Log-concavity property for some well-known distributions

• Main Authors: G. R. Mohtashami Borzadaran, H. A. Mohtashami Borzadaran
• Format: Article
• Language: English
• Published: University Constantin Brancusi of Targu-Jiu 2011-12-01
• Series: Surveys in Mathematics and its Applications
• Subjects: Log-concavity; Log-convexity; Continuous distributions; Pearson family; Burr family; Generalized gamma distributions; Generalized inverse Gaussian
• Online Access: http://www.utgjiu.ro/math/sma/v06/p15.pdf
• ISSN: 1843-7265; 1842-6298

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 (1988,1989), Bagnoli and Khanna (1989) and Bagnoli and Bergstrom (1989 , 1989, 2005) 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 [<A HREF="#b6">6</A>]) 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.