On the Generalized Lognormal Distribution

Hindawi Publishing Corporation Journal of Probability and Statistics Volume 2013, Article ID 432642, 15 pages http://dx.doi.org/10.1155/2013/432642

Research Article On the Generalized Lognormal Distribution Thomas L. Toulias and Christos P. Kitsos Technological Educational Institute of Athens, Department of Mathematics, Ag. Spyridonos & Palikaridi Street, 12210 Egaleo, Athens, Greece Correspondence should be addressed to Thomas L. Toulias; [email protected] Received 11 April 2013; Revised 4 June 2013; Accepted 5 June 2013 Academic Editor: Mohammad Fraiwan Al-Saleh Copyright © 2013 T. L. Toulias and C. P. Kitsos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 𝛾GLD 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 N𝑝𝛾 (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 𝑓𝑋 of a 𝑝-variate, 𝛾-order, normally distributed random variable 𝑌 ∼ N𝑝𝛾 (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]. 𝑓𝑌 (𝑦) = 𝑓𝑌 (𝑦; 𝜇, Σ, 𝛾) = 𝐶𝛾𝑝 |det Σ|−1/2 exp {−

𝛾−1 𝛾/2(𝛾−1) }, 𝑄𝜃 (𝑦) 𝛾

(1)

𝑦 ∈ R𝑝 , where 𝑄𝜃 is the quadratic form 𝑄𝜃 (𝑦) = (𝑦 − 𝜇)T Σ−1 (𝑦 − 𝜇), 𝜃 = (𝜇, Σ), while 𝐶𝛾𝑝 being the normalizing factor 𝐶𝛾𝑝 = 𝜋−𝑝/2

Γ (𝑝/2 + 1) 𝛾 − 1 𝑝((𝛾−1)/𝛾)−1 . ( ) 𝛾 Γ (𝑝 ((𝛾 − 1) /𝛾))

(2)

2

Journal of Probability and Statistics

From (1), notice that the second-ordered normal is the 𝑝 known multivariate normal distribution; that is, N2 (𝜇, Σ) = 𝑝 N (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 N𝛾 (𝜇, 𝜎2 ) = N1𝛾 (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 LogUniform and Log-Laplace distributions, is presented. The generalized error function, that is briefly provided here, plays an important role in the development of LN𝛾 (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 Erf𝑎 (𝑥) :=

Γ (𝑎 + 1) 𝑥 −𝑡𝑎 ∫ 𝑒 𝑑𝑡, √𝜋 0

𝑥 ∈ R.

(3)

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 Erf𝑎 (𝑥) =

1 Γ (𝑎) Γ (𝑎) 1 𝑎 1 𝛾( ,𝑥 ) = [Γ ( ) − Γ ( , 𝑥𝑎 )] , √𝜋 √𝜋 𝑎 𝑎 𝑎 𝑥 ∈ R; (4)

see [16]. Moreover, adopting the series expansion form of the lower incomplete gamma function, 𝑥

∞

(−1) 𝑥𝑎+𝑘 , 𝑘! + 𝑘) (𝑎 𝑘=0

𝛾 (𝑎, 𝑥) := ∫ 𝑡𝑎−1 𝑒−𝑡 𝑑𝑡 = ∑ 0

𝑘

𝑥, 𝑎 ∈ R+ , (5)

a series expansion form of the generalized error function is extracted: Erf𝑎 (𝑥) =

Γ (𝑎 + 1) ∞ (−1)𝑘 𝑥𝑘𝑎+1 , ∑ √𝜋 𝑘=0 𝑘! (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) = √𝜋 erf √𝑥 and Γ(1/2, 𝑥) = √𝜋(1 − erf √𝑥) = √𝜋 erfc √𝑥, 𝑥 ≥ 0 are obtained. Moreover, Erf𝑎 0 = 0 for all 𝑎 ∈ R+ and lim Erf𝑎 𝑥 = ±

𝑥 → ±∞

1 1 Γ (𝑎) Γ ( ) , √𝜋 𝑎

as 𝛾(𝑎, 𝑥) → Γ(𝑎) when 𝑥 → +∞.

