Modern Stochastics: Theory and Applications logo


  • Help
Login Register

  1. Home
  2. Issues
  3. Volume 4, Issue 3 (2017)
  4. Quantifying non-monotonicity of function ...

Modern Stochastics: Theory and Applications

Submit your article Information Become a Peer-reviewer
  • Article info
  • Full article
  • Cited by
  • More
    Article info Full article Cited by

Quantifying non-monotonicity of functions and the lack of positivity in signed measures
Volume 4, Issue 3 (2017), pp. 219–231
Youri Davydov   Ričardas Zitikis  

Authors

 
Placeholder
https://doi.org/10.15559/17-VMSTA84
Pub. online: 28 September 2017      Type: Research Article      Open accessOpen Access

Received
16 June 2017
Revised
5 September 2017
Accepted
5 September 2017
Published
28 September 2017

Abstract

In various research areas related to decision making, problems and their solutions frequently rely on certain functions being monotonic. In the case of non-monotonic functions, one would then wish to quantify their lack of monotonicity. In this paper we develop a method designed specifically for this task, including quantification of the lack of positivity, negativity, or sign-constancy in signed measures. We note relevant applications in Insurance, Finance, and Economics, and discuss some of them in detail.

References

[1] 
Bebbington, M., Lai, C.D., Zitikis, R.: Modelling deceleration in senescent mortality. Math. Popul. Stud. 18, 18–37 (2011). MR2770696. doi:10.1080/08898480.2011.540173
[2] 
Bernardo, A.E., Ledoit, O.: Gain, loss, and asset pricing. J. Polit. Econ. 108, 144–172 (2000). doi:10.1086/262114
[3] 
Brazauskas, V., Jones, B.L., Zitikis, R.: Trends in disguise. Ann. of Actuar. Sci. 9, 58–71 (2015). doi:10.1017/S1748499514000232
[4] 
Chakravarty, S.R.: Extended Gini indices of inequality. Int. Econ. Rev. 29, 147–156 (1988). MR0954119. doi:10.2307/2526814
[5] 
Davydov, Y., Zitikis, R.: An index of monotonicity and its estimation: a step beyond econometric applications of the Gini index. Metron – International Journal of Statistics 63 (special issue in memory of Corrado Gini), 351–372 (2005). MR2276056
[6] 
Denuit, M., Dhaene, J., Goovaerts, M., Kaas, R.: Actuarial Theory for Dependent Risks: Measures, Orders and Methods. Wiley, New York (2005)
[7] 
Furman, E., Zitikis, R.: Weighted premium calculation principles. Insur. Math. Econ. 42, 459–465 (2008). MR2392102. doi:10.1016/j.insmatheco.2007.10.006
[8] 
Furman, E., Zitikis, R.: Weighted pricing functionals with application to insurance: an overview. N. Am. Actuar. J. 13, 483–496 (2009). MR2595160. doi:10.1080/10920277.2009.10597570
[9] 
Gillen, B., Markowitz, H.M.: A taxonomy of utility functions. In: Aronson, J.R., Parmet, H.L., Thornton, R.J. (eds.) Variations in Economic Analysis: Essays in Honor of Eli Schwartz. Springer, New York (2009)
[10] 
Keating, C., Shadwick, W.F.: In: A Universal Performance Measure, The Finance Development Centre Limited. London, UK (2002)
[11] 
Lehmann, E.L.: Some concepts of dependence.. Ann. Math. Stat. 37, 1137–1153 (1966). MR202228. doi:10.1214/1177699260.1966.10597570
[12] 
Levy, H.: Stochastic Dominance: Investment Decision Making under Uncertainty. Springer, New York (2006). MR2239375
[13] 
Li, H., Li, X.: Stochastic Orders in Reliability and Risk. In: Honor of Professor Moshe Shaked. Springer, New York (2013). MR3157470. doi:10.1007/978-1-4614-6892-9
[14] 
von Neuman, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton, NJ (1944). MR0011937
[15] 
Qoyyimi, D.T., Zitikis, R.: Measuring the lack of monotonicity in functions. Math. Sci. 39, 107–117 (2014). MR3307986
[16] 
Qoyyimi, D.T., Zitikis, R.: Measuring association via lack of co-monotonicity: the LOC index and a problem of educational assessment. Dependence Modeling 3, 83–97 (2015). MR3418658. doi:10.1515/demo-2015-0006
[17] 
Quiggin, J.: Generalized Expected Utility Theory. Kluwer, Dordrecht (1993).
[18] 
Wang, S.S.: Insurance pricing and increased limits ratemaking by proportional hazards transforms. Insur. Math. Econ. 17, 43–54 (1995). MR1363642. doi:10.1016/0167-6687(95)91054-P
[19] 
Wang, S.S.: Premium calculation by transforming the layer premium density. ASTIN Bull. 26, 71–92 (1996). doi:10.2143/AST.26.1.563234
[20] 
Yaari, M.E.: The dual theory of choice under risk. Econometrica 55, 95–115 (1987). MR0875518. doi:10.2307/1911158
[21] 
Yitzhaki, S., Schechtman, E.: The Gini Methodology: A Primer on a Statistical Methodology. Springer, New York (2013). MR3012052. doi:10.1007/978-1-4614-4720-7
[22] 
Zitikis, R.: Asymptotic estimation of the E-Gini index. Econom. Theory 19, 587–601 (2003). MR1997934. doi:10.1017/S0266466603194042
[23] 
Zitikis, R., Gastwirth, J.L.: Asymptotic distribution of the S-Gini index. Aust. N. Z. J. Stat. 44, 439–446 (2002). MR1934733. doi:10.1111/1467-842X.00245

Full article Cited by PDF XML
Full article Cited by PDF XML

Copyright
© 2017 The Author(s). Published by VTeX
by logo by logo
Open access article under the CC BY license.

Keywords
Non-monotonic functions signed measures Hahn and Jordan decompositions weighted premium risk measure gain–loss ratio

MSC2010
28E05 26A48 62P05 97M30

Metrics
since March 2018
1201

Article info
views

705

Full article
views

449

PDF
downloads

221

XML
downloads

Export citation

Copy and paste formatted citation
Placeholder

Download citation in file


Share


RSS

MSTA

MSTA

  • Online ISSN: 2351-6054
  • Print ISSN: 2351-6046
  • Copyright © 2018 VTeX

About

  • About journal
  • Indexed in
  • Editors-in-Chief

For contributors

  • Submit
  • OA Policy
  • Become a Peer-reviewer

Contact us

  • ejournals-vmsta@vtex.lt
  • Mokslininkų 2A
  • LT-08412 Vilnius
  • Lithuania
Powered by PubliMill  •  Privacy policy