# On the Generalized Lognormal Distribution

Authors: Thomas L. Toulias; Christos P. Kitsos
Journal: Journal of Probability and Statistics
 (2013)

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## Abstract

This paper introduces, investigates, and discusses theγ-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 the members 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 γ-GLD are also studied.


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## Body

## 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 generalizedγ-order Lognormal distribution (γ-GLD) is the distribution of a random vector whose logarithm follows the γ-order normal distribution, an exponential power generalization of the usual normal distribution, introduced by [11, 12]. This family of p-dimensional generalized normal distributions, denoted by 𝒩γp(μ,Σ), is equipped with an extra shape parameter γ and constructed to play the role of normal distribution for the generalized Fisher’s entropy type of information; see also [13, 14].The density functionfX of a p-variate, γ-order, normally distributed random variable Y~𝒩γp(μ,Σ), with location vector μ∈ℝp, positive definite scale matrix Σ∈ℝp×p, and shape parameter γ∈ℝ∖[0,1], is given by [11]. 						(1)fY(y)=fY(y;μ,Σ,γ)=Cγp|detΣ|-1/2exp{-γ-1γQθ(y)γ/2(γ-1)},y∈ℝp,
where Qθ is the quadratic form Qθ(y)=(y-μ)TΣ-1(y-μ), θ=(μ,Σ), while Cγp being the normalizing factor
(2)Cγp=π-p/2Γ(p/2+1)Γ(p((γ-1)/γ))(γ-1γ)p((γ-1)/γ)-1.From (1), notice that the second-ordered normal is the known multivariate normal distribution; that is, 𝒩2p(μ,Σ)=𝒩p(μ,Σ); see also [13, 15].In Section2, a generalized form of the Lognormal distribution is introduced, which is derived from the univariate family of 𝒩γ(μ,σ2)=𝒩γ1(μ,σ2) distributions, denoted by ℒ𝒩γ(μ,σ), and includes the Log-Laplace distribution as well as the Log-Uniform distribution. The shape of the ℒ𝒩γ(μ,σ) members is extensively discussed while it is connected to the tailing behavior of ℒ𝒩γ 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 ofℒ𝒩γ(μ,σ); see Section 2. The generalized error function denoted by Erfa and the generalized complementary error function Erfca=1-Erfa, a≥0 [16], are defined, respectively, as
(3)Erfa(x):=Γ(a+1)π∫0xe-tadt,x∈ℝ.
The generalized error function can be expressed (changing to ta variable), through the lower incomplete gamma function γ(a,x) or the upper (complementary) incomplete gamma function Γ(a,x)=Γ(a)-γ(a,x), in the form
(4)Erfa(x)=Γ(a)πγ(1a,xa)=Γ(a)π[Γ(1a)-Γ(1a,xa)],x∈ℝ;
see [16]. Moreover, adopting the series expansion form of the lower incomplete gamma function,
(5)γ(a,x):=∫0xta-1e-tdt=∑k=0∞‍(-1)kk!(a+k)xa+k,x,a∈ℝ+,
a series expansion form of the generalized error function is extracted:
(6)Erfa(x)=Γ(a+1)π∑k=0∞‍(-1)kk!(ka+1)xka+1,x,a∈ℝ+.
Notice that, Erf2 is the known error function erf, that is, Erf2(x)=erf(x), while Erf0 is the function of a straight line through the origin with slope (eπ)-1. Applying a=2, the known incomplete gamma function identities such as γ(1/2,x)=πerfx and Γ(1/2,x)=π(1-erfx)=πerfcx,x≥0 are obtained. Moreover, Erfa0=0 for all a∈ℝ+ and
(7)limx→±∞Erfax=±1πΓ(a)Γ(1a),a∈ℝ+,
as γ(a,x)→Γ(a) when x→+∞.


## 2. Theγ-Order Lognormal Distribution

The generalized univariate Lognormal distribution is defined, through the univariate generalizedγ-order normal distribution, as follows.Definition 1.
When the logarithm of a random variableX follows the univariate γ-order normal distribution, that is, logX~𝒩γ(μ,σ2), then X is said to follow the generalized Lognormal distribution, denoted by ℒ𝒩γ(μ,σ); that is, X~ℒ𝒩γ(μ,σ).Theℒ𝒩γ(μ,σ)  is referred to  as the (generalized) γ-order Lognormal distribution (γ-GLD). Like the usual Lognormal distribution, the parameter μ∈ℝ is considered to be log-scaled, while the non log-scaled μ (i.e. eμ when μ is assumed log-scaled) is referred to as the location parameter of ℒ𝒩γ(μ,σ). Hence, if X~ℒ𝒩γ(μ,σ), then logX is a γ-order normally distributed variable; that is, logX~𝒩γ(μ,σ2). Therefore, the location parameter μ∈ℝ of X is in fact the mean of X’s natural logarithm, that is, E[logX]=μ, while
(8)Var[logX]=(γγ-1)2((γ-1)/γ)Γ(3((γ-1)/γ))Γ(((γ-1)/γ))σ2,(9)Kurt[logX]=Γ(γ-1/γ)Γ(5(γ-1/γ))Γ2(3((γ-1)/γ));
see [15] for details on 𝒩γ.LetY:=logX~𝒩γ(μ,σ2) with density function as in (1) and X=g(Y)=eY. Then, the density function fX of X~ℒ𝒩γ(μ,σ) can be written, through (1), as
(10)fX(x)=fX(x;μ,σ,γ)=fY(g-1(x))|ddxg-1(x)|=fY(logx)1x=exp{-((γ-1)/γ)|(logx-μ)/σ|γ/(γ-1)}2σ((γ-1)/γ)1/γΓ((γ-1)/γ)x.
The probability density function fX, as in (10), is defined in ℝ+*=ℝ+∖0; that is, ℒ𝒩γ(μ,σ) has zero threshold. Therefore, the following definition extends Definition 1.Definition 2.