𝑎 ∈ R+ ,

(7)

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, log 𝑋 ∼ N𝛾 (𝜇, 𝜎2 ), then 𝑋 is said to follow the generalized Lognormal distribution, denoted by LN𝛾 (𝜇, 𝜎); that is, 𝑋 ∼ LN𝛾 (𝜇, 𝜎). The LN𝛾 (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 log 𝑋 is a 𝛾-order normally distributed variable; that is, log 𝑋 ∼ N𝛾 (𝜇, 𝜎2 ). Therefore, the location parameter 𝜇 ∈ R of 𝑋 is in fact the mean of 𝑋’s natural logarithm, that is, E[log 𝑋] = 𝜇, while Var [log 𝑋] = (

𝛾 2((𝛾−1)/𝛾) Γ (3 ((𝛾 − 1) /𝛾)) 2 ) 𝜎, 𝛾−1 Γ (((𝛾 − 1) /𝛾))

Kurt [log 𝑋] =

Γ (𝛾 − 1/𝛾) Γ (5 (𝛾 − 1/𝛾)) ; Γ2 (3 ((𝛾 − 1)/𝛾))

(8)

(9)

see [15] for details on N𝛾 . Let 𝑌 := log 𝑋 ∼ 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 󵄨󵄨 𝑑 󵄨󵄨 󵄨 󵄨 𝑓𝑋 (𝑥) = 𝑓𝑋 (𝑥; 𝜇, 𝜎, 𝛾) = 𝑓𝑌 (𝑔−1 (𝑥)) 󵄨󵄨󵄨 𝑔−1 (𝑥)󵄨󵄨󵄨 󵄨󵄨 𝑑𝑥 󵄨󵄨 = 𝑓𝑌 (log 𝑥) =

1 𝑥

(10)

󵄨𝛾/(𝛾−1) 󵄨 exp {− ((𝛾 − 1) /𝛾) 󵄨󵄨󵄨(log 𝑥 − 𝜇) /𝜎󵄨󵄨󵄨 } 2𝜎 ((𝛾 − 1)/𝛾)

1/𝛾

Γ ((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 variable 𝑋 + 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 when 𝑋 ∼ 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[log 𝑋] is the same as in (8). Let 𝑌 = 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.


3

Let 𝑧 = (log(𝑥 − 𝜗) − 𝜇)/𝜎. Then, the limiting threshold density value of 𝑓𝑋 (𝑥) with 𝑥 → 𝜗+ implies that lim 𝑓𝑋 (𝑥)

𝑥 → 𝜗+

= 𝜎−1 𝐶𝛾1 lim exp {−𝑧 (𝜎 + 𝑧 → −∞

𝜇 𝛾 − 1 1/(𝛾−1) )} (11) − |𝑧| 𝑧 𝛾

= 𝜎−1 𝐶𝛾1 𝑒(sgn 𝛾)(−∞) ,

(ii) The “normal” case 𝛾 = 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 case 𝛾 = ±∞: we have LN±∞ (𝜇, 𝜎) := lim𝛾 → ±∞ LN𝛾 (𝜇, 𝜎) with 𝑓𝑋±∞ (𝑥) := lim 𝑓𝑋𝛾 (𝑥) = 𝛾 → ±∞

and therefore

󵄨󵄨 log 𝑥 − 𝜇 󵄨󵄨 1 󵄨 󵄨󵄨 exp {− 󵄨󵄨󵄨 󵄨󵄨} 󵄨 󵄨󵄨 2𝜎𝑥 𝜎 󵄨

−𝜇/𝜎

0, 𝛾 ∈ (1, +∞) , lim+ 𝑓𝑋 (𝑥) = { 𝑥→𝜗 +∞, 𝛾 ∈ (−∞, 0) ;

𝑒 { (1−𝜎)/𝜎 { , 𝑥 ∈ (0, 𝑒𝜇 ] , { { 2𝜎 𝑥 = { 𝜇/𝜎 { {𝑒 { 𝑥−(𝜎+1)/𝜎 , 𝑥 > 𝑒𝜇 , { 2𝜎

(12)

(15)

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 distributions LN𝛾 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.

which coincides with the density function of the known Log-Laplace distribution (symmetric logexponential distribution) LL(𝜇󸀠 , 𝛼, 𝛽) with 𝜇󸀠 = 𝑒𝜇 and 𝛼 = 𝛽 = 1/𝜎; see [8]. Therefore, the infiniteordered log-normal distribution is in fact the LogLaplace distribution, with threshold density

