Systems and Means of Informatics
2019, Volume 29, Issue 1, pp 140-163CONDITIONAL BOUNDS OF EXPECTED MAXIMA OF RANDOM VARIABLES AND THEIR REACHABILITY
- D. V. Ivanov
Abstract
The subject of this article is the expected maxima of an arbitrary number n of independent and identically distrubuted random variables. Probability distributions with zero mean and variance of 1 and with given value of the expected maximum of m independent random variables of this distribution are taken into consideration. The question of reachability of the boundaries obtained by other authors is investigated. In the cases of failure to derive the answer to this question, the obtained boundaries are specified. The problem might have various applications in queuing theory, insurance, finance, and other fields.[+] References (28)
- Volkova, M.A., and I.V. Kuzhevskaya. 2011. Klimatologia. Teoreticheskie i pri- kladnye aspecty [Climatology. Theoretical and applied aspects]. Tomsk: Tomsk State University. Electronic edition.
- Arnold, B. C. 1988. Bounds on the expected maximum. Commun. Stat. Theor. M. 17 (7):2135-2150.
- Ross, A. M. 2010. Computing bounds on the expected maximum of correlated normal variables. Methodol. Comput. Appl. 12(1):111-138.
- Borovkov, K., Yu. Mishura, A. Novikov, and M. Zhitlukhin. 2017. Bounds for expected maxima of Gaussian processes and their discrete approximations. Stochastics. 89(1):21-37.
- Cherny, A. S., andD. B. Madan. 2006. Coherent measurement of factor risks. Available at: http://arxiv.org/abs/math/0605062v1 (accessed April 5, 2019).
- Orlov, D. 2009. On two estimates of a risk measure. Theor. Probab Appl. 53(1): 169-173.
- Cherny, A., and D. Orlov. 2011. On two approaches to coherent risk contribution. Math. Financ. 21(3): 557-571.
- Wang, S. S. 1996. Premium calculation by transforming the layer premium density. Astin Bull. 26(1):71-92.
- Irkhina, N. 2010. Printsyp Wanga v matematicheskoy teorii strakhovaniya [Wang's principle in mathematical insurance theory]. Moscow: MSU. PhD Diss. 137 p.
- Gnedenko, B. V., Yu. K. Belyayev, and A. D. Solovyev. 1969. Mathematical methods of reliability theory. 1st ed. New York, NY: Academic Press. 518 p.
- Thomasian, A. 2014. Analysis of fork/join and related queueing systems. ACM Comput. Surv. 47. 17 p.
- Fiorini, P.M., and L. Lipsky. 2015. Exact analysis of some split-merge queues. Perf. E. R. 43(2):51-53.
- Gorbunova, A., I. Zaryadov, S. Matyushenko, K. Samouylov, and S. Shorgin. 2015. Approksimatsiya vremeni otklika sistemy oblachnykh vychisleniy [The approximation of response time of a cloud computing system]. Informatika i ee Primeneniya - Inform. Appl. 9(3):32-38.
- Zaryadov, I., A. Kradenyh, and A. Gorbunova. 2017. The analysis of cloud computing system as a queueing system with several servers and a single buffer. Analytical and computational methods in probability theory. Eds. D.B. Gnedenko, S. S. Demidov, A. M. Zubkov, and V. A. Kashtanov. Lecture notes in computer science ser. Springer. 10684:11-22.
- Meykhanadzhyan, L., S. Matyushenko, D. Pyatkina, and R. Razumchik. 2017. Revisiting joint stationary distribution in two finite capacity queues operating in parallel. Informatika i ee Primeneniya - Inform. Appl. 11 (3): 106-112.
- Osipov, O. A., and I. E. Tananko. 2017. Seti massovogo obsluzhivaniya proizvol'noy topologii s deleniem i sliyaniem trebovaniy: sluchay beskonechnopribornykh sistem obsluzhivaniya [Fork-join queueing networks with an arbitrary topology: The infinite server case]. Herald of Tver State University. Ser. Appl. Math. 4:43-58.
- Patterson, D. A., G. Gibson, and R.H. Katz. 1987. A case of redundant arrays of inexpensive disks. Berkley, CA: University of California. Technical Report CSD-87391. 26 p.
- Gumbel, E. J. 1954. The maxima of the mean largest value and of the range. Ann. Math. Stat. 25(1):76-84.
- Hartley, H. O., and H. A. David. 1954. Universal bounds for mean range and extreme observation. Ann. Math. Stat. 25(1):85-99.
- Crow, C.S., D. Goldberg, and W. Whitt. 2007. Two-moment approximations for maxima. Oper. Res. 55(3):532-548.
- Arnold, B. C. 1985. p-Norm bounds on the expectation of the maximum of possibly dependent sample. J. Multivariate Anal. 17(3):316-332.
- Bertsimas, D. 2006. Tight bounds on expected order statistics. Probab. Eng. Inform. Sc. 20(4):667-686.
- Rychlik, T. 2014. Maximal expectations of extreme order statistics from increasing density and failure rate populations. Commun. Stat. Theor. M. 43(10-12):2199-2213.
- Goroncy, A., and T. Rychlik. 2016. Evaluations of expectations of order statistics and spacings based on IFR distributions. Metrika 79(6):635-657.
- Balakrishnan, N. 1990. Improving the Hartley-David-Gumbel bound for the mean of extreme order statistics. Stat. Probabil. Lett. 9(4): 291-294.
- Grigoryeva, M. 2015. Uslovnye granitsy mer riska v finansovoy matematike [Conditional boundaries of risk measure in financial mathematics]. Sovremennye problemy matematiki i mekhaniki [Modern Problems of Mathematics and Mechanics]. Moscow: MSU. 10(3):63-81.
- Huang, J. S. 1998. Sequence of expectations of maximum-order statistics. Stat. Probabil. Lett. 38(2):117-123.
- Ruben, H. 1962. The moments of the order statistics and of the range in samples from normal populations. Contributions to order statistics. Eds. A. E. Sarhan and B. G. Greenberg. New York, NY: Wiley. Ch. 10A.
[+] About this article
Title
CONDITIONAL BOUNDS OF EXPECTED MAXIMA OF RANDOM VARIABLES AND THEIR REACHABILITYJournal
Systems and Means of InformaticsVolume 29, Issue 1, pp 140-163
Cover Date
2019-03-30DOI
10.14357/08696527190112Print ISSN
0869-6527Publisher
Institute of Informatics Problems, Russian Academy of SciencesAdditional Links
Key words
expected maximum; reachabilityAuthors
D. V. Ivanov
Author Affiliations
Faculty of Mechanics and Mathematics, M. V. Lomonosov Moscow State University, Main Building, 1 Leninskiye Gory, Moscow 119991, Russian Federation