On the Generalized Lognormal Distribution

Abstract

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 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.

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 𝛾-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 𝒩^𝑝_𝛾(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 function 𝑓_𝑋 of a 𝑝-variate, 𝛾-order, normally distributed random variable 𝑌∼𝒩^𝑝_𝛾(𝜇,Σ), with location vector 𝜇∈ℝ^𝑝, positive definite scale matrix Σ∈ℝ^𝑝×𝑝, and shape parameter 𝛾∈ℝ⧵[0,1], is given by [11]. 𝑓_𝑌𝑦=𝑓_𝑌𝑦;𝜇,Σ,𝛾=𝐶^𝑝_𝛾|detΣ|^−1/2exp−𝛾−1𝛾𝑄_𝜃𝑦^{𝛾/2(𝛾−1)},𝑦∈ℝ^𝑝, (1) where 𝑄_𝜃 is the quadratic form 𝑄_𝜃(𝑦)=(𝑦−𝜇)^TΣ⁻¹(𝑦−𝜇), 𝜃=(𝜇,Σ), while 𝐶^𝑝_𝛾 being the normalizing factor 𝐶^𝑝_𝛾=𝜋^−𝑝/2Γ𝑝/2+1Γ𝑝𝛾−1/𝛾𝛾−1𝛾^{𝑝((𝛾−1)/𝛾)−1}. (2)

From (1), notice that the second-ordered normal is the known multivariate normal distribution; that is, 𝒩^𝑝₂(𝜇,Σ)=𝒩^𝑝(𝜇,Σ); see also [13, 15].

In Section 2, a generalized form of the Lognormal distribution is introduced, which is derived from the univariate family of 𝒩_𝛾(𝜇,𝜎²)=𝒩¹_𝛾(𝜇,𝜎²) distributions, denoted by ℒ𝒩_𝛾(𝜇,𝜎), and includes the Log-Laplace distribution as well as the Log-Uniform distribution. The shape of the ℒ𝒩_𝛾(𝜇,𝜎) members is extensively discussed while it is connected to the tailing behavior of ℒ𝒩_𝛾 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_𝛾(𝜇,𝜎); see Section 2. The generalized error function denoted by Erf_𝑎 and the generalized complementary error function Erfc_𝑎=1−Erf_𝑎, 𝑎≥0 [16], are defined, respectively, as Erf_𝑎(𝑥)Γ()∶=𝑎+1√𝜋^𝑥₀𝑒^−𝑡𝑎𝑑𝑡,𝑥∈ℝ. (3) The generalized error function can be expressed (changing to 𝑡^𝑎 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 Erf_𝑎(𝑥)=Γ(𝑎)√𝜋𝛾1𝑎,𝑥^𝑎=Γ(𝑎)√𝜋Γ1𝑎1−Γ𝑎,𝑥^𝑎,𝑥∈ℝ; (4) see [16]. Moreover, adopting the series expansion form of the lower incomplete gamma function, 𝛾(𝑎,𝑥)∶=^𝑥₀𝑡^𝑎−1𝑒^−𝑡𝑑𝑡=^∞_𝑘=0(−1)^𝑘()𝑥𝑘!𝑎+𝑘^𝑎+𝑘,𝑥,𝑎∈ℝ₊, (5) a series expansion form of the generalized error function is extracted: Erf_𝑎(𝑥)=Γ(𝑎+1)√𝜋^∞_𝑘=0(−1)^𝑘()𝑥𝑘!𝑘𝑎+1^𝑘𝑎+1,𝑥,𝑎∈ℝ₊. (6) 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 (𝑒√𝜋)⁻¹. Applying 𝑎=2, the known incomplete gamma function identities such as 𝛾(1/2,𝑥)=√𝜋erf√𝑥 and Γ(1/2,𝑥)=√𝜋(1−erf√𝑥)=√𝜋erfc√𝑥,𝑥≥0 are obtained. Moreover, Erf_𝑎0=0 for all 𝑎∈ℝ₊ and lim_𝑥→±∞Erf_𝑎1𝑥=±√𝜋Γ(𝑎)Γ1𝑎,𝑎∈ℝ₊, (7) as 𝛾(𝑎,𝑥)→Γ(𝑎) when 𝑥→+∞.

2. The 𝛾-Order Lognormal Distribution

The generalized univariate Lognormal distribution is defined, through the univariate generalized 𝛾-order normal distribution, as follows.

Definition 1. When the logarithm of a random variable 𝑋 follows the univariate 𝛾-order normal distribution, that is, log𝑋∼𝒩_𝛾(𝜇,𝜎²), then 𝑋 is said to follow the generalized Lognormal distribution, denoted by ℒ𝒩_𝛾(𝜇,𝜎); that is, 𝑋∼ℒ𝒩_𝛾(𝜇,𝜎).

The ℒ𝒩_𝛾(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. 𝑒^𝜇 when 𝜇 is assumed log-scaled) is referred to as the location parameter of ℒ𝒩_𝛾(𝜇,𝜎). Hence, if 𝑋∼ℒ𝒩_𝛾(𝜇,𝜎), then log𝑋 is a 𝛾-order normally distributed variable; that is, log𝑋∼𝒩_𝛾(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 =𝛾Varlog𝑋𝛾−1^{2((𝛾−1)/𝛾)}Γ3𝛾−1/𝛾Γ𝜎𝛾−1/𝛾²,=ΓΓ5Kurtlog𝑋𝛾−1/𝛾𝛾−1/𝛾Γ²3;𝛾−1/𝛾 (8)(9) see [15] for details on 𝒩_𝛾.

Let 𝑌∶=log𝑋∼𝒩_𝛾(𝜇,𝜎²) with density function as in (1) and 𝑋=𝑔(𝑌)=𝑒^𝑌. Then, the density function 𝑓_𝑋 of 𝑋∼ℒ𝒩_𝛾(𝜇,𝜎) can be written, through (1), as 𝑓_𝑋(𝑥)=𝑓_𝑋𝑥;𝜇,𝜎,𝛾=𝑓_𝑌𝑔⁻¹(𝑥)𝑑𝑔𝑑𝑥⁻¹(𝑥)=𝑓_𝑌1log𝑥𝑥=exp−𝛾−1/𝛾log𝑥−𝜇/𝜎^{𝛾/(𝛾−1)}2𝜎(𝛾−1)/𝛾^1/𝛾Γ𝑥.𝛾−1/𝛾 (10) The probability density function 𝑓_𝑋, as in (10), is defined in ℝ^∗₊=ℝ₊⧵0; that is, ℒ𝒩_𝛾(𝜇,𝜎) has zero threshold. Therefore, the following definition extends Definition 1.

Definition 2. When the logarithm of a random variable 𝑋+𝜗 follows the univariate 𝛾-order normal distribution, that is, log(𝑋+𝜗)∼𝒩_𝛾(𝜇,𝜎²), then 𝑋 is said to follow the generalized Lognormal distribution with threshold 𝜗∈ℝ; that is, 𝑋∼ℒ𝒩_𝛾(𝜇,𝜎;𝜗).

It is clear that when 𝑋∼ℒ𝒩_𝛾(𝜇,𝜎;𝜗,), log(𝑋−𝜗) is a 𝛾-order normally distributed variable, that is, log(𝑋−𝜗)∼𝒩_𝛾(𝜇,𝜎²), and thus, 𝜇 is the mean of (𝑋−𝜗)’s natural logarithm while Var[log𝑋] is the same as in (8).

Let 𝑌=log(𝑋+𝜗)∼𝒩_𝛾(𝜇,𝜎²). The density function of 𝑋=𝑒^𝑌−𝜗∼ℒ𝒩_𝛾(𝜇,𝜎;𝜗) is given by 𝑓_𝑋(𝑥)=𝑓_𝑋(𝑥−𝜗), 𝑥>0.

