# A Neyman-Pearson Proper Way to Universal Testing of Multiple Hypotheses Formed by Groups of Distributions

## DOI:

https://doi.org/10.51408/1963-0056## Keywords:

Statistical hypotheses testing, Families of hypotheses, Optimal detection, Test with no match detection, Neyman-Pearson approach, Neyman-Pearson Lemma, Principle of maximum of Kullback-Leibler distance, Error exponent## Abstract

The asymptotically optimal Neyman-Pearson procedures of detection for models characterized by M discrete probability distributions arranged into K, 2 ≤ K ≤ M groups considered as hypotheses are investigated. The sequence of tests based on a growing number of observations is logarithmically asymptotically optimal (LAO) when a certain part of the given error probability exponents (reliabilities) provides positives values for all other reliabilities. LAO tests sequences for some models of objects, including cases, when rejection of decision may be permitted, and when part, or all given error probabilities decrease subexponentially with an increase in the of number of experiments, are desined. For all reliabilities of such tests single-letter characterizations are obtained. A simple case with three distributions and two hypotheses is considered.

## References

R. F. Ahlswede and E. A. Haroutunian, “On logarithmically asymptotically optimal testing of hypotheses and identification”, Lecture Notes in Computer Science, volume 4123, ”General Theory of Information Transfer and Combinatorics”, Springer, pp. 462– 478, 2006.

R. F. Ahlswede, E. Aloyan E. A. Haroutunian, “On logarithmically asymptotically optimal hypothesis testing for arbitrarily varying source with side information”, Lecture Notes in Computer Science, volume 4123, ”General Theory of Information Transfer and Combinatorics”, Springer, pp. 457-461, 2006.

L. Birg´e, “Vitesses maximals de d´ecroissence des erreurs et tests optimaux associ´es”. Z. Wahrsch. Verw. Gebiete, vol. 55, pp. 261–273, 1981.

R. E. Blahut, “Hypotheses testing and information theory” IEEE Trans. Inform Theory, vol. 20, no 4, pp. 405-417, 1974.

R. E. Blahut, Principles and Practice of Information Theory, Addison-Wesley, Reading, MA, 1987.

A. A. Borovkov Mathematical Statistics (in Russian). Nauka, Novosibirsk, 1997.

T. M. Cover and J. A. Thomas, Elements of Information Theory. Second Edition, New York, Wiley, 2006.

D. R. Cox, “Tests of separate families of hypotheses” In Proceeding 4th Berkley Simp. Math. Statist. Prob., University of Callifornia Press, Berkelly, pp. 105-123, 1961.

D. R. Cox, “Furthe results on tests of separate families of hypotheses”, Journal of the Royal Statist. Society: Serie B, vol. 24, no. 2, pp. 406-424, 1962.

D. R. Cox, “A return to an old paper: Tests of separate families of hypotheses”, Journal of the Royal Statist. Society: Serie B, vol. 75, no. 2, pp. 207-2015, 2013.

I. Csisz´ar and J. K¨orner, Information Theory: Coding Theorems for Discrete Memoryless Systems, Academic press., New York, 1981.

I. Csisz´ar and G. Longo, “On the error exponent for source coding and for testing simple statistical hypotheses”, Studia Sc. Math. Hungarica, vol. 6, pp. 181–191, 1971.

I. Csisz´ar and P. Shields, “ Information theory and statistics: A tutorial”, Foundations and Trends in Communications and Information Theory, vol. 1, no. 4, 2004.

M. Feder and N. Merhav, “ Universal composite hypotheses testing: A competetive minimax approach,” IEEE Trans. Inform. Theory, vol. 48, pp. 1504-1517, June 2002.

F. W. Fu and S. Y. Shan, “Hypotheses testing for arbitrarily varying source with exponential type constraint”, IEEE Trans. on Inform. Theory, vol. 14, no. 2, pp. 892-895, 1998.

M. Gutman, “Asymptotically optimal classification for multiple tests with empirically observed statistics”, IEEE Trans. Inform. Theory, vol. 35, no. 2, pp. 401–408, 1989.

T. S. Han and K. Kobayashi , “Exponential-type error probabilities for multiterminal hypotheses testing”, IEEE Trans. Inform. Theory, vol. 35, no. 1, pp. 2-13, 1989.