When the logarithm of a random variableX+ϑ follows the univariate γ-order normal distribution, that is, log(X+ϑ)~𝒩γ(μ,σ2), then X is said to follow the generalized Lognormal distribution with threshold ϑ∈ℝ; that is, X~ℒ𝒩γ(μ,σ;ϑ).It is clear that whenX~ℒ𝒩γ(μ,σ;ϑ,), log(X-ϑ) is a γ-order normally distributed variable, that is, log(X-ϑ)~𝒩γ(μ,σ2), and thus, μ is the mean of (X-ϑ)’s natural logarithm while Var[logX] is the same as in (8).LetY=log(X+ϑ)~𝒩γ(μ,σ2). The density function of X=eY-ϑ~ℒ𝒩γ(μ,σ;ϑ) is given by fX(x)=fX(x-ϑ), x>0.Letz=(log(x-ϑ)-μ)/σ. Then, the limiting threshold density value of fX(x) with x→ϑ+ implies that
(11)limx→ϑ+fX(x)=σ-1Cγ1limz→-∞exp{-z(σ+μz-γ-1γ|z|1/(γ-1))}=σ-1Cγ1e(sgnγ)(-∞),
and therefore
(12)limx→ϑ+fX(x)={0,γ∈(1,+∞),+∞,γ∈(-∞,0);
that is, the fX’s defining domain, for the positive-ordered Lognormal random variable X, can be extended to include threshold point ϑ by letting fX(ϑ)=0.The generalized Lognormal family of distributionsℒ𝒩γ 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 distributionℒ𝒩γ(μ,σ), for order values of γ=0,1,2,±∞, is reduced to
(13)ℒ𝒩γ(μ,σ)={𝒟(eμ),γ=0,ℒ𝒰(eμ-σ,eμ+σ),γ=1,ℒ𝒩(μ,σ),γ=2,ℒℒ(eμ,1σ,1σ),γ=±∞.Proof.
From definition (1) of 𝒩γ the order γ value is a real number outside the closed interval [0,1]. Let Xγ~ℒ𝒩γ(μ,σ) with density function fXγ as in (10). We consider the following cases.(i)
The limiting caseγ=1: let x∈ℝ+* such that |logx-μ|≤1. Using the gamma function additive identity Γ(z+1)=zΓ(z), z∈ℝ+, in (10), we have ℒ𝒩1(μ,σ)=limγ→1+ℒ𝒩γ(μ,σ) with
(14)fX1(x)∶=limγ→1+fXγ(x)={12σx,x∈[eμ-σ,eμ+σ],0,x∈(-∞,eμ-σ)∪(eμ+σ,+∞),
which is the density function of the Log-Uniform distribution ℒ𝒰(a,b), 0<a<b, with a=eμ-σ and b=eμ+σ; that is, μ=(1/2)log(ab) and σ=(1/2)log(b/a). Therefore, first-ordered Lognormal distribution is in fact the Log-Uniform distribution, with vanishing threshold density, fX1(0)=limx→0+fX1(x)=0. For the purposes of statistical application the Log-Uniform moments are not the same as the model parameters; that is, although μX=μ, σX=σ/3.(ii)
The “normal” caseγ=2: it is clear that  ℒ𝒩2(μ,σ)=ℒ𝒩(μ,σ), as fX2 coincides with the Lognormal density function, and therefore the second-ordered Lognormal distribution is in fact the usual Lognormal distribution.(iii)
The limiting caseγ=±∞: we have ℒ𝒩±∞(μ,σ):=limγ→±∞ℒ𝒩γ(μ,σ) with
(15)fX±∞(x):=limγ→±∞fXγ(x)=12σxexp{-|logx-μσ|}={e-μ/σ2σx(1-σ)/σ,x∈(0,eμ],eμ/σ2σx-(σ+1)/σ,x>eμ,
which coincides with the density function of the known Log-Laplace distribution (symmetric log-exponential distribution) ℒℒ(μ′,α,β) with μ′=eμ and α=β=1/σ; see [8]. Therefore, the infinite-ordered log-normal distribution is in fact the Log-Laplace distribution, with threshold density
(16)fX±∞(0)=limx→0+fX±∞(x)={0,σ<1,1,σ=1,+∞,σ>1.
For the purposes of statistical application, the Log-Laplace moments are not the same as the model parameters; that is, although μX=μ, σX=2σ.(iv)
The limiting caseγ=0:  we have
(17)limγ→0-fXγ(x)=limk≔[(γ-1)/γ]→∞fXγ(x),x∈ℝ+*,
where [a] is the integer value of a∈ℝ. For the value x=eμ, the p.d.f. as in (10) implies
(18)limγ→0-fXγ(eμ)=12σeμ(limk→∞kkk!)·e0=+∞,
through Stirling’s asymptotic formula k!≈2πk(k/e)k as k→∞. Assuming now x≠eμ, (10), through (18), implies
(19)fX0(x):=limγ→0-fXγ(x)=|logx-μ|22πσ2x·0·1e=0;
that is, ℒ𝒩0(μ,σ):=limγ→0-ℒ𝒩γ(μ,σ)=𝒟(eμ) as fX0 coincides with the Dirac density function, with the (non-log-scaled) location parameter eμ of ℒ𝒩0(μ,σ) being the singular (infinity) point. Therefore, the zero-ordered Lognormal distribution ℒ𝒩0 is in fact the degenerate Dirac distribution with pole at the location parameter of ℒ𝒩γ→0- (with vanishing threshold density fX0(0)=limx→0+fX0(x)=0). From the above limiting cases (i), (iii), and (iv), the defining domain ℝ∖[0,1] of the order values γ, used in (1), is safely extended to include the values γ=0,1,±∞; that is, γ can now be defined outside the open interval (0,1). Eventually, the family of the γ-order normals can include the Log-Uniform, Lognormal, Log-Laplace, and the degenerate Dirac distributions as (13) holds.From Theorem3, (12), and (15), the domain of the density functions fXγ(x), x>0, can also be extended to include the threshold point x=0 by setting fXγ(0):=0 for all non-negative-ordered Lognormals, that is, for all γ∈0∪[1,+∞), while for the Log-Laplace case of γ=+∞ with σ=1 by setting fX+∞(0):=(1/2)e-μ.From the fact that𝒩0(μ,·)=𝒟(μ), see [15], one can say that the degenerate log-Dirac distribution, say ℒ𝒟(μ), equals ℒ𝒩0(μ,·), and hence, through Theorem 3, we can write ℒ𝒟(μ)=𝒟(eμ).Proposition 4.
The mode of the positive-ordered Lognormal random variableXγ~ℒ𝒩γ(μ,σ), X<eμγ∈(1,+∞), is given by
(20)ModeXγ=eμ-σγ,
with corresponding maximum density value,
(21)maxfXγ=fXγ(ModeXγ)=exp{(σγ)/γ-μ}2σ((γ-1)/γ)1/γΓ((γ-1)/γ).Proof.