Let 𝑧=(log(𝑥−𝜗)−𝜇)/𝜎. Then, the limiting threshold density value of 𝑓_𝑋(𝑥) with 𝑥→𝜗⁺ implies that lim_𝑥→𝜗+𝑓_𝑋(𝑥)=𝜎⁻¹𝐶¹_𝛾lim_{𝑧→−∞}𝜇exp−𝑧𝜎+𝑧−𝛾−1𝛾|𝑧|^1/(𝛾−1)=𝜎⁻¹𝐶¹_𝛾𝑒^{(sgn𝛾)(−∞)}, (11) and therefore lim_𝑥→𝜗+𝑓_𝑋(𝑥)=(),();0,𝛾∈1,+∞+∞,𝛾∈−∞,0 (12) that is, the 𝑓_𝑋’s defining domain, for the positive-ordered Lognormal random variable 𝑋, can be extended to include threshold point 𝜗 by letting 𝑓_𝑋(𝜗)=0.

The generalized Lognormal family of distributions ℒ𝒩_𝛾 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_𝛾(𝜇,𝜎), for order values of 𝛾=0,1,2,±∞, is reduced to ℒ𝒩_𝛾=𝒟𝑒𝜇,𝜎^𝜇𝑒,𝛾=0,ℒ𝒰^𝜇−𝜎,𝑒^𝜇+𝜎𝑒,𝛾=1,ℒ𝒩𝜇,𝜎,𝛾=2,ℒℒ^𝜇,1𝜎,1𝜎,𝛾=±∞. (13)

Proof. From definition (1) of 𝒩_𝛾 the order 𝛾 value is a real number outside the closed interval [0,1]. Let 𝑋_𝛾∼ℒ𝒩_𝛾(𝜇,𝜎) with density function 𝑓_𝑋𝛾 as in (10). We consider the following cases.(i) The limiting case 𝛾=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)=lim_𝛾→1⁺ℒ𝒩_𝛾(𝜇,𝜎) with 𝑓_𝑋1(𝑥)∶=lim_𝛾→1+𝑓_𝑋𝛾(𝑥)=1𝑒2𝜎𝑥,𝑥∈^𝜇−𝜎,𝑒^𝜇+𝜎,0,𝑥∈−∞,𝑒^𝜇−𝜎∪𝑒^𝜇+𝜎,+∞, (14) which is the density function of the Log-Uniform distribution ℒ𝒰(𝑎,𝑏), 0<𝑎<𝑏, with 𝑎=𝑒^𝜇−𝜎 and 𝑏=𝑒^𝜇+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, 𝑓_𝑋1(0)=lim_𝑥→0⁺𝑓_𝑋1(𝑥)=0. For the purposes of statistical application the Log-Uniform moments are not the same as the model parameters; that is, although 𝜇_𝑋=𝜇, 𝜎_𝑋=𝜎/√3.(ii) The “normal” case 𝛾=2: it is clear that ℒ𝒩₂(𝜇,𝜎)=ℒ𝒩(𝜇,𝜎), as 𝑓_𝑋2 coincides with the Lognormal density function, and therefore the second-ordered Lognormal distribution is in fact the usual Lognormal distribution.(iii) The limiting case 𝛾=±∞: we have ℒ𝒩_±∞(𝜇,𝜎)∶=lim_𝛾→±∞ℒ𝒩_𝛾(𝜇,𝜎) with 𝑓_𝑋±∞(𝑥)∶=lim_𝛾→±∞𝑓_𝑋𝛾(𝑥)=1−2𝜎𝑥explog𝑥−𝜇𝜎=𝑒^−𝜇/𝜎𝑥2𝜎^{(1−𝜎)/𝜎},𝑥∈0,𝑒^𝜇,𝑒^𝜇/𝜎𝑥2𝜎^{−(𝜎+1)/𝜎},𝑥>𝑒^𝜇, (15) which coincides with the density function of the known Log-Laplace distribution (symmetric log-exponential distribution) ℒℒ(𝜇^′,𝛼,𝛽) with 𝜇^′=𝑒^𝜇 and 𝛼=𝛽=1/𝜎; see [8]. Therefore, the infinite-ordered log-normal distribution is in fact the Log-Laplace distribution, with threshold density 𝑓_𝑋±∞(0)=lim_𝑥→0+𝑓_𝑋±∞(𝑥)=0,𝜎<1,1,𝜎=1,+∞,𝜎>1. (16) For the purposes of statistical application, the Log-Laplace moments are not the same as the model parameters; that is, although 𝜇_𝑋=𝜇, 𝜎_𝑋=√2𝜎.(iv) The limiting case 𝛾=0: we have lim_𝛾→0−𝑓_𝑋𝛾(𝑥)=lim_{𝑘∶=(𝛾−1)/𝛾→∞}𝑓_𝑋𝛾(𝑥),𝑥∈ℝ^∗₊, (17) where [𝑎] is the integer value of 𝑎∈ℝ. For the value 𝑥=𝑒^𝜇, the p.d.f. as in (10) implies lim_𝛾→0−𝑓_𝑋𝛾𝑒^𝜇=12𝜎𝑒^𝜇lim_𝑘→∞𝑘^𝑘𝑘!⋅𝑒⁰=+∞, (18) through Stirling’s asymptotic formula 𝑘!≈√2𝜋𝑘(𝑘/𝑒)^𝑘 as 𝑘→∞. Assuming now 𝑥≠𝑒^𝜇, (10), through (18), implies 𝑓_𝑋0(𝑥)∶=lim_𝛾→0−𝑓_𝑋𝛾(𝑥)=log𝑥−𝜇2√2𝜋𝜎²𝑥1⋅0⋅𝑒=0; (19) that is, ℒ𝒩₀(𝜇,𝜎)∶=lim_𝛾→0⁻ℒ𝒩_𝛾(𝜇,𝜎)=𝒟(𝑒^𝜇) as 𝑓_𝑋0 coincides with the Dirac density function, with the (non-log-scaled) location parameter 𝑒^𝜇 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 ℒ𝒩_𝛾→0⁻ (with vanishing threshold density 𝑓_𝑋0(0)=lim_𝑥→0⁺𝑓_𝑋0(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 Theorem 3, (12), and (15), the domain of the density functions 𝑓_𝑋𝛾(𝑥), 𝑥>0, can also be extended to include the threshold point 𝑥=0 by setting 𝑓_𝑋𝛾(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 𝑓_𝑋+∞(0)∶=(1/2)𝑒^−𝜇.

From the fact that 𝒩₀(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(𝑒^𝜇).

Proposition 4. The mode of the positive-ordered Lognormal random variable 𝑋_𝛾∼ℒ𝒩_𝛾(𝜇,𝜎), 𝑋<𝑒^𝜇𝛾∈(1,+∞), is given by 𝑀𝑜𝑑𝑒𝑋_𝛾=𝑒^{𝜇−𝜎𝛾}, (20) with corresponding maximum density value, max𝑓_𝑋𝛾=𝑓_𝑋𝛾𝑀𝑜𝑑𝑒𝑋_𝛾=exp𝜎^𝛾/𝛾−𝜇2𝜎(𝛾−1)/𝛾^1/𝛾Γ.𝛾−1/𝛾 (21)

