Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 6 (2011), 203 -- 219

LOG-CONCAVITY PROPERTY FOR SOME WELL-KNOWN DISTRIBUTIONS

G. R. Mohtashami Borzadaran and H. A. Mohtashami Borzadaran

Abstract. Interesting properties and propositions, in many branches of science such as economics have been obtained according to the property of cumulative distribution function of a random variable as a concave function. Caplin and Nalebuff (1988 [10],1989 [11]), Bagnoli and Khanna (1989 [7]) and Bagnoli and Bergstrom (1989 [4], 1989 [5], 2005 [6]) have discussed the log-concavity property of probability distributions and their applications, especially in economics.
Log-concavity concerns twice differentiable real-valued function g whose domain is an interval on extended real line. g as a function is said to be log-concave on the interval (a,b) if the function ln(g) is a concave function on (a,b). Log-concavity of g on (a,b) is equivalent to g'/g being monotone decreasing on (a,b) or (ln(g))" <0. (2005 BERGSTROM AND BAGNOLI HREF="#b6" [6]) have obtained log-concavity for distributions such as normal, logistic, extreme-value, exponential, Laplace, Weibull, power function, uniform, gamma, beta, Pareto, log-normal, Student's t, Cauchy and F distributions. We have discussed and introduced the continuous versions of the Pearson family, also found the log-concavity for this family in general cases, and then obtained the log-concavity property for each distribution that is a member of Pearson family. For the Burr family these cases have been calculated, even for each distribution that belongs to Burr family. Also, log-concavity results for distributions such as generalized gamma distributions, Feller-Pareto distributions, generalized Inverse Gaussian distributions and generalized Log-normal distributions have been obtained.

2010 Mathematics Subject Classification: 91B02; 62H10; 62P20.
Keywords: Log-concavity; Log-convexity; Continuous distributions; Pearson family; Burr family; Generalized gamma distributions; Generalized inverse Gaussian.

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References

  1. M.Y. An, Duration dependence, endogenous search, and structural analysis of labor market histories, Paper presented at the International Conference on the Econometrics of Dynamic Decision Making, Tilberg, the Netherlands, 1994.

  2. M.Y. An, Log-concave probability distributions: Theory and statistical testing, Duke University, 1995.

  3. M.Y. An, Log-concavity and statistical inference of linear index modes, Manuscript, Duke University, 1997.

  4. M. Bagnoli and T. Bergstrom, Courtship as a waiting game, University of Michigan, Working Paper, 1989.

  5. M. Bagnoli and T. Bergstrom, Signalling and costly appraisals, University of Michigan, Working Paper, 1989.

  6. M. Bagnoli and T. Bergstrom, Log-concave probability and it's applications, Economic Theory, Springer, 26(2)  (2005), 445-469. MR2213177. Zbl 1077.60012.

  7. M. Bagnoli and N. Khanna, Why are buyers represented by sellers agents when buying a house?, University of Michigan, Working Paper, 1989.

  8. David P. Baron and Roger B. Myerson, Regulating a monopolist with unknown costs, Econometrica, Econometric Society, \ 50 (4) (1982), 911-930. MR0666117(83i:90039). Zbl 0483.90019.

  9. I. W. Burr, Cumulative frequency functions, Annals of Mathematical Statistics, 13 (1942), 215-232. MR0006644(4,19f). Zbl 0060.29602.

  10. A. Caplin and B. Nalebuff, After Hotelling: Existence of equilibrium for an imperfectly competitive market, Princeton University, Working Paper, 1988.

  11. A. Caplin and B. Nalebuff, Aggregation and social choice: A mean voter theorem. Princeton University, Working Paper, 1989.

  12. C. Flinn and J. Heckman, Are unemployment and out of the labor force behaviorally distinct labor force states?, Journal of Labor Economics, 1 (1983), 28-43.

  13. H. W. Hager and L. J. Bain, Inferential procedures for the generalized gamma distribution, Journal of the American Statistical Association, 65 (1970), 334-342. Zbl 0224.62014.

  14. N. Jegadeesh and B. Chowdhry, Optimal pre-tender offer share acquisition strategy in takeovers, UCLA Working Paper, 1989.

