On the Generalized Lognormal Distribution

This paper introduces, investigates, and discusses the -order generalized lognormal distribution ( -GLD). Under certain values of the extra shape parameter , the usual lognormal, log-Laplace, and log-uniform distribution, are obtained, as well as the degenerate Dirac distribution. The shape of all the members of the -GLD family is extensively discussed. The cumulative distribution function is evaluated through the generalized error function, while series expansion forms are derived. Moreover, the moments for the -GLD are also studied. 1. Introduction Lognormal distribution has been widely applied in many different aspects of life sciences, including biology, ecology, geology, and meteorology as well as in economics, finance, and risk analysis, see [1]. Also, it plays an important role in Astrophysics and Cosmology; see [2–4] among others, while for Lognormal expansions see [5]. In principle, the lognormal distribution is defined as the distribution of a random variable whose logarithm is normally distributed, and usually it is formulated with two parameters. Furthermore, log-uniform and log-laplace distributions can be similarly defined with applications in finance; see [6, 7]. Specifically, the power-tail phenomenon of the Log-Laplace distributions [8] attracts attention quite often in environmental sciences, physics, economics, and finance as well as in longitudinal studies [9]. Recently, Log-Laplace distributions have been proposed for modeling growth rates as stock prices [10] and currency exchange rates [7]. In this paper a generalized form of Lognormal distribution is introduced, involving a third shape parameter. With this generalization, a family of distributions is emerged, which combines theoretically all the properties of Lognormal, Log-Uniform, and Log-Laplace distributions, depending on the value of this third parameter. The generalized -order Lognormal distribution ( -GLD) is the distribution of a random vector whose logarithm follows the -order normal distribution, an exponential power generalization of the usual normal distribution, introduced by [11, 12]. This family of -dimensional generalized normal distributions, denoted by , is equipped with an extra shape parameter and constructed to play the role of normal distribution for the generalized Fisher’s entropy type of information; see also [13, 14]. The density function of a -variate, -order, normally distributed random variable , with location vector , positive definite scale matrix , and shape parameter , is given by [11]. where is the quadratic form , , while being the normalizing