Proof. Recall the density function of 𝑋_𝛾∼ℒ𝒩_𝛾(𝜇,𝜎) as in (10), and let 𝑚=Mode𝑋_𝛾>0. Then it holds that (𝑑/𝑑𝑥)𝑓_𝑋𝛾(𝑚;𝜇,𝜎,𝛾)=0; that is, 1𝜎log𝑚−𝜇^1/(𝛾−1)𝑑𝑑𝑥log𝑥−𝜇_𝑥=𝑚𝜎=−𝑚. (22) From (𝑑/𝑑𝑥)|𝑥|=sgn𝑥, 𝑥∈ℝ, we have 𝑚=𝑒^{𝜇−𝜎𝛾}, (23) provided that 𝑥<𝑒^𝜇. Otherwise, (23) holds trivially, as (22) implies 𝜎=0; that is, 𝑚=𝑒^𝜇. Moreover, (𝑑/𝑑𝑥)𝑓_𝑋𝛾(𝑥)>0 when 1+sgnlog𝑥−𝜇𝜎^{𝛾/(1−𝛾)}log𝑥−𝜇^1/(𝛾−1)>0,𝑥>0, (24) and thus, 𝑓_𝑋𝛾 is a strictly ascending density function on (0,𝑒^{𝜇−𝜎𝛾}) when 𝛾>1 and also on (𝑒^{𝜇−𝜎𝛾},𝑒^𝜇) when 𝛾<0. Similarly, with (𝑑/𝑑𝑥)𝑓_𝑋𝛾(𝑥)<0, 𝑓_𝑋 is a strictly descending density function on (𝑒^{𝜇−𝜎𝛾},+∞) when 𝛾>1 and also on (0,𝑒^{𝜇−𝜎𝛾})∪(𝑒^𝜇,+∞) when 𝛾<0. Specifically, for 𝛾<0, the point 𝑒^𝜇 is a nonsmooth point of 𝑓_𝑋, as lim_{𝑥→𝑒𝜇}𝑑𝑓𝑑𝑥^𝑋𝛾(𝑥)𝐶=−^1𝛾(𝜎𝑒^𝜇)²𝜎+lim_{𝑥→𝑒𝜇}×𝜎sgnlog𝑥−𝜇log𝑥−𝜇^1/(1−𝛾)=+∞. (25) 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 functions 𝑓𝑋𝛾, 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 variable 𝑋_𝛾∼ℒ𝒩_𝛾(𝜇,𝜎), 𝛾∈(−∞,0), is (in limit) the threshold 0 which is an infinite (probability) density point. Moreover, the location parameter (i.e., 𝑒^𝜇) of ℒ𝒩_𝛾(𝜇,𝜎) is a nonsmooth (local) mode point for all 𝑋_𝛾<0 that corresponds to locally maximum density 𝑓_𝑋𝛾𝑒^𝜇=12𝜎(𝛾−1)/𝛾^1/𝛾Γ𝛾−1/𝛾𝑒^𝜇, (26) while 𝑒^{𝜇−𝜎𝛾} is a local minimum (probability) density point with corresponding locally minimum density 𝑓_𝑋𝛾(𝑒^{𝜇−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 𝑓_𝑋𝛾 in Proposition 4, all the negative-ordered Lognormals admit also 𝑒^𝜇 as a local nonsmooth mode point and exp{𝜇−𝜎^𝛾} as a local minimum density point, with densities as in (26) and 𝑓_𝑋𝛾(𝑒^{𝜇−𝜎𝛾}), respectively; see Figures 1(b1), 1(b2), and 1(b3).

Furthermore, Proposition 5 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∶=Mode𝑋_𝛾→1⁺=𝑒^𝜇−𝜎=𝑎 with corresponding maximum density (i.e., the maximum value of the density function), max𝑓_𝑋1=𝑓_𝑋1(𝑎)=𝑒^𝜎−𝜇2𝜎lim_𝛾→1+𝛾−1/𝛾^{(𝛾−1)/𝛾}Γ=1𝛾−1/𝛾+1().𝑎log𝑏/𝑎 (27) Moreover, the nonzero minimum density (i.e., the minimum, but not zero, value of the density function) is obtained at 𝑥=𝑏 with min𝑓_𝑋1=𝑓_𝑋1(𝑏)=1/2𝜎𝑏=1/(𝑏log(𝑏/𝑎)). These results are in accordance with the Log-Uniform density function in (14).

For 𝑋_±∞∼ℒ𝒩_±∞(log𝜇,1/𝜎)=ℒℒ(𝜇,𝜎,𝜎), we evaluate, through (20) and (21), that Mode𝑋_+∞=𝜇0,𝜎<1,𝑒𝜎=1,𝜇,𝜎>1, (28) with the corresponding maximum density value being infinite; that is, max𝑓_𝑋𝛾=𝑓_𝑋𝛾(0)=+∞, provided 𝜎<1, and max𝑓_𝑋+∞=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 parameter 𝛾 is formed in the following propositions.

Proposition 6. Consider the positive-ordered Lognormal family of distributions ℒ𝒩_𝛾(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 ℒ𝒩_𝛾 family, the mode points of ℒ𝒩_𝛾 are(i) strictly increasing from 𝑒^𝜇−𝜎 (Log-Uniform case) to 𝑒^𝜇 (Log-Laplace case) provided that 𝜎<1 (with their corresponding maximum density values moving smoothly from (1/2𝜎)𝑒^𝜎−𝜇 to +∞), (ii) fixed at 𝑒^𝜇−1 for all ℒ𝒩_𝛾≥1(𝜇,𝜎=1) (with the corresponding maximum density values moving smoothly from (1/2)𝑒^1−𝜇 to (1/2)𝑒^−𝜇),  (iii)strictly decreasing from 𝑒^𝜇−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)𝑒^𝜎−𝜇 to (1/2𝜎)𝑒^−𝜇).

Proof. Let 𝑋_𝛾∼ℒ𝒩_𝛾(𝜇,𝜎), Mode𝑋_𝛾 is a smooth monotonous function of 𝛾∈(−∞,0)∪(1,+∞) for positive-and negative-ordered 𝑋_𝛾, as 𝑑𝑑𝛾Mode𝑋_𝛾=−𝜎^𝛾log(𝜎)𝑒^{𝜇−𝜎𝛾}. (29) For 𝑋_±∞∼ℒ𝒩_±∞(𝜇,𝜎) we evaluate, through (20) and (21), that Mode𝑋_+∞=𝑒^𝜇𝑒,𝜎<1,_𝜇−1,𝜎=1,0,𝜎>1, (30) with the corresponding maximum density value being infinite; that is, max𝑓_𝑋𝛾=𝑓_𝑋𝛾(0)=+∞, provided 𝜎>1, and max𝑓_𝑋+∞=1/(2𝜎𝑒^𝜇), provided 𝜎≤1.
Assume that 𝛾≥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 distributions ℒ𝒩_𝛾(𝜇,𝜎) with 𝛾<0, when 𝛾 rises, that is, when one moves from Log-Laplace to degenerate Dirac distribution inside the ℒ𝒩_𝛾 family, the local minimum (probability) density points of ℒ𝒩_𝛾 are(i) strictly increasing from threshold 0 (Log-Laplace case) to 𝑒^𝜇−1 (Dirac case) provided that 𝜎<1,(ii) fixed at 𝑒^𝜇−1 for all ℒ𝒩_𝛾(𝜇,𝜎=1),(iii) strictly decreasing from 𝑒^𝜇 (Log-Laplace case) to 𝑒^𝜇−1 (Dirac case) provided that 𝜎>1.

Proof. Assume now that 𝛾<0. From (29) we have (𝑑/𝑑𝛾)(𝑒^{𝜇−𝜎𝛾})<0 when 𝜎>1 and (𝑑/𝑑𝛾)(𝑒^{𝜇−𝜎𝛾})>0 when 𝜎<1. Therefore, the local minimum density point 𝑒^{𝜇−𝜎𝛾} (see Proposition 5) for 𝜎>1 is decreasing from 𝑒^{𝜇−𝜎𝛾}∣_{𝛾→−∞}=𝑒^𝜇 to Mode𝑋₀=𝑒^𝜇−1 through (20). When 𝜎=1, 𝑒^{𝜇−𝜎𝛾}=𝑒^𝜇−1 for all 𝛾<0, while for 𝜎<1, 𝑒^{𝜇−𝜎𝛾} increases from 𝑒^{𝜇−𝜎𝛾}∣_{𝛾→−∞}=0 to Mode𝑋₀=𝑒^𝜇−1 through (20).

It is easy to see that for the Log-Laplace case ℒℒ(𝜇,𝜎,𝜎), the local minimum density point 𝑒^{𝜇−𝜎𝛾} of 𝑋_𝛾<0 with 𝜎>1 coincides (in limit) with the local nonsmooth mode point 𝑒^𝜇 of 𝑋_𝛾; see Figure 1(b3). Also, notice that the local minimum density point 𝑒^{𝜇−𝜎𝛾}, 𝛾<0, for the Dirac case 𝒟(𝑒^𝜇), is the limiting point 𝑒^𝜇−1 although the (probability) density in 𝒟(𝑒^𝜇) case vanishes everywhere except at the infinite pole 𝑒^𝜇.

