirical estimates with minimal d-risk for discrete exponential families,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 8, 89–98 (2010).
E. D. Sherman and I. N. Volodin, “Empirical estimate with uniformly minimal d-risk for Bernoulli trials success probability,” Mathematical and Statisical Models and Methods of Reliability, Birkhauser (2011), pp. 297–306.
D. S. Simushkin, “Comparative analysis of two sequential d-guaranteed procedures with respect to the number of observations,” Obozrenie Prikladnoi i promyshlennoi Matematiki 18, No. 1, 91–93 (2011).
S. V. Simushkin, “Optimal d-guaranteed procedures for discrimination of two hypotheses,” deposited at VINITI AN SSSR, 5547–81Dep. (1981).
S. V. Simushkin, “Optimal sample size for d-guaranteed discrimination of hypotheses,” J. Sov. Math., No. 5, 47–52 (1982).
S. V. Simushkin, “The empirical d-posterior approach to the problem of statistical inference,” J. Soviet. Math., No. 11, 47–67 (1983).
S. V. Simushkin and I. N. Volodin, “Statistical inference with a minimal d-risk,” in: Lecture Notes in Math. 1021, Springer, Berlin–New York, 629–636 (1983).
P. Vaitkus, “Studies in mathematical statistics and its applications,” Litov. Mat. Sb., 20, No. 3, 117–128 (1980).
I. N. Volodin, “Guaranteed statistical inference procedure (determination of the sample size),” J. Sov. Math., 44, No. 5, 568–600 (1989).
I. N. Volodin and An. A. Novikov, “Asymptotic behavior of the necessary sample size in d-guaranteed discrimination of two close hypotheses,” J. Sov. Math., 27, No. 11, 68–78 (1983).
I. N. Volodin and An. A. Novikov, “Statistical estimates with asymptotically minimal d-risk,” Theory Probab. Appl., 38, No. 1, 118–128 (1993).
I. N. Volodin and An. A. Novikov, “Asymptotic behavior of the necessary sample size under severe constraints on the d-risk,” Aktual’nye voprosy sovremennoi matematiki. Sbornik nauchnykh trudov, NII MIOO NGU, Novosibirsk, 1, 52–59 (1993).
I. N. Volodin and An. A. Novikov, “Local asymptotic efficiency of sequential probability ratio test under d-guaranteed discrimination of composite hypotheses,” Theory Probab. Appl. 43, no 2, 209–225 (1998).
I. N. Volodin and An. A. Novikov, “Asymptotics of the necessary sample size in testing parametric hypothesis: d-posterior approach,” Math. Methods Statist., 7, No. 1, 111–121 (1998).
I. N. Volodin, An. A. Novikov, and S. V. Simushkin, “Guaranteed Statistical Quality Control: Posterior Approach,” Obozrenie Prikladnoi i Promyshlennoi Matematiki, 1, No. 2, 1–32 (1994).
I. N. Volodin and S. V. Simushkin, “On d-posterior approach to the problem of statistical inference,” in: Abstracts of papers at 3d Intern. Vilnius Conf. on Probab. Theory and Math. Statist., Vilnius, 1, (1981), pp. 101–102.
I. N. Volodin and S. V. Simushkin, “Unbiasedness and the Bayes property,” J. Sov. Math., 31, No. 1, 1–8 (1987).
I. N. Volodin and S. V. Simushkin, “Confidence estimation in the d-posterior approach,” Theory Probab. Appl., 35, No. 2, 318–329 (1990).
I. N. Volodin and S. V. Simushkin, “D-posterior concept of p-value,” Math. Methods Stat., 13, 108–121 (2004).
A. Wald, Sequential Analysis, Wiley, New York (1947).
Author information
Authors and Affiliations
Corresponding author
Additional information
Proceedings of the XIX International Seminar on Stability Problems for Stochastic Models, Svetlogorsk, Russia, October 10–16, 2011
Rights and permissions
About this article
Cite this article
Simushkin, D.S., Simushkin, S.V. & Volodin, I.N. D-Guaranteed Discrimination of Statistical Hypotheses: a Review of Results and Unsolved Problems. J Math Sci 228, 543–565 (2018). https://doi.org/10.1007/s10958-017-3643-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-017-3643-6