E. A. Haroutunian, “Many statistical hypotheses: interdependence of optimal test’s error probabilities exponents”, (In Russian), Abstract of the report on the 3rd AllUnion school-seminar, “Program-algorithmical software for applied multi-variate statistical analysis”, Tsakhkadzor, Part 2, pp. 177–178, 1988.

E. A. Haroutunian, “On asymptotically optimal testing of many statistical hypotheses conserning Markov chain”, (in Russian), Isvestia Akademii Nauk Armenii, Mathematika, vol. 23, no. 1, pp. 76-80, 1988.

E. A. Haroutunian, “Logarithmically asymptotically optimal testing of multiple statistical hypotheses”, Problems of Control and Information Theory, vol. 19(5-6), pp. 413–421, 1990.

E. A. Haroutunian, “Reliability in multiple hypotheses testing and identification problems” Nato Science Series III, Computer and System Sciences, vol.198, IOS Press, pp. 189-201, 2003.

E. A. Haroutunian, “On three hypotheses robust detection design under mismatch”, Proceedings of International Conference CSIT 2019, pp. 125 – 128, Yerevan 2019.

E. A. Haroutunian and P. M. Hakobyan, “On LAO testing of multiple hypotheses for pair of objects”, Mathematical Problems of Computer Sciences, vol. 25, pp. 93-101, 2006.

E. A. Haroutunian and P. M. Hakobyan, “Multiple hypotheses LAO testing for many independent objects”, International Journal “Scholarly Research Exchange”, vol. 2009, pp. 1-6, 2009.

E. A. Haroutunian and P. M. Hakobyan, “On Neyman-Pearson principle in multiple hypotheses testing”, Mathematical Problems of Computer Sciences, vol. 40, pp. 34-37, 2013.

E. A. Haroutunian and P. M. Hakobyan, “Multiple objects: Error exponents in hypotheses testing and identification”, Lecture Notes in Computer Science, volume 7777, ”Ahlswede Festschrift”, Springer , pp. 313-345, 2013.

E. A. Haroutunian, P. M. Hakobyan and F. Hormosi-nejad, “On two-stage LAO testing of multiple hypotheses for the pair of families of distributions”, Journal of Statistics and Econometrics Methods , vol. 2, no. 2, pp. 127-156, 2013.

E. A. Haroutunian, P. M. Hakobyan and A. O. Yessayan, “On multiple hypotheses LAO testing with rejection of decision for many independent objects”, Proceedings of International Conference CSIT, pp. 117 – 120, 2011.

E. A. Haroutunian, P. M. Hakobyan and A. O. Yesayan, “Many hypotheses LAO testing with rejection of decision for arbitrarily varying object”, Mathematical Problems of Computer Sciences, vol. 35, pp. 77-85, 2011.

E. A. Haroutunian, P. M. Hakobyan and A. O. Yessayan, “On multiple hypotheses LAO testing with liberty of rejection of decision for two independent objects”, International Journal, Information theories and applications, vol. 25 , no. 1, pp. 38-46, 2018.

E. A. Haroutunian, M. E. Haroutunian and A. N. Harutyunyan, “Reliability criteria in information theory and in statistical hypothesis testing”, Foundations and Trends in Communications and Information Theory, vol. 4, no. 2-3, 2008.

W. Hoeffding, “Asymptotically optimal tests for multinomial distributions,” The Annals of Mathematical Statistics, vol. 36, pp. 369–401, 1965.

S. Kullback, Information Theory and Statistics, New York, Wiley, 1959.

E. Levitan and N. Merhav, “A competitive Neyman-Pearson approach to universal hypothesis testing with applications”, IEEE Trans. Inform. Theory, vol. 48, no. 8, pp. 2215-2229, 2002.

B. C. Levy, Principles of Signal Detection and Parameter Estimation, Springer, 2008.

G. Longo and A. Sgarro, “ The error exponent for the testing of simple statistical hypotheses: A combinatorial approach”, Journal of Combinatorics and Informational System Science, vol. 5, no. 1, pp. 58-67, 1980.

V. Poor, An Introduction to Signal Deyection and Estimation, Springer, 1994.

E. Tuncel, “On error exponents in hypothesis testing’, IEEE Trans. Inform. Theory, vol. 51, mo.8, pp. 2945-2950, 2005.

G. Tusn´ady, “On asymptotically optimal tests,” Annals of Statatistics, vol. 5, no. 2, pp. 385-393, 1977.

H. van Trees, Detection, Estimation and Modulation Theory, pt.1 New York, Wiley, 1968.

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