Figure 1 illustrates the probability density functions 𝑓_𝑋𝛾 curves for scale parameters 𝜎=2/3,1,3/2 of the positive-ordered lognormally distributed 𝑋_𝛾∼ℒ𝒩_𝛾≥1(0,𝜎) in Figures 1(a1)–1(a3), respectively, while the p.d.f. of negative-ordered lognormally distributed 𝑋_𝛾∼ℒ𝒩_𝛾<0(0,𝜎) are depicted in Figures 1(b1)–1(b3), respectively. Moreover, the density points 𝑒^{𝜇−𝜎𝛾} on 𝑓_𝑋𝛾 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 𝑋_𝛾≥1, these density points represent the mode points on 𝑓_𝑋𝛾 while in Figures 1(b1)–1(b3), that is, for negative-ordered 𝑋_𝛾<0, represent the local minimum density points on 𝑓_𝑋𝛾 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. 𝐹_𝑋𝛾 of a 𝛾-order Lognormal random variable 𝑋_𝛾∼ℒ𝒩_𝛾(𝜇,𝜎) is given by 𝐹_𝑋𝛾(𝑥)=12+√𝜋2Γ𝛾−1/𝛾Γ𝛾/𝛾−1×Erf_{𝛾/(𝛾−1)}𝛾−1𝛾^{(𝛾−1)/𝛾}log𝑥−𝜇𝜎1=1−Γ2Γ𝛾−1/𝛾𝛾−1𝛾,𝛾−1𝛾log𝑥−𝜇𝜎^{𝛾/(𝛾−1)},𝑥∈ℝ^∗₊. (31)(32)

Proof. From density function 𝑓_𝑋𝛾, as in (10), we have 𝐹_𝑋𝛾(𝑥)=𝐹_𝑋𝛾𝑥;𝜇,𝜎,𝛾=^𝑥₀𝑓_𝑋𝛾(𝑡)𝑑𝑡=𝜎⁻¹𝐶¹_𝛾^𝑥₀𝑡⁻¹exp−𝛾−1𝛾log𝑡−𝜇𝜎^{𝛾/(𝛾−1)}𝑑𝑡. (33) Applying the transformation 𝑤=(log𝑡−𝜇)/𝜎, 𝑡>0, the above c.d.f. is reduced to 𝐹_𝑋𝛾(𝑥)=𝐶¹_𝛾^{(log𝑥−𝜇)/𝜎}_−∞exp−𝛾−1𝛾|𝑤|^{𝛾/(𝛾−1)}𝑑𝑤=Φ_𝑍𝛾log𝑥−𝜇𝜎, (34) where Φ_𝑍𝛾 is the c.d.f. of the standardized 𝛾-order normal distribution 𝑍_𝛾=(1/𝜎)(log𝑋_𝛾−𝜇)∼𝒩_𝛾(0,1). Moreover, Φ_𝑍𝛾 can be expressed in terms of the generalized error function. In particular, Φ_𝑍𝛾(𝑧)=𝐶¹_𝛾^𝑧_−∞exp−𝛾−1𝛾|𝑤|^{𝛾/(𝛾−1)}𝑑𝑤=Φ_𝑍𝛾(0)+𝐶¹_𝛾^𝑧₀exp−𝛾−1𝛾|𝑤|^{𝛾/(𝛾−1)}𝑑𝑤, (35) and as 𝑓_𝑍𝛾 is a symmetric density function around zero, we have Φ_𝑍𝛾(𝑧)=12+𝐶¹_𝛾^𝑧₀exp−𝛾−1𝛾|𝑤|^{𝛾/(𝛾−1)}=1𝑑𝑤2+𝐶¹_𝛾^𝑧₀exp−𝛾−1𝛾^{(𝛾−1)/𝛾}𝑤^{𝛾/(𝛾−1)}𝑑𝑤, (36) and thus Φ_𝑍𝛾(𝑧)=12+𝐶¹_𝛾𝛾𝛾−1^{(𝛾−1)/𝛾}×^{((𝛾−1)/𝛾)(𝛾−1)/𝛾𝑧}₀exp−𝑢^{𝛾/(𝛾−1)}𝑑𝑢. (37) Substituting the normalizing factor, as in (2), and using (3), we obtain Φ_𝑍𝛾(𝑧)=12+√𝜋2Γ𝛾−1/𝛾+1Γ2𝛾−1/𝛾−1×Erf_{𝛾/(𝛾−1)}𝛾−1𝛾^{(𝛾−1)/𝛾}𝑧,𝑧∈ℝ, (38) 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 𝐹_𝑋𝛾(𝑥)=12+sgnlog𝑥−𝜇√𝜋2Γ𝛾−1/𝛾Γ𝛾/𝛾−1×Erf_{𝛾/(𝛾−1)}𝛾−1𝛾^{(𝛾−1)/𝛾}log𝑥−𝜇𝜎, (39) while applying (4) into (39) it is obtained that 𝐹_𝑋𝛾(𝑥)=1+sgnlog𝑥−𝜇2−sgnlog𝑥−𝜇2Γ𝛾−1/𝛾×Γ𝛾−1𝛾,𝛾−1𝛾log𝑥−𝜇𝜎^{𝛾/(𝛾−1)}. (40)

As the generalized error function Erf_𝑎 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. 𝐹_𝑋𝛾 can be expressed in the series expansion form 𝐹_𝑋𝛾(𝑥)=12+(𝛾−1)/𝛾^{(𝛾−1)/𝛾}2/𝛾Γ𝛾−1/𝛾log𝑥−𝜇𝜎×^∞_𝑘=0(1−𝛾)/𝛾(log𝑥−𝜇)/𝜎^{𝛾/(𝛾−1)𝑘}),𝑘!(𝑘+1𝛾−1𝑥∈ℝ^∗₊. (41)

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 variable 𝑋_𝛾 with 𝛾=1/(1−𝑛)∈ℝ₋, 𝑛∈ℕ, 𝑛≥2, the finite expansion is obtained as 𝐹_𝑋𝛾(𝑥)=12+12sgnlog𝑥−𝜇−sgnlog𝑥−𝜇2exp𝑛(log𝑥−𝜇)/𝜎^1/𝑛×^𝑛−1_𝑘=0𝑛^𝑘𝑘!log𝑥−𝜇𝜎^𝑘/𝑛. (42)

Proof. Applying the following finite expansion form of the upper incomplete gamma function, Γ(𝑛,𝑥)=(𝑛−1)!𝑒^{−𝑥𝑛−1}_𝑘=0𝑥^𝑘𝑘,𝑥∈ℝ,𝑛∈ℕ^∗=ℕ⧵0, (43) into (40), we readily get (42).

Example 11. For the (−1)-ordered lognormally distributed 𝑋₋₁ (i.e., for 𝑛=2), we have 𝐹_𝑋−1(𝑥)=12+12×sgnlog𝑥−𝜇−sgnlog𝑥−𝜇1+2log𝑥−𝜇/𝜎2exp2,log𝑥−𝜇/𝜎 (44) while for the (−1/2)-ordered lognormally distributed 𝑋_−1/2 (i.e., for 𝑛=3), we have 𝐹_𝑋−1/2(𝑥)=12+12×sgnlog𝑥−𝜇−sgnlog𝑥−𝜇1+3³log𝑥−𝜇/𝜎+9³(log𝑥−𝜇)/𝜎²2exp3³.log𝑥−𝜇/𝜎 (45)