Recall the density function ofXγ~ℒ𝒩γ(μ,σ) as in (10), and let m=ModeXγ>0. Then it holds that (d/dx)fXγ(m;μ,σ,γ)=0; that is,
(22)(1σ|logm-μ|)1/(γ-1)ddx|logx-μ|x=m=-σm.
From (d/dx)|x|=sgnx, x∈ℝ,  we have
(23)m=eμ-σγ,
provided that x<eμ. Otherwise, (23) holds trivially, as (22) implies σ=0; that is, m=eμ. Moreover, (d/dx)fXγ(x)>0 when
(24)1+sgn(logx-μ)σγ/(1-γ)|logx-μ|1/(γ-1)>0,x>0,
and thus, fXγ is a strictly ascending density function on (0,eμ-σγ) when γ>1 and also on (eμ-σγ,eμ) when γ<0. Similarly, with (d/dx)fXγ(x)<0, fX is a strictly descending density function on (eμ-σγ,+∞) when γ>1 and also on (0,eμ-σγ)∪(eμ,+∞) when γ<0.  Specifically, for γ<0, the point eμ is a nonsmooth point of fX, as
(25)limx→eμddxfXγ(x)=-Cγ1(σeμ)2[|σlogx-μ|1/(1-γ)σ+limx→eμsgn(logx-μ)×|σlogx-μ|1/(1-γ)]=+∞.
Therefore, the positive-ordered Lognormals are formed by a unimodal density function with mode as in (20) and corresponding maximum density as in (21); see Figures 1(a1), 1(a2), and 1(a3).Figure 1
Graphs of the density functionsfXγ, Xγ~ℒ𝒩γ(0,σ), for σ=2/3,1,3/2, and various positive (left subfigures) and negative (right subfigures) γ values.Proposition 5.
The global mode point of the negative-ordered Lognormal random variableXγ~ℒ𝒩γ(μ,σ), γ∈(-∞,0), is (in limit) the threshold 0 which is an infinite (probability) density point. Moreover, the location parameter (i.e., eμ) of ℒ𝒩γ(μ,σ) is a nonsmooth (local) mode point for all Xγ<0 that corresponds to locally maximum density
(26)fXγ(eμ)=12σ((γ-1)/γ)1/γΓ((γ-1)/γ)eμ,
while eμ-σγ is a local minimum (probability) density point with corresponding locally minimum density fXγ(eμ-σγ).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 fXγ in Proposition 4, all the negative-ordered Lognormals admit also eμ as a local nonsmooth mode point and exp{μ-σγ} as a local minimum density point, with densities as in (26) and fXγ(eμ-σγ), respectively; see Figures 1(b1), 1(b2), and 1(b3).Furthermore, Proposition5 holds (in limit) for random variables X1 and X±∞ which provide Log-Uniform and Log-Laplace distributions, respectively. Indeed, for given a,b∈ℝ+*, X1~ℒ𝒰(a,b)=ℒ𝒩1(μ,σ) with μ=(1/2)log(ab) and σ=(1/2)log(b/a), we get, through (20) and (21), that ModeX1:=ModeXγ→1+=eμ-σ=a with corresponding maximum density (i.e., the maximum value of the density function),
(27)maxfX1=fX1(a)=eσ-μ2σlimγ→1+((γ-1)/γ)(γ-1)/γΓ(((γ-1)/γ)+1)=1alog(b/a).
Moreover, the nonzero minimum density (i.e., the minimum, but not zero, value of the density function) is obtained at x=b with minfX1=fX1(b)=1/2σb=1/(blog(b/a)). These results are in accordance with the Log-Uniform density function in (14).ForX±∞~ℒ𝒩±∞(logμ,1/σ)=ℒℒ(μ,σ,σ), we evaluate, through (20) and (21), that
(28)ModeX+∞={0,σ<1,μeσ=1,μ,σ>1,
with the corresponding maximum density value being infinite; that is, maxfXγ=fXγ(0)=+∞, provided σ<1, and maxfX+∞=σ/(2μ), provided σ≥1. The same result can also be derived through (26) as γ→-∞. These results are in accordance with the Log-Laplace density function in (15), although for σ=1, ModeX±∞ can be defined, through (15), for any value inside the interval (0,μ].The above discussion on behavior of the modes with respect to shape parameterγ is formed in the following propositions.Proposition 6.
Consider the positive-ordered Lognormal family of distributionsℒ𝒩γ(μ,σ) with fixed parameters μ, σ and γ≥1. When γ rises, that is, when one moves from Log-Uniform to Log-Laplace distribution inside the ℒ𝒩γ family, the mode points of ℒ𝒩γ are(i)
strictly increasing fromeμ-σ (Log-Uniform case) to eμ (Log-Laplace case) provided that σ<1 (with their corresponding maximum density values moving smoothly from (1/2σ)eσ-μ to +∞),(ii)
fixed ateμ-1 for all ℒ𝒩γ≥1(μ,σ=1) (with the corresponding maximum density values moving smoothly from (1/2)e1-μ to (1/2)e-μ),(iii)
strictly decreasing fromeμ-σ (Log-Uniform case) to threshold 0 (Log-Laplace case) provided that σ>1 (with their corresponding maximum density values moving smoothly from (1/2σ)eσ-μ to (1/2σ)e-μ).Proof.
LetXγ~ℒ𝒩γ(μ,σ), ModeXγ is a smooth monotonous function of γ∈(-∞,0)∪(1,+∞) for positive-and negative-ordered Xγ, as
(29)ddγModeXγ=-σγlog(σ)eμ-σγ.
For X±∞~ℒ𝒩±∞(μ,σ) we evaluate, through (20) and (21), that
(30)ModeX+∞={eμ,σ<1,eμ-1,σ=1,0,σ>1,
with the corresponding maximum density value being infinite; that is, maxfXγ=fXγ(0)=+∞, provided σ>1, and maxfX+∞=1/(2σeμ), provided σ≤1.
Assume thatγ≥1. Considering (27), (30) with (29) and Proposition 4, the results for the positive-ordered Lognormals hold.Proposition 7.
For the negative-ordered Lognormal family of distributionsℒ𝒩γ(μ,σ) with γ<0, when γ rises, that is, when one moves from Log-Laplace to degenerate Dirac distribution inside the ℒ𝒩γ family, the local minimum (probability) density points of ℒ𝒩γ are(i)
strictly increasing from threshold 0 (Log-Laplace case) toeμ-1(Dirac case) provided thatσ<1,(ii)
fixed ateμ-1for allℒ𝒩γ(μ,σ=1),(iii)
strictly decreasing fromeμ(Log-Laplace case) toeμ-1(Dirac case) provided thatσ>1.Proof.
