Surveys in Mathematics and its Applications

ISSN1842-6298 (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. Bagnoli and Bergstrom (2005 [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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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