Example 12. For the second-ordered Lognormal random variable 𝑋₂∼ℒ𝒩₂(𝜇,𝜎), we immediately derive, from (31), that 𝐹_𝑋2(𝑥)=Φ_𝑋2log𝑥−𝜇𝜎=12+12Erf₂log𝑥−𝜇√=12𝜎2+12erflog𝑥−𝜇√;2𝜎 (46) 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 Lognormal 𝑋_±∞∼ℒ𝒩_±∞(𝜇,𝜎), setting (𝛾−1)/𝛾=1, we obtain through (41) and the exponential series expansion that 𝐹_𝑋±∞(𝑥)=12+12√𝜋Erf₁log𝑥−𝜇𝜎=12−12sgnlog𝑥−𝜇^∞_𝑘=01()!−𝑘+1log𝑥−𝜇𝜎^𝑘+1=12+12−1sgnlog𝑥−𝜇2−sgnlog𝑥−𝜇×explog𝑥−𝜇𝜎, (47) and hence 𝐹_𝑋±∞(𝑥)=12𝑒^𝜇/𝜎𝑥^1/𝜎,𝑥∈0,𝑒^𝜇,𝑒1−^𝜇/𝜎2𝑥^1/𝜎,𝑥∈𝑒^𝜇,+∞, (48) which is the c.d.f. of the Log-Laplace distribution as in (15). This is expected as ℒ𝒩_±∞(𝜇,𝜎)=ℒℒ(𝑒^𝜇,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 variable 𝑋₁∼ℒ𝒩₁(𝜇,𝜎), the expansion (41) can be written as 𝐹_𝑋𝛾(𝑥)=12+(𝛾−1)/𝛾^{(𝛾−1)/𝛾}2Γ𝛾−1/𝛾+1log𝑥−𝜇𝜎×1+𝛾−1^∞_𝑘=1(1−𝛾)/𝛾(log𝑥−𝜇)/𝜎^{𝛾/(𝛾−1)𝑘}),𝑘!(𝑘+1𝛾−1 (49) and provided that (log𝑥−𝜇)/𝜎≤1, we obtain 𝐹_𝑋1(𝑥)=lim_𝛾→1+𝐹_𝑋𝛾(𝑥)=12+log𝑥−𝜇()=2𝜎1+0log𝑥−𝜇+𝜎,2𝜎 (50) with 𝐹_𝑋1(𝑒^𝜇−𝜎)=0 and 𝐹_𝑋1(𝑒^𝜇+𝜎)=1. Therefore, 𝐹_𝑋1(𝑥)=0,𝑥∈0,𝑒^𝜇−𝜎,1𝑒2𝜎log𝑥−𝜇+𝜎,𝑥∈^𝜇−𝜎,𝑒^𝜇+𝜎,1,𝑥∈𝑒^𝜇+𝜎,+∞, (51) coincides with the c.d.f. of the Log-Uniform distribution ℒ𝒰(𝑎=𝑒^𝜇−𝜎,𝑏=𝑒^𝜇+𝜎) as in (15). This is expected as ℒ𝒩₁(𝜇,𝜎)=ℒ𝒰(𝑒^𝜇−𝜎,𝑒^𝜇+𝜎); see Theorem 3.

Table 1 provides the probability values 𝑃_𝛾;1=Pr{𝑋_𝛾≤𝑖}, 𝑖=1/2,1,2,…,5, for various 𝑋_𝛾∼ℒ𝒩_𝛾(0,1). Notice that 𝑃_𝛾;1=1/2 for all 𝛾 values due to the fact that 1=𝑒^𝜇∣_𝜇=0=Med𝑋_𝛾 (see Theorem 8); that is, the point 1 coincides with the 𝛾-invariant median of the ℒ𝒩_𝛾(0,1) family discussed previously. Moreover, the last two columns provide also the 1st and 3rd quartile points 𝑞_1;𝛾 and 𝑞_3;𝛾 of 𝑋_𝛾; that is, Pr{𝑋_𝛾≤𝑞_𝑘;𝛾}=𝑘/4, 𝑘=1,3, for various 𝛾 values. These quartiles are evaluated using the quantile function Q_𝑋𝛾 of r.v. 𝑋_𝛾; that is, Q_𝑋𝛾(𝑃)∶=inf𝑥∈ℝ^∗₊∣𝐹_𝑋𝛾(𝑥)()𝜎×𝛾≥𝑃=expsgn2𝑃−1Γ𝛾−1⁻¹𝛾−1𝛾,|2𝑃−1|^{(𝛾−1)/𝛾},𝑃∈(0,1), (52) 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 values 𝑃_𝛾;𝑖=Pr{𝑋_𝛾≤𝑖}, 𝑖=1/2,1,2,…,5, and the 1st and 3rd quartiles 𝑞_1;𝛾, 𝑞_3;𝛾 for various generalized lognormally distributed 𝑋_𝛾∼ℒ𝒩_𝛾(0,1).

Figure 2 illustrates the c.d.f. 𝐹_𝑋𝛾 curves, as in (39), for certain r.v. 𝑋_𝛾∼ℒ𝒩_𝛾(0,𝜎) and for scale parameters 𝜎=2/3,1,3/2 in the 3 subfigures, respectively. Moreover, the 1st and 3rd quartile points Q_𝑋𝛾(1/4) and Q_𝑋𝛾(3/4) are also depicted (small circles over c.d.f. curves with their corresponding ticks on 𝑥-axis).

(a)

(b)

(c)

Figure 2 

Graphs of the c.d.f. 𝐹𝑋𝛾, 𝑋𝛾∼ℒ𝒩𝛾(0,𝜎), for 𝜎=2/3,1,3/2, and various 𝛾 values.

Theorem 15. The (non-log-scaled) location parameter 𝑒^𝜇 is in fact the geometric mean as well as the median for all generalized lognormally distributed 𝑋_𝛾∼ℒ𝒩_𝛾(𝜇,𝜎). Moreover, this median is also characterized by vanishing median absolute deviation.

Proof. Considering (39) and the fact that Erf_𝑎0=0, 𝑎∈ℝ^∗₊, it holds that Med𝑋_𝛾=𝐹_𝑋⁻¹_𝛾(1/2)=𝑒^𝜇. For the geometric mean (𝜇_𝑔)_𝑋𝛾=𝑒^{E[log𝑋𝛾]}, we readily obtain (𝜇_𝑔)_𝑋𝛾=𝑒^𝜇 as log𝑋_𝛾∼𝒩_𝛾(𝜇,𝜎²) with E[𝑋_𝛾]=𝜇. A dispersion measure for the median is the so-called median absolute deviation or MAD, defined by MAD(𝑋_𝛾)=Med∣𝑋_𝛾−Med𝑋_𝛾∣. For 𝑋_𝛾∼ℒ𝒩_𝛾(𝜇,𝜎), we have 𝑋_𝛾−Med𝑋_𝛾=𝑋_𝛾−𝑒^𝜇∼ℒ𝒩_𝛾(𝜇,𝜎;−𝑒^𝜇); that is, 𝑋_𝛾−𝑒^𝜇 follows the generalized Lognormal distribution with threshold −𝑒^𝜇. Furthermore, |𝑌| is the “folded distribution” case of 𝑌∶=𝑋_𝛾−𝑒^𝜇 which is distributed through p.d.f. of the form 𝑓_|𝑌|(𝑥)=𝑓_𝑌(−𝑥)+𝑓_𝑌(𝑥),𝑥∈ℝ₊, (53) where 𝑓_𝑌 is the p.d.f. of 𝑌. For example, see [17] on the folded normal distribution. However, the density function 𝑓_𝑌 is defined in (−𝑒^𝜇,+∞) due to threshold −𝑒^𝜇, while it vanishes elsewhere; that is, 𝑓_|𝑌|(𝑥)=𝑓_𝑌(−𝑥)+𝑓_𝑌(𝑥),0≤𝑥≤𝑒^𝜇,𝑓_𝑌(𝑥),𝑥>𝑒^𝜇. (54) Therefore, the c.d.f. of |𝑌| is given by 𝐹_|𝑌|(𝑥)=^𝑥₀𝑓_|𝑌|(𝑡)𝑑𝑡=^𝑒𝜇₀𝑓_𝑌(−𝑡)𝑑𝑡+^𝑒𝜇₀𝑓_𝑌(𝑡)+𝑑𝑡_𝑥𝑒𝜇𝑓_𝑌(𝑡)𝑑𝑡,𝑥∈ℝ₊. (55) Applying the transformation 𝑤=(1/𝜎)[log(𝑒^𝜇−𝑡)−𝜇], 𝑡<𝑒^𝜇, into the first integral of (55) and 𝑧=(1/𝜎)[log(𝑡+𝑒^𝜇)−𝜇], 𝑡>−𝑒^𝜇, into the other two integrals, we obtain 𝐹_|𝑌|(𝑥)=𝐶¹_𝛾⁰_−∞exp−𝛾−1𝛾|𝑤|^{𝛾/(𝛾−1)}𝑑𝑤+𝐶¹_𝛾×₀^(1/𝜎)log2exp−𝛾−1𝛾|𝑧|^{𝛾/(𝛾−1)}𝑑𝑧+𝐶¹_𝛾()^𝑔(𝑥)_1/𝜎log2exp−𝛾−1𝛾|𝑧|^{𝛾/(𝛾−1)}𝑔(𝑥)𝑑𝑧,∶=log𝑥+𝑒^𝜇−𝜇𝜎,𝑥∈ℝ₊, (56) and hence 𝐹_|𝑌|(𝑥)=Φ_𝑍(0)+Φ_𝑍log2𝜎−Φ_𝑍(0+Φ_𝑍)𝑔(𝑥)−Φ_𝑍log2𝜎=Φ_𝑍log𝑥+𝑒^𝜇−𝜇𝜎, (57) with Φ_𝑍 being the c.d.f. of the standardized r.v. 𝑍∼𝒩_𝛾(0,1). From (38) and the fact that Erf_𝑎0=0, 𝑎∈ℝ^∗₊, it is clear that (57) implies MAD𝑋_𝛾=Med|𝑋_𝛾−𝑒^𝜇|=𝐹⁻¹_{|𝑋𝛾−𝑒𝜇|}(1/2)=0, for every 𝑋_𝛾∼ℒ𝒩_𝛾(𝜇,𝜎), and the theorem has been proved.