Assume now thatγ<0. From (29)  we have (d/dγ)(eμ-σγ)<0 when σ>1 and (d/dγ)(eμ-σγ)>0 when σ<1. Therefore, the local minimum density point eμ-σγ (see Proposition 5) for σ>1 is decreasing from eμ-σγ∣γ→-∞=eμ to ModeX0=eμ-1 through (20). When σ=1, eμ-σγ=eμ-1 for all γ<0, while for σ<1, eμ-σγ increases from eμ-σγ∣γ→-∞=0 to ModeX0=eμ-1 through (20).It is easy to see that for the Log-Laplace caseℒℒ(μ,σ,σ), the local minimum density point eμ-σγ of Xγ<0 with σ>1 coincides (in limit) with the local nonsmooth mode point eμ of Xγ; see Figure 1(b3). Also, notice that the local minimum density point eμ-σγ, γ<0, for the Dirac case 𝒟(eμ), is the limiting point eμ-1 although the (probability) density in 𝒟(eμ) case vanishes everywhere except at the infinite pole eμ.Figure1 illustrates the probability density functions fXγ curves for scale parameters σ=2/3,1,3/2 of the positive-ordered lognormally distributed Xγ~ℒ𝒩γ≥1(0,σ) in Figures 1(a1)–1(a3), respectively, while the p.d.f. of negative-ordered lognormally distributed Xγ~ℒ𝒩γ<0(0,σ) are depicted in Figures 1(b1)–1(b3),  respectively. Moreover, the density points eμ-σγ on fXγ are also depicted (small circles over p.d.f. curves with their corresponding ticks on x-axis). According to Proposition 7, in Figures 1(a1)–1(a3), that is, for positive-ordered Xγ≥1, these density points represent the mode points on fXγ while in Figures 1(b1)–1(b3), that is, for negative-ordered Xγ<0, represent the local minimum density points on fXγ curves.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.FXγ of a γ-order Lognormal random variable Xγ~ℒ𝒩γ(μ,σ) is given by
(31)FXγ(x)=12+π2Γ((γ-1)/γ)Γ(γ/(γ-1))×Erfγ/(γ-1){(γ-1γ)(γ-1)/γlogx-μσ}(32)=1-12Γ((γ-1)/γ)Γ(γ-1γ,γ-1γ(logx-μσ)γ/(γ-1)),x∈ℝ+*.Proof.
From density functionfXγ, as in (10), we have
(33)FXγ(x)=FXγ(x;μ,σ,γ)=∫0xfXγ(t)dt=σ-1Cγ1∫0xt-1exp{-γ-1γ|logt-μσ|γ/(γ-1)}dt.
Applying the transformation w=(logt-μ)/σ, t>0, the above c.d.f. is reduced to
(34)FXγ(x)=Cγ1∫-∞(logx-μ)/σexp{-γ-1γ|w|γ/(γ-1)}dw=ΦZγ(logx-μσ),
where ΦZγ is the c.d.f. of the standardized γ-order normal distribution Zγ=(1/σ)(logXγ-μ)~𝒩γ(0,1). Moreover, ΦZγ can be expressed in terms of the generalized error function. In particular,
(35)ΦZγ(z)=Cγ1∫-∞zexp{-γ-1γ|w|γ/(γ-1)}dw=ΦZγ(0)+Cγ1∫0zexp{-γ-1γ|w|γ/(γ-1)}dw,
and as fZγ is a symmetric density function around zero, we have
(36)ΦZγ(z)=12+Cγ1∫0zexp{-γ-1γ|w|γ/(γ-1)}dw=12+Cγ1∫0zexp{-|(γ-1γ)(γ-1)/γw|γ/(γ-1)}dw,
and thus
(37)ΦZγ(z)=12+Cγ1(γγ-1)(γ-1)/γ×∫0((γ-1)/γ)(γ-1)/γzexp{-uγ/(γ-1)}du.
Substituting the normalizing factor, as in (2), and using (3), we obtain
(38)ΦZγ(z)=12+π2Γ(((γ-1)/γ)+1)Γ((2γ-1)/(γ-1))×Erfγ/(γ-1){(γ-1γ)(γ-1)/γz},z∈ℝ,
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
(39)FXγ(x)=12+sgn(logx-μ)π2Γ((γ-1)/γ)Γ(γ/(γ-1))×Erfγ/(γ-1){(γ-1γ)(γ-1)/γ|logx-μσ|},
while applying (4) into (39) it is obtained that
(40)FXγ(x)=1+sgn(logx-μ)2-sgn(logx-μ)2Γ((γ-1)/γ)×Γ(γ-1γ,γ-1γ|logx-μσ|γ/(γ-1)).As the generalized error functionErfa is defined in (4), through the upper incomplete gamma function Γ(a-1,·), series expansions can be used for a more “numerical-oriented” form of (4). Here some expansions of the c.d.f. of the generalized Lognormal distribution are presented.Corollary 9.
The c.d.f.FXγ can be expressed in the series expansion form
(41)FXγ(x)=12+((γ-1)/γ)(γ-1)/γ(2/γ)Γ((γ-1)/γ)(logx-μσ)×∑k=0∞‍(((1-γ)/γ)|(logx-μ)/σ|γ/(γ-1))kk![(k+1)γ-1],x∈ℝ+*.Proof.
Substituting the series expansion form of (6) into (39) and expressing the infinite series using the integer powers k, the series expansion as in (41) is derived.Corollary 10.
For the negative-ordered lognormally distributed random variableXγ with γ=1/(1-n)∈ℝ-, n∈ℕ, n≥2, the finite expansion is obtained as
(42)FXγ(x)=12+12sgn(logx-μ)-sgn(logx-μ)2exp{n|(logx-μ)/σ|1/n}×∑k=0n-1‍nkk!|logx-μσ|k/n.Proof.
Applying the following finite expansion form of the upper incomplete gamma function,(43)Γ(n,x)=(n-1)!e-x∑k=0n-1‍xkk,x∈ℝ,n∈ℕ*=ℕ∖0,
into (40), we readily get (42).Example 11.