Theorem 3. The generalized Lognormal distribution LN𝛾 (𝜇, 𝜎), for order values of 𝛾 = 0, 1, 2, ±∞, is reduced to

(16)

D (𝑒𝜇 ) , { { 𝜇−𝜎 𝜇+𝜎 { { {LU (𝑒 , 𝑒 ) , LN𝛾 (𝜇, 𝜎) = {LN (𝜇, 𝜎) , { { { 1 1 { LL (𝑒𝜇 , , ) , { 𝜎 𝜎

𝛾 = 0, 𝛾 = 1, 𝛾 = 2,

(13)

(i) The limiting case 𝛾 = 1: let 𝑥 ∈ R∗+ such that | log 𝑥 − 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 𝑓𝑋1 (𝑥) := lim+ 𝑓𝑋𝛾 (𝑥) 𝛾→1

𝑥 ∈ [𝑒𝜇−𝜎 , 𝑒𝜇+𝜎 ] ,

For the purposes of statistical application, the LogLaplace moments are not the same as the model parameters; that is, although 𝜇𝑋 = 𝜇, 𝜎𝑋 = √2𝜎. (iv) The limiting case 𝛾 = 0: we have

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.

{ 1 , = { 2𝜎𝑥 {0,

0, 𝜎 < 1, { { 𝑓𝑋±∞ (0) = lim+ 𝑓𝑋±∞ (𝑥) = {1, 𝜎 = 1, { 𝑥→0 {+∞, 𝜎 > 1.

(14)

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 LogUniform moments are not the same as the model parameters; that is, although 𝜇𝑋 = 𝜇, 𝜎𝑋 = 𝜎/√3.

lim 𝑓𝑋𝛾 (𝑥) =

𝛾 → 0−

lim

𝑓𝑋𝛾 (𝑥) ,

𝑘:=[(𝛾−1)/𝛾] → ∞

𝑥 ∈ R∗+ ,

(17)

where [𝑎] is the integer value of 𝑎 ∈ R. For the value 𝑥 = 𝑒𝜇 , the p.d.f. as in (10) implies lim− 𝑓𝑋𝛾 (𝑒𝜇 ) =

𝛾→0

𝑘𝑘 1 ( lim ) ⋅ 𝑒0 = +∞, 2𝜎𝑒𝜇 𝑘 → ∞ 𝑘!

(18)

through Stirling’s asymptotic formula 𝑘! ≈ √2𝜋𝑘(𝑘/ 𝑒)𝑘 as 𝑘 → ∞. Assuming now 𝑥 ≠ 𝑒𝜇 , (10), through (18), implies 󵄨 󵄨󵄨 󵄨log 𝑥 − 𝜇󵄨󵄨󵄨 1 ⋅ 0 ⋅ = 0; 𝑓𝑋0 (𝑥) := lim− 𝑓𝑋𝛾 (𝑥) = 󵄨 2 𝛾→0 𝑒 2√2𝜋𝜎 𝑥

(19)

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

4


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 LogUniform, Lognormal, Log-Laplace, and the degenerate Dirac distributions as (13) holds. From Theorem 3, (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 nonnegative-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 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(𝑒𝜇 ). Proposition 4. The mode of the positive-ordered Lognormal random variable 𝑋𝛾 ∼ LN𝛾 (𝜇, 𝜎), 𝑋 < 𝑒𝜇 𝛾 ∈ (1, +∞), is given by 𝛾

Mode 𝑋𝛾 = 𝑒𝜇−𝜎 ,

(20)

with corresponding maximum density value, max 𝑓𝑋𝛾 = 𝑓𝑋𝛾 (Mode 𝑋𝛾 ) =

exp {(𝜎𝛾 ) /𝛾 − 𝜇} 1/𝛾

2𝜎((𝛾 − 1)/𝛾)

Γ ((𝛾 − 1) /𝛾)

.

