Skip to main content
SpringerLink
Log in

q-log-distributions: Log-concavity and log-convexity

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract.

Tsallis statistics along with Tsallis distributions have been attracting considerable attention of statisticians in recent years, since applications of the q-distributions can be found in various fields. However, only until recently, in 2018, a q -logarithm transformation has been considered in order to obtain new Tsallis distributions. In this paper, we introduce the q-log-distributions defined for \(q > 1\). We propose these models to be considered for modeling demographic data. We discuss the log-concavity and log-convexity of these models with emphasis on log-concavity. We extend the notions of log-concavity and log-convexity. We explore the shapes of the hazard rate functions and give bounds for the mean residual life function.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. W. Shaw, I. Buckley, The alchemy of probability distributions: beyond Gram-Charlier expansions and a skew-kurtotic-normal distribution from a rank transmutation map, arXiv:0901.0434 (2007)

  2. V. Preda, I. Băncescu, Int. J. Risk Theory 1, 13 (2016)

    Google Scholar 

  3. I. Băncescu, An. Stiint. Univ. Constanţa 26, 43 (2018)

    Google Scholar 

  4. S.A. Khan, Lifetime Data Anal. 24, 328 (2017)

    Article  Google Scholar 

  5. R.R. Officer, J. Am. Stat. Assoc. 67, 807 (1972)

    Article  Google Scholar 

  6. K.T. Tallakstad, R. Toussaint, S. Santucci, K.J. Maloy, Phys. Rev. Lett. 110, 145501 (2013)

    Article  ADS  Google Scholar 

  7. C. Tsallis, Quim. Nova 17, 468 (1994)

    Google Scholar 

  8. C. Tsallis, J. Stat. Phys. 52, 479 (1998)

    Article  ADS  Google Scholar 

  9. C. Tsallis, C. Anteneodo, L. Borland, R. Osorio, Physica A 324, 89 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  10. C. Tsallis, R.S. Mendes, A.R. Plastino, Physica A 261, 534 (1998)

    Article  ADS  Google Scholar 

  11. S. Furuichi, J. Math. Phys. 47, 023302 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  12. S. Furuichi, IEEE Trans. Inf. Theory 51, 3638 (2005)

    Article  MathSciNet  Google Scholar 

  13. B. Trivellato, Entropy 15, 3471 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  14. B. Trivellato, Int. J. Theor. Appl. Fin. 15, 1250038 (2012)

    Article  Google Scholar 

  15. E. Moretto, S. Pasquali, B. Trivellato, Physica A 446, 246 (2016)

    Article  MathSciNet  Google Scholar 

  16. V. Preda, S. Dedu, C. Gheorghe, Physica A 436, 925 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  17. V. Preda, S. Dedu, M. Sheraz, Physica A 407, 350 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  18. T. Yamano, Physica A 305, 486 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  19. F. Zhang, Y. Shi, H.K.T. Ng, R. Wang, Eur. Phys. J. Plus 131, 379 (2016)

    Article  Google Scholar 

  20. J. Luckstead, S. Devadoss, Econ. Lett. 124, 290 (2014)

    Article  Google Scholar 

  21. P. Krugman, The Self-Organizing Economy (Blackwell Publishers Cambridge, Massachusetts, 1996)

  22. X. Gabaix, Quart. J. Econ. 114, 739 (1999)

    Article  Google Scholar 

  23. J. Eeckhout, Am. Econ. Rev. 94, 1429 (2004)

    Article  Google Scholar 

  24. W.J. Reed, M. Jorgensen, Commun. Stat. - Theory Methods 33, 1733 (2004)

    Article  Google Scholar 

  25. Y. Ioannides, S. Skouras, J. Urban Econ. 73, 18 (2013)

    Article  Google Scholar 

  26. M.Y. An, J. Econ. Theory 80, 350 (1998)

    Article  Google Scholar 

  27. M.C. Jones, A. Noufaily, Electron. J. Stat. 9, 2732 (2015)

    Article  MathSciNet  Google Scholar 

  28. C. Gigliarano, U. Basellini, M. Bonetti, Lifetime Data Anal. 23, 254 (2017)

    Article  MathSciNet  Google Scholar 

  29. M. Ahsanullah, Ann. Inst. Stat. Math. 32, 363 (1980)

    Article  Google Scholar 

  30. D. Cousineau, IEEE Trans. Dielectr. Electr. Insul. 16, 281 (2009)

    Article  Google Scholar 

  31. S. Shang, X. Mao, Chin. Geogr. Sci. 24, 575 (2014)

    Article  Google Scholar 

  32. J. Yang, C. Zhao, X. Li, F. Wang, Open Mech. Eng. J. 9, 630 (2015)

    Article  Google Scholar 

  33. P. Wais, Renew. Energy 103, 15 (2017)

    Article  Google Scholar 

  34. S.S. Sanni, W.I.E. Chukwu, Appl. Math. Model. 37, 9698 (2013)

    Article  MathSciNet  Google Scholar 

  35. E.P. Borges, Physica A 31, 5281 (1998)

    Google Scholar 

  36. A.W. Marshall, I. Olkin, Life distributions

  37. F. Zhang, Hon Keung Tony Ng, Yimin Shi, Physica A 490, 1171 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  38. L. Montrucchio, L. Peccati, Insurance: Math. Econ. 10, 125 (1991)