For the(-1)-ordered lognormally distributed X-1 (i.e., for n=2), we have
(44)FX-1(x)=12+12sgn(logx-μ)-sgn(logx-μ)×1+2|(logx-μ)/σ|2exp{2|(logx-μ)/σ|},
while for the (-1/2)-ordered lognormally distributed X-1/2 (i.e., for n=3), we have
(45)FX-1/2(x)=12+12sgn(logx-μ)-sgn(logx-μ)×1+3|(logx-μ)/σ|3+9((logx-μ)/σ)232exp{3|(logx-μ)/σ|3}.Example 12.
For the second-ordered Lognormal random variableX2~ℒ𝒩2(μ,σ), we immediately derive, from (31), that
(46)FX2(x)=ΦX2(logx-μσ)=12+12Erf2(logx-μ2σ)=12+12erf(logx-μ2σ);
that is, the c.d.f. of the usual Lognormal is derived, as it is expected, due to ℒ𝒩2=ℒ𝒩; see Theorem 3.Example 13.
For the infinite-ordered LognormalX±∞~ℒ𝒩±∞(μ,σ), setting (γ-1)/γ=1, we obtain through (41) and the exponential series expansion that
(47)FX±∞(x)=12+12πErf1(logx-μσ)=12-12sgn(logx-μ)∑k=0∞‍1(k+1)!(-|logx-μσ|)k+1=12+12sgn(logx-μ)-12sgn(logx-μ)×exp{-|logx-μσ|},
and hence
(48)FX±∞(x)={12eμ/σx1/σ,x∈(0,eμ],1-eμ/σ2x1/σ,x∈(eμ,+∞),
which is the c.d.f. of the Log-Laplace distribution as in (15). This is expected as ℒ𝒩±∞(μ,σ)=ℒℒ(eμ,1/σ,1/σ); see Theorem 3.It is interesting to mention here that the same result can also be derived through (42), as this finite expansion can be extended for n=1, which provides (in limit) the c.d.f. of the infinite-ordered Lognormal distribution.Example 14.
Similarly, for the first-ordered random variableX1~ℒ𝒩1(μ,σ), the expansion (41) can be written as
(49)FXγ(x)=12+((γ-1)/γ)(γ-1)/γ2Γ(((γ-1)/γ)+1)(logx-μσ)×[1+(γ-1)∑k=1∞‍(((1-γ)/γ)|(logx-μ)/σ|γ/(γ-1))kk![(k+1)γ-1]],
and provided that (logx-μ)/σ≤1, we obtain
(50)FX1(x)=limγ→1+FXγ(x)=12+logx-μ2σ(1+0)=logx-μ+σ2σ,
with FX1(eμ-σ)=0 and FX1(eμ+σ)=1. Therefore,
(51)FX1(x)={0,x∈(0,eμ-σ),12σ(logx-μ+σ),x∈[eμ-σ,eμ+σ],1,x∈(eμ+σ,+∞),
coincides with the c.d.f. of the Log-Uniform distribution ℒ𝒰(a=eμ-σ,b=eμ+σ) as in (15). This is expected as ℒ𝒩1(μ,σ)=ℒ𝒰(eμ-σ,eμ+σ); see Theorem 3.Table1 provides the probability values Pγ;1=Pr{Xγ≤i}, i=1/2,1,2,…,5, for various Xγ~ℒ𝒩γ(0,1). Notice that Pγ;1=1/2 for all γ values due to the fact that 1=eμ∣μ=0=MedXγ (see Theorem 8); that is, the point 1 coincides with the γ-invariant median of the ℒ𝒩γ(0,1) family discussed previously. Moreover, the last two columns provide also the 1st and 3rd quartile points q1;γ and q3;γ of Xγ; that is, Pr{Xγ≤qk;γ}=k/4, k=1,3, for various γ values. These quartiles are evaluated using the quantile function QXγ of r.v. Xγ; that is,
(52)QXγ(P)∶=inf{x∈ℝ+*∣FXγ(x)≥P}=exp{(γ-1γ,|2P-1|)](γ-1)/γsgn(2P-1)σ×[γγ-1Γ-1(γ-1γ,|2P-1|)](γ-1)/γ},P∈(0,1),
for P=1/4,3/4, that is derived through (40). The values of the inverse upper incomplete gamma function Γ-1((γ-1)/γ,·) were numerically calculated.Table 1
Probability mass valuesPγ;i=Pr{Xγ≤i}, i=1/2,1,2,…,5, and the 1st and 3rd quartiles q1;γ, q3;γ for various generalized lognormally distributed Xγ~ℒ𝒩γ(0,1).
γ
P
γ
;
1
/
2
P
γ
;
1
P
γ
;
2
P
γ
;
3
P
γ
;
4
P
γ
;
5
q
1
;
γ
q
3
;
γ
-
50
0.2501
0.5000
0.7499
0.8326
0.8739
0.8987
0.4998
2.0008
-
10
0.2505
0.5000
0.7495
0.8297
0.8698
0.8940
0.4990
2.0038
-
5
0.2508
0.5000
0.7492
0.8264
0.8652
0.8887
0.4982
2.0071
-
2
0.2515
0.5000
0.7485
0.8187
0.8539
0.8756
0.4964
2.0145
-
1
0.2521
0.5000
0.7479
0.8097
0.8408
0.8601
0.4945
2.0223
-
1
/
2
0.2524
0.5000
0.7476
0.7989
0.8248
0.8410
0.4925
2.0303
-
1
/
10
0.2528
0.5000
0.7482
0.7757
0.7895
0.7984
0.4986
2.0426
1
0.1534
0.5000
0.8466
1.0000
1.0000
1.0000
0.6065
1.6487
3
/
2
0.2381
0.5000
0.7619
0.8848
0.9437
0.9721
0.5172
1.9334
2
0.2441
0.5000
0.7559
0.8640
0.9172
0.9462
0.5094
1.9630
3
0.2472
0.5000
0.7528
0.8505
0.8989
0.9267
0.5049
1.9804
4
0.2481
0.5000
0.7519
0.8452
0.8917
0.9188
0.5034
1.9867
5
0.2486
0.5000
0.7514
0.8425
0.8878
0.9145
0.5025
1.9899
10
0.2494
0.5000
0.7506
0.8375
0.8810
0.9068
0.5011
1.9954
50
0.2499
0.5000
0.7501
0.8341
0.8761
0.9013
0.5002
1.9992
±
∞
0.2500
0.5000
0.7500
0.8333
0.8750
0.9000
0.5000
2.0000Figure2 illustrates the c.d.f. FXγ curves, as in (39), for certain r.v. Xγ~ℒ𝒩γ(0,σ) and for scale parameters σ=2/3,1,3/2 in the 3 subfigures, respectively. Moreover, the 1st and 3rd quartile points QXγ(1/4) and QXγ(3/4) are also depicted (small circles over c.d.f. curves with their corresponding ticks on x-axis).Graphs of the c.d.f.FXγ, Xγ~ℒ𝒩γ(0,σ), for σ=2/3,1,3/2, and various γ values.