(21)

Proof. Recall the density function of 𝑋𝛾 ∼ LN𝛾 (𝜇, 𝜎) as in (10), and let 𝑚 = Mode 𝑋𝛾 > 0. Then it holds that (𝑑/𝑑𝑥) 𝑓𝑋𝛾 (𝑚; 𝜇, 𝜎, 𝛾) = 0; that is, 1/(𝛾−1) 𝑑 󵄨󵄨 𝜎 1󵄨 󵄨 󵄨 ( 󵄨󵄨󵄨log 𝑚 − 𝜇󵄨󵄨󵄨) 󵄨󵄨log 𝑥 − 𝜇󵄨󵄨󵄨𝑥=𝑚 = − . 𝜎 𝑑𝑥 𝑚 From (𝑑/𝑑𝑥)|𝑥| = sgn 𝑥, 𝑥 ∈ R, we have 𝛾

𝑚 = 𝑒𝜇−𝜎 ,

(22)

provided that 𝑥 < 𝑒 . Otherwise, (23) holds trivially, as (22) implies 𝜎 = 0; that is, 𝑚 = 𝑒𝜇 . Moreover, (𝑑/𝑑𝑥)𝑓𝑋𝛾 (𝑥) > 0 when 󵄨1/(𝛾−1) 󵄨 1 + sgn (log 𝑥 − 𝜇) 𝜎𝛾/(1−𝛾) 󵄨󵄨󵄨log 𝑥 − 𝜇󵄨󵄨󵄨 > 0, 𝑥 > 0, (24) and thus, 𝑓𝑋𝛾 is a strictly ascending density function on 𝛾

(0, 𝑒𝜇−𝜎 ) when 𝛾 > 1 and also on (𝑒𝜇−𝜎 , 𝑒𝜇 ) when 𝛾 < 0. Similarly, with (𝑑/𝑑𝑥)𝑓𝑋𝛾 (𝑥) < 0, 𝑓𝑋 is a strictly descending 𝛾

density function on (𝑒𝜇−𝜎 , +∞) when 𝛾 > 1 and also on 𝛾 (0, 𝑒𝜇−𝜎 ) ∪ (𝑒𝜇 , +∞) when 𝛾 < 0. Specifically, for 𝛾 < 0, the 𝜇 point 𝑒 is a nonsmooth point of 𝑓𝑋, as lim

𝑑

𝑥 → 𝑒𝜇 𝑑𝑥

𝑓𝑋𝛾 (𝑥) = −

𝐶𝛾1 (𝜎𝑒𝜇 )2

Proposition 5. The global mode point of the negative-ordered Lognormal random variable 𝑋𝛾 ∼ 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 1,

with the corresponding maximum density value being infinite; that is, max 𝑓𝑋𝛾 = 𝑓𝑋𝛾 (0) = +∞, provided 𝜎 < 1, and


e−2/3

e2/3

1.5 fX𝛾 ∼ 𝒩𝛾 1 is decreasing from 𝑒𝜇−𝜎 |𝛾 → −∞ = 𝑒𝜇

Proposition 6. Consider the positive-ordered Lognormal family of distributions LN𝛾 (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

It is easy to see that for the Log-Laplace case LL(𝜇, 𝜎, 𝜎), 𝛾 the local minimum density point 𝑒𝜇−𝜎 of 𝑋𝛾 1 coincides (in limit) with the local nonsmooth mode point 𝑒𝜇 of 𝑋𝛾 ; see Figure 1(b3). Also, notice that the local minimum

(i) strictly increasing from 𝑒𝜇−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 at 𝑒𝜇−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 from 𝑒𝜇−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. Let 𝑋𝛾 ∼ LN𝛾 (𝜇, 𝜎), Mode 𝑋𝛾 is a smooth monotonous function of 𝛾 ∈ (−∞, 0) ∪ (1, +∞) for positive-and negative-ordered 𝑋𝛾 , as 𝛾 𝑑 Mode 𝑋𝛾 = −𝜎𝛾 log (𝜎) 𝑒𝜇−𝜎 . 𝑑𝛾

(29)

For 𝑋±∞ ∼ LN±∞ (𝜇, 𝜎) we evaluate, through (20) and (21), that Mode 𝑋+∞

𝑒𝜇 , { { 𝜇−1 = {𝑒 , { {0,

𝜎 < 1, 𝜎 = 1, 𝜎 > 1,

(30)

with the corresponding maximum density value being infinite; that is, max 𝑓𝑋𝛾 = 𝑓𝑋𝛾 (0) = +∞, provided 𝜎 > 1, and max 𝑓𝑋+∞ = 1/(2𝜎𝑒𝜇 ), 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 LN𝛾 (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) to 𝑒𝜇−1 (Dirac case) provided that 𝜎 < 1, (ii) fixed at 𝑒𝜇−1 for all LN𝛾 (𝜇, 𝜎 = 1), (iii) strictly decreasing from 𝑒𝜇 (Log-Laplace case) to 𝑒𝜇−1 (Dirac case) provided that 𝜎 > 1.