3. Moments of the 𝛾-Order Lognormal Distribution

For the evaluation of the moments of the generalized Lognormal distribution, the following holds.

Proposition 16. The 𝑡th raw moment 𝜇_𝑋^(𝑡) of a generalized lognormally distributed random variable 𝑋∼ℒ𝒩_𝛾(𝜇,𝜎) is given by 𝜇_𝑋^(𝑡)=𝑒^𝑡𝜇Γ𝛾−1/𝛾^∞_𝑛=0(𝑡𝜎)^2𝑛()!𝛾2𝑛𝛾−1^{2𝑛((𝛾−1)/𝛾)}×Γ(2𝑛+1)𝛾−1𝛾 (58) and coincides with the moment generating function of the 𝛾-order normally distributed log𝑋; that is, 𝑀_log𝑋(𝑡)=𝜇_𝑋^(𝑡).

Proof. From the definition of the 𝑡th raw moment 𝜇_𝑋^(𝑡), we have 𝜇_𝑋^(𝑡)=E𝑋^𝑡=_ℝ+𝑥^𝑡𝑓_𝑋(𝑥)=1𝑑𝑥𝜎𝐶¹_𝛾ℝ+𝑥^𝑡−1exp−𝛾−1𝛾log𝑥−𝜇𝜎^{𝛾/(𝛾−1)}𝑑𝑥, (59) and applying the transformation 𝑧=((𝛾−1)/𝛾)^{(𝛾−1)/𝛾}(1/𝜎)(log𝑥−𝜇), 𝑥>0, we get 𝜇_𝑋^(𝑡)=𝐶¹_𝛾𝛾𝛾−1^{(𝛾−1)/𝛾}_ℝ𝛾exp𝑡𝜇+𝑛𝛾−1^{(𝛾−1)/𝛾}−|𝑧|𝜎𝑧×exp^{𝛾/(𝛾−1)}𝑑𝑧. (60) Through the exponential series expansion 𝑡𝛾exp𝛾−1^{(𝛾−1)/𝛾}𝜎𝑧=^∞_𝑛=0(𝑡𝜎)^𝑛𝛾𝑛!𝛾−1^{𝑛((𝛾−1)/𝛾)}𝑧^𝑛, (61) it is obtained that 𝜇_𝑋^(𝑡)=2𝐶¹_𝛾𝛾𝛾−1^{(𝛾−1)/𝛾}𝑒^∞𝑡𝜇_𝑛=0(𝑡𝜎)^2𝑛()!𝛾2𝑛𝛾−1^{2𝑛((𝛾−1)/𝛾)}×_ℝ+𝑧^2𝑛exp−𝑧^{𝛾/(𝛾−1)}𝑑𝑧. (62) Finally, substituting the normalizing factor 𝐶¹_𝛾 as in (2) into (62) and utilizing the known integral [16], _ℝ+𝑥^𝑚𝑒^{−𝑏𝑥𝑛}Γ))𝑑𝑥=((𝑚+1/𝑛𝑛𝑏^{(𝑚+1)/𝑛},𝑛,𝑚,𝑏∈ℝ^∗₊, (63) we obtain (58).
Moreover, for 𝑌∶=log𝑋∼𝒩_𝛾(𝜇,𝜎²) we have 𝑀_𝑌(𝑡)=E[𝑒^𝑡𝑌]=E[𝑋^𝑡]=𝜇_𝑋^(𝑡), and the proposition has been proved.

Example 17. For the second-ordered lognormally distributed 𝑋∼ℒ𝒩₂(𝜇,𝜎), (58) implies 𝜇_𝑋^(𝑡)=𝑒^𝑡𝜇√𝜋^∞_𝑛=02𝑡²𝜎^2𝑛()!Γ12𝑛𝑛+2,𝑡∈ℝ₊, (64) and through the gamma function identity Γ1𝑛+2=(2𝑛)!2^2𝑛√𝑛!𝜋,𝑛∈ℕ, (65) we have 𝜇_𝑋^(𝑡)=𝑒^∞𝑡𝜇_𝑛=0(𝑡𝜎)^2𝑛2^𝑛𝑛!=𝑒^∞𝑡𝜇_𝑛=011𝑛!2𝑡²𝜎^2𝑛=𝑒^{𝑡𝜇+(1/2)(𝑡𝜎)2},𝑡∈ℝ₊, (66) which is the 𝑡th raw moment of the usual lognormally distributed 𝑋∼ℒ𝒩(𝜇,𝜎), with mean 𝜇_𝑋∶=𝜇_𝑋⁽¹⁾=E[𝑋]=exp{𝜇+(1/2)𝜎²}. This is true as 𝑀_log𝑋(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. The 𝑘th central moment (about the mean) 𝜇_𝑋^(𝑡) of a generalized lognormally distributed random variable 𝑋∼ℒ𝒩_𝛾(𝜇,𝜎) is given by 𝜇_𝑋^(𝑘)=𝑒^𝑘𝜇Γ𝛾−1/𝛾^𝑘_𝑛=0𝑘𝑛−𝜇_𝑋𝑒^𝜇𝑛𝑆_𝑘−𝑛,𝑘∈ℕ, (67) where 𝑆_𝑘=^∞_𝑚=0(𝑘𝜎)^2𝑚()!𝛾2𝑚𝛾−1^{2𝑚((𝛾−1)/𝛾)}Γ(2𝑚+1)𝛾−1𝛾,𝑘∈ℕ. (68)

Proof. From the definition of the 𝑘th central moment 𝜇_𝑋^(𝑘) we have 𝜇_𝑋^(𝑘)∶=E𝑋−𝜇_𝑋^𝑘=_ℝ+𝑥−𝜇_𝑋^𝑘𝑓_𝑋𝑥;𝜇,𝜎,𝛾𝑑𝑥, (69) while using the binomial identity we get 𝜇_𝑋^(𝑘)=^𝑘_𝑛=0𝑘𝑛−𝜇_𝑋^𝑛_ℝ+𝑥^𝑘−𝑛𝑓_𝑋(𝑥)=𝑑𝑥_𝑘𝑛=0𝑘𝑛−𝜇_𝑋^𝑛𝜇_𝑋^{(𝑘−𝑛)}. (70) Applying Proposition 16, (70) implies that 𝜇_𝑋^(𝑘)=𝑒^𝑘𝜇Γ𝛾−1/𝛾^𝑘_𝑛=0𝑘𝑛−𝜇_𝑋𝑒^𝜇𝑛∞_𝑚=0[(𝑘−𝑛)𝜎]^2𝑚()!×𝛾2𝑚𝛾−1^{2𝑚((𝛾−1)/𝛾)}Γ(2𝑚+1)𝛾−1/𝛾, (71) while taking the summation index 𝑛 until 𝑘−1, we finally obtain (67), and the theorem has been proved.