(a)
(b)
(c)Theorem 15.
The (non-log-scaled) location parametereμ is in fact the geometric mean as well as the median for all generalized lognormally distributed Xγ~ℒ𝒩γ(μ,σ). Moreover, this median is also characterized by vanishing median absolute deviation.Proof.
Considering (39) and the fact that Erfa0=0, a∈ℝ+*, it holds that MedXγ=FXγ-1(1/2)=eμ. For the geometric mean (μg)Xγ=eE[logXγ], we readily obtain (μg)Xγ=eμ as logXγ~𝒩γ(μ,σ2) with E[Xγ]=μ. A dispersion measure for the median is the so-called median absolute deviation or MAD, defined by MAD(Xγ)=Med∣Xγ-MedXγ∣. For Xγ~ℒ𝒩γ(μ,σ), we have Xγ-MedXγ=Xγ-eμ~ℒ𝒩γ(μ,σ;-eμ); that is, Xγ-eμ follows the generalized Lognormal distribution with threshold -eμ. Furthermore, |Y| is the “folded distribution” case of Y:=Xγ-eμ which is distributed through p.d.f. of the form
(53)f|Y|(x)=fY(-x)+fY(x),x∈ℝ+,
where fY is the p.d.f. of Y. For example, see [17] on the folded normal distribution. However, the density function fY is defined in (-eμ,+∞) due to threshold -eμ, while it vanishes elsewhere; that is,
(54)f|Y|(x)={fY(-x)+fY(x),0≤x≤eμ,fY(x),x>eμ.
Therefore, the c.d.f. of |Y| is given by
(55)F|Y|(x)=∫0xf|Y|(t)dt=∫0eμfY(-t)dt+∫0eμfY(t)dt+∫eμxfY(t)dt,x∈ℝ+.
Applying the transformation w=(1/σ)[log(eμ-t)-μ], t<eμ, into the first integral of (55) and z=(1/σ)[log(t+eμ)-μ], t>-eμ, into the other two integrals, we obtain
(56)F|Y|(x)=Cγ1∫-∞0exp{-γ-1γ|w|γ/(γ-1)}dw+Cγ1×∫0(1/σ)log2exp{-γ-1γ|z|γ/(γ-1)}dz+Cγ1∫(1/σ)log2g(x)exp{-γ-1γ|z|γ/(γ-1)}dz,g(x):=log(x+eμ)-μσ,x∈ℝ+,
and hence
(57)F|Y|(x)=ΦZ(0)+[ΦZ(log2σ)-ΦZ(0)]+[ΦZ(g(x))-ΦZ(log2σ)]=ΦZ(log(x+eμ)-μσ),
with ΦZ being the c.d.f. of the standardized r.v. Z~𝒩γ(0,1). From (38) and the fact that Erfa0=0, a∈ℝ+*, it is clear that (57) implies MADXγ=Med|Xγ-eμ|=F|Xγ-eμ|-1(1/2)=0, for every Xγ~ℒ𝒩γ(μ,σ), and the theorem has been proved.


## 3. Moments of theγ-Order Lognormal Distribution

For the evaluation of the moments of the generalized Lognormal distribution, the following holds.Proposition 16.
Thetth raw moment μ~X(t) of a generalized lognormally distributed random variable X~ℒ𝒩γ(μ,σ) is given by
(58)μ~X(t)=etμΓ((γ-1)/γ)∑n=0∞‍(tσ)2n(2n)!(γγ-1)2n((γ-1)/γ)×Γ((2n+1)γ-1γ)
and coincides with the moment generating function of the γ-order normally distributed logX; that is, MlogX(t)=μ~X(t).Proof.
From the definition of thetth raw moment μ~X(t), we have
(59)μ~X(t)=E[Xt]=∫ℝ+xtfX(x)dx=1σCγ1∫ℝ+xt-1exp{-γ-1γ|logx-μσ|γ/(γ-1)}dx,
and applying the transformation z=((γ-1)/γ)(γ-1)/γ(1/σ)(logx-μ), x>0, we get
(60)μ~X(t)=Cγ1(γγ-1)(γ-1)/γ∫ℝexp{tμ+n(γγ-1)(γ-1)/γσz}×exp{-|z|γ/(γ-1)}dz.
Through the exponential series expansion
(61)exp{t(γγ-1)(γ-1)/γσz}=∑n=0∞‍(tσ)nn!(γγ-1)n((γ-1)/γ)zn,
it is obtained that
(62)μ~X(t)=2Cγ1(γγ-1)(γ-1)/γetμ∑n=0∞‍(tσ)2n(2n)!(γγ-1)2n((γ-1)/γ)×∫ℝ+z2nexp{-zγ/(γ-1)}dz.
Finally, substituting the normalizing factor Cγ1 as in (2) into (62) and utilizing the known integral [16],
(63)∫ℝ+xme-bxndx=Γ((m+1)/n)nb(m+1)/n,n,m,b∈ℝ+*,
we obtain (58).
Moreover, forY:=logX~𝒩γ(μ,σ2) we have MY(t)=E[etY]=E[Xt]=μ~X(t), and the proposition has been proved.Example 17.
For the second-ordered lognormally distributedX~ℒ𝒩2(μ,σ), (58) implies
(64)μ~X(t)=etμπ∑n=0∞‍(2t2σ2)n(2n)!Γ(n+12),t∈ℝ+,
and through the gamma function identity
(65)Γ(n+12)=(2n)!22nn!π,n∈ℕ,
we have
(66)μ~X(t)=etμ∑n=0∞‍(tσ)2n2nn!=etμ∑n=0∞‍1n!(12t2σ2)n=etμ+(1/2)(tσ)2,t∈ℝ+,
which is the tth raw moment of the usual lognormally distributed X~ℒ𝒩(μ,σ), with mean μX:=μ~X(1)=E[X]=exp{μ+(1/2)σ2}. This is true as MlogX(t)=μ~X(t)=exp{tμ+(1/2)(tσ)2} is the known moment-generating function of the normally distributed logX~𝒩(μ,σ2).Theorem 18.