𝛾

to Mode 𝑋0 = 𝑒𝜇−1 through (20). When 𝜎 = 1, 𝑒𝜇−𝜎 = 𝛾 𝑒𝜇−1 for all 𝛾 < 0, while for 𝜎 < 1, 𝑒𝜇−𝜎 increases from 𝛾 𝑒𝜇−𝜎 |𝛾 → −∞ = 0 to Mode 𝑋0 = 𝑒𝜇−1 through (20).

𝛾

density point 𝑒𝜇−𝜎 , 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 . Figure 1 illustrates the probability density functions 𝑓𝑋𝛾 curves for scale parameters 𝜎 = 2/3, 1, 3/2 of the positiveordered lognormally distributed 𝑋𝛾 ∼ LN𝛾≥1 (0, 𝜎) in Figures 1(a1)–1(a3), respectively, while the p.d.f. of negativeordered lognormally distributed 𝑋𝛾 ∼ LN𝛾 −𝑒𝜇 , into the other two integrals, we obtain 0

𝐹|𝑌| (𝑥) = 𝐶𝛾1 ∫

−∞

×∫

exp {−

(1/𝜎) log 2

0

𝛾 − 1 𝛾/(𝛾−1) } 𝑑𝑤 + 𝐶𝛾1 |𝑤| 𝛾 exp {−

𝑔(𝑥)

+ 𝐶𝛾1 ∫

(1/𝜎) log 2

𝑔 (𝑥) :=

𝜇

log (𝑥 + 𝑒 ) − 𝜇 , 𝜎

𝛾 − 1 𝛾/(𝛾−1) } 𝑑𝑧 |𝑧| 𝛾

exp {−

𝛾 − 1 𝛾/(𝛾−1) } 𝑑𝑧, |𝑧| 𝛾

𝑥 ∈ R+ , (56)

(53) and hence

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, 𝑓 (−𝑥) + 𝑓𝑌 (𝑥) , 0 ≤ 𝑥 ≤ 𝑒𝜇 , 𝑓|𝑌| (𝑥) = { 𝑌 𝑓𝑌 (𝑥) , 𝑥 > 𝑒𝜇 .

(54)

𝐹|𝑌| (𝑥) = Φ𝑍 (0) + [Φ𝑍 (

log 2 ) − Φ𝑍 (0)] 𝜎

+ [Φ𝑍 (𝑔 (𝑥)) − Φ𝑍 ( = Φ𝑍 (

log 2 )] 𝜎

log (𝑥 + 𝑒𝜇 ) − 𝜇 ), 𝜎

(57)

10

Journal of Probability and Statistics e−2/3

e2/3

0.9

0.9

0.8

0.8

0.7

0.7

0.6 0.5 0.4 0.3

0.5 0.4 0.3 0.2

0.1

0.1 1

2

x

X1 ∼ ℒ𝒰(e−2/3 , e2/3 ) X1.1,1.2,...,1.9 X2 ∼ℒ𝒩(0, 2/3) X3,4,...,10 X+∞ ∼ ℒℒ(1, 3/2, 3/2)

0

3

e1

0.6

0.2 0 0

e−1

1

FX𝛾 ∼ 𝒩𝛾 (0, 1)

FX𝛾 ∼ 𝒩𝛾 (0, 2/3)

1

1

0

x

X1 ∼ ℒ𝒰(e−1 , e1 ) X1.1,1.2,...,1.9 X2 ∼ℒ𝒩(0, 1) X3,4,...,10 X+∞ ∼ ℒℒ(1, 1, 1)

X−∞ ∼ ℒℒ(1, 3/2, 3/2) X−10,−9,...,−1 X−0.9,−0.8,...,−0.1

2

3

X−∞ ∼ ℒℒ(1, 1, 1) X−10,−9,...,−1 X−0.9,−0.8,...,−0.1

(b)

(a) e−3/2

1 0.9

FX𝛾 ∼ 𝒩𝛾 (0, 3/2)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

x X1 ∼ ℒ𝒰(e−3/2 , e3/2 ) X1.1,1.2,...,1.9 X2 ∼ℒ𝒩(0, 3/2) X3,4,...,10 X+∞ ∼ ℒℒ(1, 2/3, 2/3)