    Google Scholar 

  39. E. Moretto, Math. Methods Econ. Finance 2, 71 (2007)

    MathSciNet  Google Scholar 

  40. K. Pearson, Philos. Trans. R. Soc. London ARRAY 186, 343 (1895)

    Article  ADS  Google Scholar 

  41. G.R. Borzadaran Mothashami, H.A. Borzadaran Mothashami, Surv. Math. Appl. 6, 203 (2011)

    MathSciNet  Google Scholar 

  42. T. Tsukatani, K. Shigemitsu, Atm. Environ. (1967) 14, 245 (1980)

    Article  Google Scholar 

  43. T.C. Sharma, U.S. Panu, Hydrol. Sci. J. 60, 1783 (2015)

    Article  Google Scholar 

  44. P.S. Prevey, Adv. X-Ray Anal. 29, 103 (1986)

    Google Scholar 

  45. M. Bebbington, C.D. Lai, R. Zitikis, IEEE Trans. Reliab. 55, 245 (2006)

    Article  Google Scholar 

  46. R.E. Barlow, P. Proschan, Statistical Theory of Reliability and Life Testing (Silver Spring, 1981)

  47. D.V. Lindley, J. R. Stat. Soc., Ser. B 20, 102 (1958)

    Google Scholar 

  48. D.V. Lindley, Introduction to Probability and Statistics from a Bayesian Viewpoint, Part II: Inference (Cambridge University Press, New York, 1965)

  49. M.E. Ghitany, B. Atieh, S. Nadarajah, Math. Comput. Simul. 78, 493 (2008)

    Article  Google Scholar 

  50. B.C. Arnold, Pareto Distributions (International Cooperative Publishing House, Fairland, 1983)

  51. R.L. Berger, D.D. Boos, F.M. Guess, Biometrics 44, 103 (1988)

    Article  MathSciNet  Google Scholar 

  52. M.J.D. Powel, A Direct Search Optimization Method that Models the Objective and Constraint Functions by Linear Interpolation, in Advances in Optimization and Numerical Analysis, edited by S. Gomez, J.P. Hennart, (Kluwer Academic Publishers, London, 1994)

  53. M.B. Subrahmanyam, J. Optim. Theory Appl. 62, 311 (1989)

    Article  MathSciNet  Google Scholar 

  54. A. Pădureanu, Metode şi tehnici evoluate utilizate in modelarea fenomenelor financiare (Evolved methods and techniques used in modeling financial phenomena), PhD Thesis, Bucharest Academy of Economic Studies, Institute of PhD Studies, PhD School of Cybernetics and Economic Statistics (2012) available on-line at http://opac.biblioteca.ase.ro/opac/bibliographic_view/191390?pn=opac%2FSearch&q=padureanu#level=all&location=0&ob=asc&q=padureanu&sb=relevance&start=0&view=CONTENT

  55. United Nations Statistics Division, http://data.un.org/Data.aspx?d=POP&f=tableCode%3A240#POP

  56. L.C. Malacarne, R.S. Mendes, E.K. Lenzi, Phys. Rev. E 65, 017106 (2001)

    Article  ADS  Google Scholar 

  57. R. Gonzalez-Val, A. Ramos, F. Sanz-Gracia, M. Vera-Cabello, Papers Regional Sci. 94, 177 (2015)

    Google Scholar 

  58. S. Devadoss, J. Luckstead, D. Danforth, S. Akhundjanov, Physica A 442, 193 (2016)

    Article  ADS  Google Scholar 

  59. H. Akaike, IEEE Trans. Autom. Control 19, 716 (1974)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. PhD School of Mathematics, University of Bucharest, Bucharest, Romania

    Irina Băncescu

  2. National Institute for Economic Research “Costin C. Kiriţescu”, Bucharest, Romania

    Irina Băncescu

Authors
  1. Irina Băncescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Irina Băncescu.

Electronic supplementary material

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Băncescu, I. q-log-distributions: Log-concavity and log-convexity. Eur. Phys. J. Plus 133, 163 (2018). https://doi.org/10.1140/epjp/i2018-12005-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/i2018-12005-3

Search

Navigation