Thekth central moment (about the mean) μX(t) of a generalized lognormally distributed random variable X~ℒ𝒩γ(μ,σ) is given by
(67)μX(k)=ekμΓ((γ-1)/γ)∑n=0k‍(kn)(-μXeμ)nSk-n,k∈ℕ,
where
(68)Sk=∑m=0∞‍(kσ)2m(2m)!(γγ-1)2m((γ-1)/γ)Γ((2m+1)γ-1γ),k∈ℕ.Proof.
From the definition of thekth central moment μX(k)  we have
(69)μX(k):=E[(X-μX)k]=∫ℝ+(x-μX)kfX(x;μ,σ,γ)dx,
while using the binomial identity we get
(70)μX(k)=∑n=0k‍(kn)(-μX)n∫ℝ+xk-nfX(x)dx=∑n=0k‍(kn)(-μX)nμ~X(k-n).
Applying Proposition 16, (70) implies that
(71)μX(k)=ekμΓ((γ-1)/γ)∑n=0k‍(kn)(-μXeμ)n∑m=0∞‍[(k-n)σ]2m(2m)!×(γγ-1)2m((γ-1)/γ)Γ((2m+1)((γ-1)/γ)),
while taking the summation index n until k-1, we finally obtain (67), and the theorem has been proved.Example 19.
Recall Example17. Substituting (66) and the mean μX=eμ+(1/2)σ2 into (70), the second-ordered lognormally distributed X~ℒ𝒩2(μ,σ) provides
(72)μX(k)=∑n=0k‍(kn)(-1)nekμ+(1/2)[n+(k-n)2]σ2,k∈ℕ,
while
(73)σX2:=Var[X]=μX(2)=e2μ+σ2(eσ2-1),
which are the kth central moment and the variance, respectively, of the usual lognormally distributed X~ℒ𝒩(μ,σ). The same result can be derived directly through (67) for γ=2 and the use of the known gamma function identity, as in (65).Theorem 20.
The meanμX:=E[X], variance σX2:=Var[X], coefficient of variation CVX, skewness λX and kurtosis κX of the generalized lognormally distributed X~ℒ𝒩γ(μ,σ) are, respectively, given by
(74)μX=eμΓ((γ-1)/γ)S1,(75)σX2=-μX2+e2μΓ((γ-1)/γ)S2,(76)CVX2=Γ(γ-1γ)S2S12-1,(77)λX=-CVX-3-CVX-1+e3μσX3Γ((γ-1)/γ)S3,(78)κX=-CVX-4-6CVX-2-4λXCVX+e4μσX4Γ((γ-1)/γ)S4,
where the sums Si,i=1,…,4, are given by (68).Proof.
From Proposition16 we easily obtain (74), as μX:=μ~X(1). From Theorem 18 we have
(79)σX2:=μX(2)=μX2+[Γ(γ-1γ)]-1(e2μS2-2eμμXS1).
Hence, substituting S1 from (74), (75) holds. Moreover, the squared coefficient of variation is readily obtained via (75) and (74). By definition, skewness λX is the standardized third (central) moment; that is, λX:=Skew[X]=μX(3)/σX3. Theorem 18 provides that
(80)λX=-CVX-3+[σX3Γ(γ-1γ)]-1×(e3μS3-3e2μμXS2+3eμμXS1).
Substituting S1 and S2 from (74) and (75), we obtain (77). Finally, kurtosis κX is (by definition) the standardized fourth (central) moment; that is, κX:=Kurt[X]=μX(4)/σX4,  which provides, through Theorem 18, that
(81)κX=CVX-4+[σX4Γ(γ-1γ)]-1×(e4μS4-4e3μμXS3+6e2μμX2S2-4eμμX3S1).
Substituting Si, i=1,2,3, from (74), (75), and (77), we obtain (78).Example 21.
For the second-ordered lognormally distributedX~ℒ𝒩2(μ,σ), utilizing (65) into (68) we get Sn=πe(n2σ2)/2, n∈ℕ*. Applying this to Theorem 20 we derive (after some algebra)
(82)μX=eμ+(1/2)σ2,σX2=e2μ+σ2(eσ2-1),CVX=eσ2-1,λX=(eσ2+2)eσ2-1,κX=e4σ2+2e3σ2+3e2σ2-3,
which are the mean, variance, coefficient of variation, skewness, and kurtosis, respectively, of usual lognormally distributed X~ℒ𝒩(μ,σ).For the usual lognormally distributed random variableX~ℒ𝒩, it is known that ModeX<MedX<μX. The following corollary examines this inequality for the ℒ𝒩γ family of distributions.Corollary 22.
For theγ-ordered lognormally distributed Xγ~ℒ𝒩γ(μ,σ), it is true that ModeXγ≤MedXγ=(μg)Xγ≤μXγ. The first equality holds for the Log-Laplace distributed X+∞ with σ<1 as well as for all the negative-ordered Xγ<0 where ModeXγ is considered to be the local (nonsmooth) mode point of Xγ. The second equality holds for the degenerate Dirac case of X0.Proof.
From (74) and Theorem 15 we have
(83)MedXγ=(μg)Xγ=eμ<μXγ,
for every Xγ~ℒ𝒩γ(μ,σ). The above inequality becomes equality for the limiting Dirac case of X0. For the relation between the mode and the median of Xγ, the following cases are considered. (i)
The positive-ordered Lognormal caseγ>1: from (20)  we have
(84)ModeXγ=eμ-σγ<eμ=MedXγ.
For the Log-Laplace case of X+∞, it holds
(85)ModeXγ=eμ=MedXγ,
provided that σ<1, while for σ≥1 we have
(86)ModeXγ=0<eμ=MedXγ.
For σ=1, the inequality (84) clearly holds.(ii)
The negative-ordered Lognormal caseγ<0: from  Proposition 4  the inequality as in (86) holds. Moreover, if ModeXγ is considered as the nonsmooth local mode point of the negative-ordered Xγ then the equality as in (85) holds.
From the above cases and (83), the corollary holds true.Corollary 23.