X−∞ ∼ ℒℒ(1, 2/3, 2/3) X−10,−9,...,−1 X−0.9,−0.8,...,−0.1

(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.

with Φ𝑍 being the c.d.f. of the standardized r.v. 𝑍 ∼ N𝛾 (0, 1). From (38) and the fact that Erf𝑎 0 = 0, 𝑎 ∈ R∗+ , it is clear that −1 (57) implies MAD𝑋𝛾 = Med|𝑋𝛾 − 𝑒𝜇 | = 𝐹|𝑋 𝜇 (1/2) = 0, for 𝛾 −𝑒 | every 𝑋𝛾 ∼ LN𝛾 (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.


11

(𝑡) Proposition 16. The 𝑡th raw moment 𝜇̃𝑋 of a generalized lognormally distributed random variable 𝑋 ∼ LN𝛾 (𝜇, 𝜎) is given by

× Γ ((2𝑛 + 1)

𝑛

2 2 𝑒𝑡𝜇 ∞ (2𝑡 𝜎 ) 1 = Γ (𝑛 + ) , ∑ √𝜋 𝑛=0 (2𝑛)! 2

(𝑡) 𝜇̃𝑋

∞ 𝛾 2𝑛((𝛾−1)/𝛾) 𝑒𝑡𝜇 (𝑡𝜎)2𝑛 = ) ( ∑ Γ ((𝛾 − 1) /𝛾) 𝑛=0 (2𝑛)! 𝛾 − 1

(𝑡) 𝜇̃𝑋

Example 17. For the second-ordered lognormally distributed 𝑋 ∼ LN2 (𝜇, 𝜎), (58) implies

(58)

𝛾−1 ) 𝛾

and coincides with the moment generating function of the 𝛾(𝑡) order normally distributed log 𝑋; that is, 𝑀log 𝑋 (𝑡) = 𝜇̃𝑋 .

R+

=

𝛾 − 1 󵄨󵄨󵄨󵄨 log 𝑥 − 𝜇 󵄨󵄨󵄨󵄨𝛾/(𝛾−1) 1 1 } 𝑑𝑥, 𝐶𝛾 ∫ 𝑥𝑡−1 exp {− 󵄨 󵄨󵄨 󵄨󵄨 𝜎 𝛾 󵄨󵄨󵄨 𝜎 R+ (59) (𝛾−1)/𝛾

and applying the transformation 𝑧 = ((𝛾 − 1)/𝛾) (log 𝑥 − 𝜇), 𝑥 > 0, we get (𝑡) 𝜇̃𝑋

=

𝐶𝛾1 (

(1/𝜎)

𝛾 (𝛾−1)/𝛾 𝛾 (𝛾−1)/𝛾 𝜎𝑧} ) ) ∫ exp {𝑡𝜇 + 𝑛( 𝛾−1 𝛾−1 R

1 (2𝑛)! Γ (𝑛 + ) = 2𝑛 √𝜋, 2 2 𝑛!

(60) Through the exponential series expansion exp {𝑡(

𝛾 ) 𝛾−1

∞

𝑛((𝛾−1)/𝛾)

𝑛

𝛾 (𝑡𝜎) ( ) 𝑛! 𝛾 − 1 𝑛=0

𝜎𝑧} = ∑

𝑧𝑛 , (61)

we have ∞

∞ 1 1 2 2 𝑛 (𝑡𝜎)2𝑛 𝑡𝜇 = 𝑒 ( 𝑡𝜎) ∑ 𝑛 𝑛=0 2 𝑛! 𝑛=0 𝑛! 2

(𝑡) 𝜇̃𝑋 = 𝑒𝑡𝜇 ∑

𝑡𝜇+(1/2)(𝑡𝜎)2

=𝑒

𝑡 ∈ R+ ,

which is the 𝑡th raw moment of the usual lognormally (1) distributed 𝑋 ∼ LN(𝜇, 𝜎), with mean 𝜇𝑋 := 𝜇̃𝑋 = E[𝑋] = (𝑡) 2 exp{𝜇 + (1/2)𝜎 }. This is true as 𝑀log 𝑋 (𝑡) = 𝜇̃𝑋 = exp{𝑡𝜇 + (1/2)(𝑡𝜎)2 } is the known moment-generating function of the normally distributed log 𝑋 ∼ N(𝜇, 𝜎2 ). (𝑡) Theorem 18. The 𝑘th central moment (about the mean) 𝜇𝑋 of a generalized lognormally distributed random variable 𝑋 ∼ LN𝛾 (𝜇, 𝜎) is given by (𝑘) 𝜇𝑋 =