Example 19. Recall Example 17. Substituting (66) and the mean 𝜇_𝑋=𝑒^{𝜇+(1/2)𝜎2} into (70), the second-ordered lognormally distributed 𝑋∼ℒ𝒩₂(𝜇,𝜎) provides 𝜇_𝑋^(𝑘)=^𝑘_𝑛=0𝑘𝑛(−1)^𝑛𝑒^{𝑘𝜇+(1/2)[𝑛+(𝑘−𝑛)2]𝜎2},𝑘∈ℕ, (72) while 𝜎²_𝑋∶=Var[𝑋]=𝜇_𝑋⁽²⁾=𝑒^2𝜇+𝜎2𝑒^𝜎2−1, (73) 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 mean 𝜇_𝑋∶=E[𝑋], variance 𝜎²_𝑋∶=Var[𝑋], coefficient of variation 𝐶𝑉_𝑋, skewness 𝜆_𝑋 and kurtosis 𝜅_𝑋 of the generalized lognormally distributed 𝑋∼ℒ𝒩_𝛾(𝜇,𝜎) are, respectively, given by 𝜇_𝑋=𝑒^𝜇Γ𝑆𝛾−1/𝛾¹,𝜎²_𝑋=−𝜇²_𝑋+𝑒^2𝜇Γ𝑆𝛾−1/𝛾²,𝐶𝑉²_𝑋=Γ𝛾−1𝛾𝑆₂𝑆²₁𝜆−1,_𝑋=−𝐶𝑉_𝑋⁻³−𝐶𝑉_𝑋⁻¹+𝑒^3𝜇𝜎³_𝑋Γ𝑆𝛾−1/𝛾³,𝜅_𝑋=−𝐶𝑉_𝑋⁻⁴−6𝐶𝑉_𝑋⁻²𝜆−4^𝑋𝐶𝑉_𝑋+𝑒^4𝜇𝜎⁴_𝑋Γ𝑆𝛾−1/𝛾⁴, (74)(75)(76)(77)(78) where the sums 𝑆_𝑖, 𝑖=1,…,4, are given by (68).

Proof. From Proposition 16 we easily obtain (74), as 𝜇_𝑋∶=𝜇_𝑋⁽¹⁾. From Theorem 18 we have 𝜎²_𝑋∶=𝜇_𝑋⁽²⁾=𝜇²_𝑋+Γ𝛾−1𝛾⁻¹𝑒^2𝜇𝑆₂−2𝑒^𝜇𝜇_𝑋𝑆₁. (79) Hence, substituting 𝑆₁ from (74), (75) holds. Moreover, the squared coefficient of variation is readily obtained via (75) and (74). By definition, skewness 𝜆_𝑋 is the standardized third (central) moment; that is, 𝜆_𝑋∶=Skew[𝑋]=𝜇_𝑋⁽³⁾/𝜎³_𝑋. Theorem 18 provides that 𝜆_𝑋=−𝐶𝑉_𝑋⁻³+𝜎³_𝑋Γ𝛾−1𝛾⁻¹×𝑒^3𝜇𝑆₃−3𝑒^2𝜇𝜇_𝑋𝑆₂+3𝑒^𝜇𝜇_𝑋𝑆₁. (80) Substituting 𝑆₁ and 𝑆₂ from (74) and (75), we obtain (77). Finally, kurtosis 𝜅_𝑋 is (by definition) the standardized fourth (central) moment; that is, 𝜅_𝑋∶=Kurt[𝑋]=𝜇_𝑋⁽⁴⁾/𝜎⁴_𝑋, which provides, through Theorem 18, that 𝜅_𝑋=𝐶𝑉_𝑋⁻⁴+𝜎⁴_𝑋Γ𝛾−1𝛾⁻¹×𝑒^4𝜇𝑆₄−4𝑒^3𝜇𝜇_𝑋𝑆₃+6𝑒^2𝜇𝜇²_𝑋𝑆₂−4𝑒^𝜇𝜇³_𝑋𝑆₁. (81) Substituting 𝑆_𝑖, 𝑖=1,2,3, from (74), (75), and (77), we obtain (78).

Example 21. For the second-ordered lognormally distributed 𝑋∼ℒ𝒩₂(𝜇,𝜎), utilizing (65) into (68) we get 𝑆_𝑛=√𝜋𝑒^{(𝑛2𝜎2)/2}, 𝑛∈ℕ^∗. Applying this to Theorem 20 we derive (after some algebra) 𝜇_𝑋=𝑒^{𝜇+(1/2)𝜎2},𝜎²_𝑋=𝑒^2𝜇+𝜎2𝑒^𝜎2−1,𝐶𝑉_𝑋=𝑒^𝜎2𝜆−1,_𝑋=𝑒^𝜎2+2𝑒^𝜎2−1,𝜅_𝑋=𝑒^4𝜎2+2𝑒^3𝜎2+3𝑒^2𝜎2−3, (82) 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 variable 𝑋∼ℒ𝒩, it is known that Mode𝑋<Med𝑋<𝜇_𝑋. The following corollary examines this inequality for the ℒ𝒩_𝛾 family of distributions.

Corollary 22. For the 𝛾-ordered lognormally distributed 𝑋_𝛾∼ℒ𝒩_𝛾(𝜇,𝜎), it is true that 𝑀𝑜𝑑𝑒𝑋_𝛾≤𝑀𝑒𝑑𝑋_𝛾=(𝜇_𝑔)_𝑋𝛾≤𝜇_𝑋𝛾. The first equality holds for the Log-Laplace distributed 𝑋_+∞ with 𝜎<1 as well as for all the negative-ordered 𝑋_𝛾<0 where 𝑀𝑜𝑑𝑒𝑋_𝛾 is considered to be the local (nonsmooth) mode point of 𝑋_𝛾. The second equality holds for the degenerate Dirac case of 𝑋₀.

Proof. From (74) and Theorem 15 we have Med𝑋_𝛾=𝜇_𝑔𝑋𝛾=𝑒^𝜇<𝜇_𝑋𝛾, (83) for every 𝑋_𝛾∼ℒ𝒩_𝛾(𝜇,𝜎). The above inequality becomes equality for the limiting Dirac case of 𝑋₀. For the relation between the mode and the median of 𝑋_𝛾, the following cases are considered. (i) The positive-ordered Lognormal case 𝛾>1: from (20) we have Mode𝑋_𝛾=𝑒^{𝜇−𝜎𝛾}<𝑒^𝜇=Med𝑋_𝛾. (84) For the Log-Laplace case of 𝑋_+∞, it holds Mode𝑋_𝛾=𝑒^𝜇=Med𝑋_𝛾, (85) provided that 𝜎<1, while for 𝜎≥1 we have Mode𝑋_𝛾=0<𝑒^𝜇=Med𝑋_𝛾. (86) For 𝜎=1, the inequality (84) clearly holds. (ii) The negative-ordered Lognormal case 𝛾<0: from Proposition 4 the inequality as in (86) holds. Moreover, if Mode𝑋_𝛾 is considered as the nonsmooth local mode point of the negative-ordered 𝑋_𝛾 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 variable 𝑋∼ℒ𝒰(𝑎,𝑏), 0<𝑎<𝑏, are given by 𝜇_𝑋^(𝑡)=𝑏^𝑡−𝑎^𝑡()𝜇𝑡log𝑏/𝑎,𝑡∈ℝ,_𝑋(𝑘)=(𝑎−𝑏)^𝑘log^𝑘()+1𝑏/𝑎()×log𝑏/𝑎_{𝑘−1𝑛=0}𝑘𝑛(𝑎−𝑏)^𝑛𝑏^𝑘−𝑛−𝑎^𝑘−𝑛(𝑘−𝑛)log^𝑛(𝑏/𝑎),𝑘∈ℕ, (87)(88) respectively, while the mean, variance, coefficient of variation, skewness, and kurtosis of 𝑋 are given, respectively, by 𝜇_𝑋=𝑏−𝑎(),𝜎log𝑏/𝑎_2𝑋=(𝑏−𝑎)²log²()+()𝑏/𝑎𝑏−𝑎)(𝑏+𝑎(),2log𝑏/𝑎𝐶𝑉_𝑋=1+𝑏+𝑎2()𝑏𝑏−𝑎log𝑎,𝜆_𝑋=1𝜎³_𝑋𝑏³−𝑎³()()3log𝑏/𝑎−3𝑏−𝑎²(𝑏+𝑎)2log²()()𝑏/𝑎+2𝑏−𝑎³log³(),𝜅𝑏/𝑎_𝑋=1𝜎⁴_𝑋𝑏⁴−𝑎⁴()(𝑏4log𝑏/𝑎−4𝑏−𝑎)³−𝑎³3log²()()𝑏/𝑎+3𝑏−𝑎³(𝑏+𝑎)log³()()𝑏/𝑎−3𝑏−𝑎⁴log⁴().𝑏/𝑎 (89)(90)(91)(92)(93)