The raw and central moments of a Log-Uniformly distributed random variableX~ℒ𝒰(a,b), 0<a<b, are given by
(87)μ~X(t)=bt-attlog(b/a),t∈ℝ,(88)μX(k)=(a-b)klogk(b/a)+1log(b/a)×∑n=0k-1‍(kn)(a-b)n(bk-n-ak-n)(k-n)logn(b/a),k∈ℕ,
respectively, while the mean, variance, coefficient of variation, skewness, and kurtosis of X are given, respectively, by
(89)μX=b-alog(b/a),(90)σX2=(b-a)2log2(b/a)+(b-a)(b+a)2log(b/a),(91)CVX=1+b+a2(b-a)logba,(92)λX=1σX3[b3-a33log(b/a)-3(b-a)2(b+a)2log2(b/a)+2(b-a)3log3(b/a)],(93)κX=1σX4[b4-a44log(b/a)-4(b-a)(b3-a3)3log2(b/a)+3(b-a)3(b+a)log3(b/a)-3(b-a)4log4(b/a)].Proof.
Recall Proposition16 with Xγ~ℒ𝒩γ(μ,σ). Through the gamma function additive identity (58) can be written as
(94)μ~Xγ(t)=etμtσΓ((γ-1)/γ+1)∑m=0∞‍(tσ)2m+1(2m+1)!(γγ-1)2m((γ-1)/γ)×Γ((2m+1)γ-1γ+1).
Thus, letting X:=X1~ℒ𝒩1(μ,σ)=ℒ𝒰(a,b) with μ=(1/2)log(ab) and σ=(1/2)log(b/a), it holds (recall the exponential odd series expansion) that
(95)μ~X(t)=limγ→1+μXγ(t)=etμtσ∑m=0∞‍(tσ)2m+1(2m+1)!=et(μ+σ)-et(μ-σ)2tσ,t∈ℝ,
and hence (87) holds. Moreover, μX:=μX(1)=E[X]=(1/2σ)(eμ+σ-eμ-σ), and therefore (89) holds.
Working similarly, (67) implies
(96)μX(k)=limγ→1+μXγ(k)=ekμ∑n=0k‍(kn)(-μXeμ)n∑m=0∞‍[(k-n)σ]2m(2m+1)!,k∈ℕ.
Using the exponential odd series expansion, the above expansion becomes
(97)μX(k)=ekμ2σ∑n=0k‍(kn)(-μXeμ)ne(k-n)σ-e-(k-n)σ(k-n),k∈ℕ,
and, through (89), we obtain (88). Moreover, for k=2, σX2:=Var[X]=μX(2)-μX2 implies (90), and hence (91) also holds. For k=3 and k=4, through μX(3) and μX(4), we obtain (92) and (93), respectively.Corollary 24.
The raw and central moments of a Log-Laplace distributed random variableX~ℒℒ(μ,σ,σ) are given by
(98)μ~X(t)=μtσ2σ2-t2>μt,σ>t,t∈ℝ,(99)μX(k)=μk∑n=0k‍(kn)σ2(n+1)(1-σ2)n[σ2-(k-n)2],σ>k,k∈ℕ.
The mean, variance, coefficient of variation, skewness, and kurtosis of X are given, respectively, by
(100)μX=μσ2σ2-1>μ,σ>1,(101)σX2=μ2σ2(2σ2+1)(σ2-4)(σ2-1)2,σ>2,(102)CVX=1σ2σ2+1σ2-4,σ>2,(103)λX=2(15σ4+7σ2+2)σ(σ2-9)σ2-4(2σ2+1)3,σ>3,(104)κX=3(8σ8+212σ6+95σ4+33σ2+12)(σ2-4)(σ2-16)(σ2-9)(2σ2+1)2,σ>4.Proof.
LetXγ~ℒℒγ(μ,σ,σ)=ℒ𝒩γ(logμ,1/σ,1/σ). For γ=±∞, that is, γ/(γ-1)=1, the raw moments as in (58) provide
(105)μ~X(t)=μ~X±∞(t)=μt∑k=0∞‍(tσ)2k,t∈ℝ,
as X=X±∞, while through the even geometric series expansion, it is
(106)μ~X±∞(t)=12μt[∑k=0∞‍(tσ)k+∑k=0∞‍(-tσ)k]=12μt(σσ-t+σσ+t),
provided that σ>t, and hence (98) holds. Moreover, μX:=μ~X(1)=E[X], and hence (100) holds.
Working similarly, (67) implies
(107)μX(k)=μkσ2∑n=0k‍(kn)(-μX/μ)nσ2-(k-n)2,k∈ℕ,
provided σ>k, and hence, through (100), the central moments (99) are obtained.
Moreover, fork=2 and due to σX2:=Var[X]=μ~X(2)-μX2, (101) holds true, while for k=3 and k=4 we derive, through μX(3)and μX(4), (103) and (104), respectively.Example 25.
For a uniformly distributed r.v.U~𝒰(a,b)=𝒩1(μ,σ) with a=μ-σ and b=μ+σ, it holds that LU:=eU~ℒ𝒰(eμ-σ,eμ+σ) due to Theorem 3, and therefore LU is a Log-Uniform distributed r.v. as LU~ℒ𝒰(ea,eb). Applying (87), the known moment-generating function of the uniformly distributed U~𝒰(a,b) is derived; that is, MU(t):=E[etU]=μ~LU(t)=(etb-eta)(1/t(b-a)).
Similarly, for a Laplace distributed r.v.L~ℒ(μ,σ)=𝒩±∞(μ,σ), it holds that LL:=eL~ℒℒ(eμ,1/σ,1/σ) due to Theorem 3, and therefore LL is a Log-Laplace distributed random variable. Applying (98), we derive the known moment-generating function of the Laplace distributed L~ℒ(μ,σ); that is, ML(t):=E[etL]=μ~LL(t)=etσ(1-t2σ2)-1.


## 4. Conclusion

The family of theγ-order Lognormal distributions was introduced, which under certain values of γ includes the Log-Uniform, Lognormal, and Log-Laplace distributions as well as the degenerate Dirac distribution. The shape of these distributions for positive and negative shape parameters γ as well as the cumulative distribution functions, was extensively discussed and evaluated through corresponding tables and figures. Moreover, a thorough study of moments was carried out, in which nonclosed forms as well as approximations were obtained and investigated in various examples. This generalized family of distributions derived through the family of the γ-order normal distribution is based on a strong theoretical background as the logarithmic Sobolev inequalities provide. Further examinations and calculations can be produced while an application to real data is upcoming.



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*Source: 432642-2013-07-29.xml*