The effect of distributions symmetrization on their peakedness
DOI:
https://doi.org/10.21638/spbu01.2021.306Abstract
Indirect transformations of a one-dimensional, two-dimensional, and multidimensional random variable are proposed. They are based on various symmetrizations of the density function. The focus is on changing peakedness of a distribution about the origin.Keywords:
peakedness, function rearrangement, continuous symmetrization, log-concave density, majorization, statistical inference
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References
Литература
1. Szego G. On a certain kind of symmetrization and its applications. Ann. Mat. Pura Appl. 40, 113–119 (1955).
2. Polya G., Szego G. Isoperimetric Inequalities in Mathematical Physics. Princeton, Princeton University Press (1951).
3. Proschan F. Peakedness of distributions of convex combinations. Ann. Math. Statist. 36, 1703–1706 (1965).
4. Hayman W.K. Multivalent Functions. Cambridge, Cambridge University Press (1958).
5. Hardy G.H., Littlewood J.E., Polya G. Inequalities. 2nd ed. Cambridge, Cambridge University Press (1952).
6. Lieb E.H., Loss M. Analysis. 2nd ed. Providence, Rhode Island, American Mathematical Society (2001).
7. Birnbaum Z.W. On random variables with comparable peakedness. Ann. Math. Statist. 19, 76–81 (1948).
8. Sherman S. A theorem on convex sets with applications. Ann. Math. Statist. 26, 763–767 (1955).
9. Azzalini A., Capitanio A. The Skew-normal and Related Families. Cambridge, Cambridge University Press (2014).
10. Ревяков М. Ранжирование и селекция популяций по выборочным средним. Зап. научн. сем. ПОМИ 454, 238–253 (2016).
11. Lehmann E.L., Romano J.P. Testing Statistical Hypotheses. 3rd ed. New York, Springer (2005).
12. Eaton M.L. Some optimum properties of ranking procedures. Ann. Math. Statist. 38, 124–137 (1967).
13. Marshall A., Olkin I., Arnold B. Inequalities: Theory of Majorization and Its Applications. 2nd ed. New York, Springer (2011).
14. An M.Y. Logconcavity versus logconvexity: a complete characterization. J. Economic Theory 80, 350–369 (1998).
References
1. Szego G. On a certain kind of symmetrization and its applications. Ann. Mat. Pura Appl. 40, 113–119 (1955).
2. Polya G., Szego G. Isoperimetric Inequalities in Mathematical Physics. Princeton, Princeton University Press (1951).
3. Proschan F. Peakedness of distributions of convex combinations. Ann. Math. Statist. 36, 1703–1706 (1965).
4. Hayman W.K. Multivalent Functions. Cambridge, Cambridge University Press (1958).
5. Hardy G.H., Littlewood J.E., Polya G. Inequalities. 2nd ed. Cambridge, Cambridge University Press (1952).
6. Lieb E.H., Loss M. Analysis. 2nd ed. Providence, Rhode Island, American Mathematical Society (2001).
7. Birnbaum Z.W. On random variables with comparable peakedness. Ann. Math. Statist. 19, 76–81 (1948).
8. Sherman S. A theorem on convex sets with applications. Ann. Math. Statist. 26, 763–767 (1955).
9. Azzalini A., Capitanio A. The Skew-normal and Related Families. Cambridge, Cambridge University Press (2014).
10. Revyakov M. Ranking and selection of populations on the base of sample means. Zap. Nauchn. Sem. POMI 454, 238–253 (2016). (In Russian) [Engl. transl.: J. Math. Sci. 229, 756–766 (2018). https://doi.org/10.1007/s10958-018-3715-2].
11. Lehmann E.L., Romano J.P. Testing Statistical Hypotheses. 3rd ed. New York, Springer (2005).
12. Eaton M.L. Some optimum properties of ranking procedures. Ann. Math. Statist. 38, 124–137 (1967).
13. Marshall A., Olkin I., Arnold B. Inequalities: Theory of Majorization and Its Applications. 2nd ed. New York, Springer (2011).
14. An M.Y. Logconcavity versus logconvexity: a complete characterization. J. Economic Theory 80, 350–369 (1998).
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Published
2021-09-26
How to Cite
Revyakov, M. I. (2021). The effect of distributions symmetrization on their peakedness. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 8(3), 442–454. https://doi.org/10.21638/spbu01.2021.306
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Mathematics
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