𝑘 𝜇 𝑛 𝑒𝑘𝜇 𝑘 ) 𝑆𝑘−𝑛 , ∑ ( ) (− 𝑋 𝑒𝜇 Γ ((𝛾 − 1) /𝛾) 𝑛=0 𝑛

𝑘 ∈ N, (67)

𝛾 2𝑚((𝛾−1)/𝛾) 𝛾−1 (𝑘𝜎)2 𝑚 ( Γ ((2 𝑚 + 1) ) ), 𝛾−1 𝛾 𝑚=0 (2 𝑚)! ∞

𝑆𝑘 = ∑

𝑘 ∈ N. (68) (𝑘) Proof. From the definition of the 𝑘th central moment 𝜇𝑋 we have

∞

2𝑛

𝑘

𝛾 (𝑡𝜎) ( ) 𝛾 − 1 (2𝑛)! 𝑛=0

𝑒𝑡𝜇 ∑

(69)

2𝑛((𝛾−1)/𝛾)

while using the binomial identity we get 𝑘

𝑘 𝑛 (𝑘) 𝜇𝑋 = ∑ ( ) (−𝜇𝑋 ) ∫ 𝑥𝑘−𝑛 𝑓𝑋 (𝑥) 𝑑𝑥 𝑛 R+ 𝑛=0

× ∫ 𝑧2𝑛 exp {−𝑧𝛾/(𝛾−1) } 𝑑𝑧. R+

(62) Finally, substituting the normalizing factor 𝐶𝛾1 as in (2) into (62) and utilizing the known integral [16],

R+

(66)

R+

𝛾 ) 𝛾−1

∫ 𝑥𝑚 𝑒−𝑏𝑥 𝑑𝑥 =

,

𝑘

(𝛾−1)/𝛾

𝑛

(65)

(𝑘) := E [(𝑋 − 𝜇𝑋 ) ] = ∫ (𝑥 − 𝜇𝑋 ) 𝑓𝑋 (𝑥; 𝜇, 𝜎, 𝛾) 𝑑𝑥, 𝜇𝑋

it is obtained that (𝑡) = 2𝐶𝛾1 ( 𝜇̃𝑋

𝑛 ∈ N,

where

× exp {−|𝑧|𝛾/(𝛾−1) } 𝑑𝑧.

(𝛾−1)/𝛾

(64)

and through the gamma function identity

(𝑡) , we Proof. From the definition of the 𝑡th raw moment 𝜇̃𝑋 have (𝑡) = E [𝑋𝑡 ] = ∫ 𝑥𝑡 𝑓𝑋 (𝑥) 𝑑𝑥 𝜇̃𝑋

𝑡 ∈ R+ ,

Γ ((𝑚 + 1) /𝑛) , 𝑛𝑏(𝑚+1)/𝑛

𝑛, 𝑚, 𝑏 ∈ R∗+ ,

(63)

we obtain (58). Moreover, for 𝑌 := log 𝑋 ∼ N𝛾 (𝜇, 𝜎2 ) we have 𝑀𝑌 (𝑡) = (𝑡) E[𝑒𝑡𝑌 ] = E[𝑋𝑡 ] = 𝜇̃𝑋 , and the proposition has been proved.

