On the Generalized Lognormal Distribution

This paper introduces, investigates, and discusses the γ-order generalized lognormal distribution (γ-GLD). Under certain values of the extra shape parameter γ, the usual lognormal, log-Laplace, and log-uniform distribution, are obtained, as well as the degenerate Dirac distribution.The shape of all themembers of the γ-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 γGLD are also studied.


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][3][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 N   (, Σ), 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].
In Section 2, a generalized form of the Lognormal distribution is introduced, which is derived from the univariate family of N  (,  2 ) = N 1  (,  2 ) distributions, denoted by LN  (, ), and includes the Log-Laplace distribution as well as the Log-Uniform distribution.The shape of the LN  (, ) members is extensively discussed while it is connected to the tailing behavior of LN  through the study of the c.d.f.In Section 3, an investigation of the moments of the generalized Lognormal distribution, as well as the special cases of Log-Uniform and Log-Laplace distributions, is presented.
The generalized error function, that is briefly provided here, plays an important role in the development of LN  (, ); see Section 2. The generalized error function denoted by Erf  and the generalized complementary error function Erfc  = 1−Erf  ,  ≥ 0 [16], are defined, respectively, as The generalized error function can be expressed (changing to   variable), through the lower incomplete gamma function (, ) or the upper (complementary) incomplete gamma function Γ(, ) = Γ() − (, ), in the form see [16].Moreover, adopting the series expansion form of the lower incomplete gamma function, a series expansion form of the generalized error function is extracted: Notice that, Erf 2 is the known error function erf, that is, Erf as (, ) → Γ() when  → +∞.

The 𝛾-Order Lognormal Distribution
The generalized univariate Lognormal distribution is defined, through the univariate generalized -order normal distribution, as follows.
Definition 1.When the logarithm of a random variable  follows the univariate -order normal distribution, that is, log  ∼ N  (,  2 ), then  is said to follow the generalized Lognormal distribution, denoted by LN  (, ); that is,  ∼ LN  (, ).
The LN  (, ) is referred to as the (generalized) -order Lognormal distribution (-GLD).Like the usual Lognormal distribution, the parameter  ∈ R is considered to be log-scaled, while the non log-scaled  (i.e.  when  is assumed log-scaled) is referred to as the location parameter of LN  (, ).Hence, if  ∼ LN  (, ), then log  is a -order normally distributed variable; that is, log  ∼ N  (,  2 ).Therefore, the location parameter  ∈ R of  is in fact the mean of 's natural logarithm, that is, E[log ] = , while see [15] for details on N  .Let  := log  ∼ N  (,  2 ) with density function as in (1) and  = () =   .Then, the density function   of  ∼ LN  (, ) can be written, through (1) The probability density function   , as in (10), is defined in R * + = R + \0; that is, LN  (, ) has zero threshold.Therefore, the following definition extends Definition 1.
Definition 2. When the logarithm of a random variable  +  follows the univariate -order normal distribution, that is, log( + ) ∼ N  (,  2 ), then  is said to follow the generalized Lognormal distribution with threshold  ∈ R; that is,  ∼ LN  (, ; ).
It is clear that when  ∼ LN  (, ; , ), log( − ) is a -order normally distributed variable, that is, log( − ) ∼ N  (,  2 ), and thus,  is the mean of ( − )'s natural logarithm while Var[log ] is the same as in (8).
Let  = (log( − ) − )/.Then, the limiting threshold density value of   () with  →  + implies that and therefore that is, the   's defining domain, for the positive-ordered Lognormal random variable , can be extended to include threshold point  by letting   () = 0.
The generalized Lognormal family of distributions LN  is a wide range family bridging the Log-Uniform LU, Lognormal LN, and Log-Laplace LL distributions, as well as the degenerate Dirac D distributions.We have the following.
Theorem 3. The generalized Lognormal distribution LN  (, ), for order values of  = 0, 1, 2, ±∞, is reduced to Proof.From definition (1) of N  the order  value is a real number outside the closed interval [0, 1].Let   ∼ LN  (, ) with density function    as in (10).We consider the following cases.
(ii) The "normal" case  = 2: it is clear that LN 2 (, ) = LN(, ), as   2 coincides with the Lognormal density function, and therefore the second-ordered Lognormal distribution is in fact the usual Lognormal distribution.
(iii) The limiting case  = ±∞: we have LN ±∞ (, ) := lim  → ±∞ LN  (, ) with which coincides with the density function of the known Log-Laplace distribution (symmetric logexponential distribution) LL(  , , ) with   =   and  =  = 1/; see [8].Therefore, the infiniteordered log-normal distribution is in fact the Log-Laplace distribution, with threshold density For the purposes of statistical application, the Log-Laplace moments are not the same as the model parameters; that is, although   = ,   = √ 2.
From the above limiting cases (i), (iii), and (iv), the defining domain R \ [0, 1] of the order values , used in (1), is safely extended to include the values  = 0, 1, ±∞; that is,  can now be defined outside the open interval (0, 1).Eventually, the family of the -order normals can include the Log-Uniform, Lognormal, Log-Laplace, and the degenerate Dirac distributions as (13) holds.
Proof.The negative-ordered Lognormals are formed by density functions admitting threshold 0 (in limit) for their global mode point (of infinite density), as shown in (12).Moreover, from the previously discussed monotonicity of    in Proposition 4, all the negative-ordered Lognormals admit also   as a local nonsmooth mode point and exp{ −   } as a local minimum density point, with densities as in (26) and    ( −  ), respectively; see Figures 1(b1), 1(b2), and 1(b3).
The above discussion on behavior of the modes with respect to shape parameter  is formed in the following propositions.
For the evaluation of the cumulative distribution function (c.d.f.) of the generalized Lognormal distribution, the following theorem is stated and proved.Theorem 8.The c.d.f.   of a -order Lognormal random variable   ∼ LN  (, ) is given by ) , Proof.From density function   , as in (10), we have } . ( Applying the transformation  = (log  − )/,  > 0, the above c.d.f. is reduced to where Φ   is the c.d.f. of the standardized -order normal distribution   = (1/)(log   − ) ∼ N  (0, 1).Moreover, Φ   can be expressed in terms of the generalized error function.In particular, and as    is a symmetric density function around zero, we have and thus Substituting the normalizing factor, as in (2), and using (3), we obtain and finally, through (34), we derive (31), which forms (32) through (4).
It is essential for numeric calculations to express (31) considering positive arguments for Erf.Indeed, through (37), we have while applying (4) into (39) it is obtained that ) . (40) As the generalized error function Erf  is defined in (4), through the upper incomplete gamma function Γ( −1 , ⋅), series expansions can be used for a more "numericaloriented" form of (4).Here some expansions of the c.d.f. of the generalized Lognormal distribution are presented.
Corollary 9.The c.d.f.   can be expressed in the series expansion form Proof.Substituting the series expansion form of ( 6) into (39) and expressing the infinite series using the integer powers , the series expansion as in (41) is derived.
It is interesting to mention here that the same result can also be derived through (42), as this finite expansion can be extended for  = 1, which provides (in limit) the c.d.f. of the infinite-ordered Lognormal distribution.

Moments of the 𝛾-Order Lognormal Distribution
For the evaluation of the moments of the generalized Lognormal distribution, the following holds.} ,

Proposition 16 .
The th raw moment μ()  of a generalized lognormally distributed random variable  ∼ LN  (, ) is