Proof. Recall Proposition 16 with 𝑋_𝛾∼ℒ𝒩_𝛾(𝜇,𝜎). Through the gamma function additive identity (58) can be written as 𝜇_𝑋^(𝑡)_𝛾=𝑒^𝑡𝜇𝑡𝜎Γ𝛾−1/𝛾+1^∞_𝑚=0(𝑡𝜎)^2𝑚+1()!𝛾2𝑚+1𝛾−1^{2𝑚((𝛾−1)/𝛾)}×Γ(2𝑚+1)𝛾−1𝛾+1. (94) 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 𝜇_𝑋^(𝑡)=lim_𝛾→1+𝜇_𝑋^(𝑡)_𝛾=𝑒^𝑡𝜇𝑡𝜎^∞_𝑚=0(𝑡𝜎)^2𝑚+1()!=𝑒2𝑚+1^{𝑡(𝜇+𝜎)}−𝑒^{𝑡(𝜇−𝜎)},2𝑡𝜎𝑡∈ℝ, (95) and hence (87) holds. Moreover, 𝜇_𝑋∶=𝜇_𝑋⁽¹⁾=E[𝑋]=(1/2𝜎)(𝑒^𝜇+𝜎−𝑒^𝜇−𝜎), and therefore (89) holds.
Working similarly, (67) implies 𝜇_𝑋^(𝑘)=lim_𝛾→1+𝜇_𝑋^(𝑘)_𝛾=𝑒^𝑘𝑘𝜇_𝑛=0𝑘𝑛−𝜇_𝑋𝑒^𝜇𝑛∞_𝑚=0[(𝑘−𝑛)𝜎]^2𝑚()!,2𝑚+1𝑘∈ℕ. (96) Using the exponential odd series expansion, the above expansion becomes 𝜇_𝑋^(𝑘)=𝑒^𝑘𝜇2𝜎^𝑘_𝑛=0𝑘𝑛−𝜇_𝑋𝑒^𝜇𝑛𝑒^{(𝑘−𝑛)𝜎}−𝑒^{−(𝑘−𝑛)𝜎}(𝑘−𝑛),𝑘∈ℕ, (97) and, through (89), we obtain (88). Moreover, for 𝑘=2, 𝜎²_𝑋∶=Var[𝑋]=𝜇_𝑋⁽²⁾−𝜇²_𝑋 implies (90), and hence (91) also holds. For 𝑘=3 and 𝑘=4, through 𝜇_𝑋⁽³⁾ and 𝜇_𝑋⁽⁴⁾, we obtain (92) and (93), respectively.

Corollary 24. The raw and central moments of a Log-Laplace distributed random variable 𝑋∼ℒℒ(𝜇,𝜎,𝜎) are given by 𝜇_𝑋^(𝑡)=𝜇^𝑡𝜎²𝜎²−𝑡²>𝜇^𝑡𝜇,𝜎>𝑡,𝑡∈ℝ,_𝑋(𝑘)=𝜇^𝑘𝑘_𝑛=0𝑘𝑛𝜎^2(𝑛+1)1−𝜎^2𝑛𝜎²−(𝑘−𝑛)²,𝜎>𝑘,𝑘∈ℕ. (98)(99) The mean, variance, coefficient of variation, skewness, and kurtosis of 𝑋 are given, respectively, by 𝜇_𝑋=𝜇𝜎²𝜎²𝜎−1>𝜇,𝜎>1,_2𝑋=𝜇²𝜎²2𝜎²+1𝜎²−4𝜎²−1²,𝜎>2,𝐶𝑉_𝑋=1𝜎2𝜎²+1𝜎²𝜆−4,𝜎>2,_𝑋=215𝜎⁴+7𝜎²+2𝜎𝜎²−9𝜎²−42𝜎²+1³𝜅,𝜎>3,_𝑋=38𝜎⁸+212𝜎⁶+95𝜎⁴+33𝜎²+12𝜎²−4𝜎²−16𝜎²−92𝜎²+1²,𝜎>4. (100)(101)(102)(103)(104)

Proof. Let 𝑋_𝛾∼ℒℒ_𝛾(𝜇,𝜎,𝜎)=ℒ𝒩_𝛾(log𝜇,1/𝜎,1/𝜎). For 𝛾=±∞, that is, 𝛾/(𝛾−1)=1, the raw moments as in (58) provide 𝜇_𝑋^(𝑡)=𝜇_𝑋^(𝑡)_±∞=𝜇^𝑡∞_𝑘=0𝑡𝜎^2𝑘,𝑡∈ℝ, (105) as 𝑋=𝑋_±∞, while through the even geometric series expansion, it is 𝜇_𝑋^(𝑡)_±∞=12𝜇^𝑡∞_𝑘=0𝑡𝜎^𝑘+^∞_𝑘=0−𝑡𝜎^𝑘=12𝜇^𝑡𝜎+𝜎𝜎−𝑡,𝜎+𝑡 (106) provided that 𝜎>𝑡, and hence (98) holds. Moreover, 𝜇_𝑋∶=𝜇_𝑋⁽¹⁾=E[𝑋], and hence (100) holds.
Working similarly, (67) implies 𝜇_𝑋^(𝑘)=𝜇^𝑘𝜎^2𝑘_𝑛=0𝑘𝑛−𝜇_𝑋/𝜇^𝑛𝜎²−(𝑘−𝑛)²,𝑘∈ℕ, (107) provided 𝜎>𝑘, and hence, through (100), the central moments (99) are obtained.
Moreover, for 𝑘=2 and due to 𝜎²_𝑋∶=Var[𝑋]=𝜇_𝑋⁽²⁾−𝜇²_𝑋, (101) holds true, while for 𝑘=3 and 𝑘=4 we derive, through 𝜇_𝑋⁽³⁾and 𝜇_𝑋⁽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∶=𝑒^𝑈∼ℒ𝒰(𝑒^𝜇−𝜎,𝑒^𝜇+𝜎) due to Theorem 3, and therefore LU is a Log-Uniform distributed r.v. as LU∼ℒ𝒰(𝑒^𝑎,𝑒^𝑏). Applying (87), the known moment-generating function of the uniformly distributed 𝑈∼𝒰(𝑎,𝑏) is derived; that is, 𝑀_𝑈(𝑡)∶=E[𝑒^𝑡𝑈]=𝜇^(𝑡)_LU=(𝑒^𝑡𝑏−𝑒^𝑡𝑎)(1/𝑡(𝑏−𝑎)).
Similarly, for a Laplace distributed r.v. 𝐿∼ℒ(𝜇,𝜎)=𝒩_±∞(𝜇,𝜎), it holds that LL∶=𝑒^𝐿∼ℒℒ(𝑒^𝜇,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, 𝑀_𝐿(𝑡)∶=E[𝑒^𝑡𝐿]=𝜇^(𝑡)_LL=𝑒^𝑡𝜎(1−𝑡²𝜎²)⁻¹.

4. Conclusion

The family of the 𝛾-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.

Acknowledgment

The authors would like to thank the referee for his valuable comments that helped improve the quality of this paper.