  15. N. L. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions, vol. 1, 2nd ed. New York: John Wiley, 1994. MR1299979(96j:62028). Zbl 0811.62001.

  16. T. Kawamura and K. Iwase, Characterizations of the distributions of power inverse Gaussian and others based on the entropy maximization principle, J. Japan Statist. Soc., 33 No. 1 (2003), 95-104. MR2021970(2005b:62032). Zbl 1023.62012.

  17. J. Laffont and J. Tirole, The dynamics of incentive contracts, Econometrica, 56 (1988), 1153-1175. MR0964150(89i:90020). Zbl 0663.90014.

  18. T. Lewis and D. Sappington, Regulating a monopolist with unknown demand, American Economic Review, 78 (1988), 986-997.

  19. E. Maskin and J. Riley, Monopoly with incomplete information, Rand. Journal of Economics, 15 (1984), 171-196. MR0755509.

  20. S. Matthews, Comparing auctions for risk-averse buyers: a buyers point of view, Econometrica,  55 (1987), Issue 3, 633-646. MR0890857(88j:90010). Zbl 0612.90018.

  21. J. B. McDonald and Y. J. Xu, A generalization of the beta distribution with applications, Journal of Econometrics, 66 (1995), 133-152. Zbl 0813.62011.

  22. R. Myerson and M. Satterthwaite, Efficient mechanisms for bilateral trading, Journal of Economic Theory, 28 (1983), 265-281. MR0707358(85b:90005). Zbl 0523.90099.

  23. K. Pearson, Contributions to the mathematical theory of evolution, II: Skew variation in homogeneous material, Philosophical Transactions of the Royal Society of London ARRAY 186 (1895), 343- 414. JFM 26.0243.03.

  24. A. Prekopa, Logarithmic concave measures with application to stochastic programming, Acta Scientiarium Mathematicarum, 32 (1971), 301- 315. MR0315079(47 #3628). Zbl 0235.90044.

  25. A. Prekopa, On the number of vertices of random convex polyhedra, Periodica Math. Hung., 2 (1972), 259-282. MR0326797(48 #5140). Zbl 0282.60007.

  26. A. Prekopa, On logarithmic concave measures and functions, Acta Scientiarium Mathematicarum, 33 (1973), 335-343. MR0404557(53 #8357). Zbl 0264.90038.

  27. R. L. Prentice, A log-gamma model and its maximum likelihood estimation, Biometrika, 61 (1974), 539-544. MR0378212(51 #14381). Zbl 0295.62034.

  28. M. Riordan and D. Sappington, Second sourcing, RAND Journal of Economics, The RAND Corporation, 20 (1) (1989), 41-58.

  29. E.W. Stacy, A generalization of the gamma distribution, The Annals of Mathematical Statistics, 33 (1962), 1187-1192. MR0143277(26 #836). Zbl 0121.36802.

  30. S. Vianelli, La misura della variabilita condizionata in uno schema generale delle curve normali di frequenza, Statistica, 23 (1963), 447-474.

  31. S. Vianelli, Sulle curve lognormali di ordine r quali famiglie di distribuzioni di errori di proporzione, Statistica, 42 (1982), 155-176. MR0685129(84i:62024). Zbl 0551.62015.

  32. S. Vianelli, Una nota sulle distribuzioni degli errori di proporzione con particolare riguardo alla distribuzione corrispondente alla prima legge degli errori additivi di Laplace, Statistica, 42 (1982), 371-380. MR0695468(84g:62024).

  33. S. Vianelli, The family of normal and lognormal distributions of order r, Metron, 41 (1983), 3-10. MR0740130. Zbl 0539.62017.



G. R. Mohtashami Borzadaran H. A. Mohtashami Borzadaran
Department of Statistics, Department of Statistics,
Faculty of Mathematical Sciences, Faculty of Mathematical Sciences,
Ferdowsi University of Mashhad, Ferdowsi University of Mashhad,
Mashhad, Iran. Mashhad, Iran.
e-mail: grmohtashami@um.ac.ir, gmb1334@yahoo.com e-mail: hmohtashami66@gmail.com

http://www.utgjiu.ro/math/sma