𝑘

(70)

𝑘 𝑛 (𝑘−𝑛) . = ∑ ( ) (−𝜇𝑋 ) 𝜇̃𝑋 𝑛 𝑛=0 Applying Proposition 16, (70) implies that (𝑘) 𝜇𝑋 =

𝑘 𝜇 𝑛 ∞ [(𝑘 − 𝑛) 𝜎]2 𝑚 𝑒𝑘𝜇 𝑘 ) ∑ ∑ ( ) (− 𝑋 𝑒𝜇 𝑚=0 (2 𝑚)! Γ ((𝛾 − 1) /𝛾) 𝑛=0 𝑛

×(

𝛾 2 𝑚((𝛾−1)/𝛾) Γ ((2 𝑚 + 1) ((𝛾 − 1) /𝛾)) , ) 𝛾−1 (71)

12


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 2 mean 𝜇𝑋 = 𝑒𝜇+(1/2)𝜎 into (70), the second-ordered lognormally distributed 𝑋 ∼ LN2 (𝜇, 𝜎) provides (𝑘) 𝜇𝑋

𝑘

2 2 𝑘 = ∑ ( ) (−1)𝑛 𝑒𝑘𝜇+(1/2)[𝑛+(𝑘−𝑛) ]𝜎 , 𝑛 𝑛=0

𝑘 ∈ N,

(72)

2

2

2 (2) := Var [𝑋] = 𝜇𝑋 = 𝑒2𝜇+𝜎 (𝑒𝜎 − 1) , 𝜎𝑋

(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). 2 := Var[𝑋], Theorem 20. The mean 𝜇𝑋 := E [𝑋], variance 𝜎𝑋 coefficient of variation 𝐶𝑉𝑋 , skewness 𝜆 𝑋 and kurtosis 𝜅𝑋 of the generalized lognormally distributed 𝑋 ∼ LN𝛾 (𝜇, 𝜎) are, respectively, given by

𝜇𝑋 =

𝑒𝜇 𝑆, Γ ((𝛾 − 1) /𝛾) 1

2 2 𝜎𝑋 = − 𝜇𝑋 +

𝐶𝑉𝑋2 = Γ (

4𝜇

𝛾 − 1 −1 )] 𝛾

3𝜇

× (𝑒 𝑆4 − 4𝑒 𝜇𝑋 𝑆3 +

(81)

2 6𝑒2𝜇 𝜇𝑋 𝑆2

−

3 4𝑒𝜇 𝜇𝑋 𝑆1 ) .

𝑒 𝑆, Γ ((𝛾 − 1) /𝛾) 2

(75)

𝛾 − 1 𝑆2 ) 2 − 1, 𝛾 𝑆1

(76) 𝑒3𝜇 𝑆, 3 𝜎𝑋 Γ ((𝛾 − 1) /𝛾) 3

𝜅𝑋 = − 𝐶𝑉𝑋−4 − 6𝐶𝑉𝑋−2 − 4

(77)

𝜆𝑋 𝑒4𝜇 + 4 𝑆, 𝐶𝑉𝑋 𝜎𝑋 Γ ((𝛾 − 1) /𝛾) 4 (78)

where the sums 𝑆𝑖 , 𝑖 = 1, . . . , 4, are given by (68). Proof. From Proposition 16 we easily obtain (74), as 𝜇𝑋 := (1) . From Theorem 18 we have 𝜇̃𝑋 (2) 𝜇𝑋

=

2 𝜇𝑋

𝛾 − 1 −1 2𝜇 + [Γ ( )] (𝑒 𝑆2 − 2𝑒𝜇 𝜇𝑋 𝑆1 ) . (79) 𝛾

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 (3) 3 third (central) moment; that is, 𝜆 𝑋 := Skew[𝑋] = 𝜇𝑋 /𝜎𝑋 . Theorem 18 provides that 𝜆𝑋 =

−𝐶𝑉𝑋−3 3𝜇

+

Example 21. For the second-ordered lognormally distributed 𝑋 ∼ LN2 (𝜇, 𝜎), utilizing (65) into (68) we get 𝑆𝑛 = 2 2 √𝜋𝑒(𝑛 𝜎 )/2 , 𝑛 ∈ N∗ . Applying this to Theorem 20 we derive (after some algebra) 2

𝜇𝑋 = 𝑒𝜇+(1/2)𝜎 ,

2

𝛾 3 [𝜎𝑋 Γ( 2𝜇

− 1 −1 )] 𝛾 𝜇

× (𝑒 𝑆3 − 3𝑒 𝜇𝑋 𝑆2 + 3𝑒 𝜇𝑋 𝑆1 ) .

2

2 𝜎𝑋 = 𝑒2𝜇+𝜎 (𝑒𝜎 − 1) ,

𝐶𝑉𝑋 = √𝑒𝜎 − 1, 2

2

2

𝜆 𝑋 = (𝑒𝜎 + 2) √𝑒𝜎 − 1,

2

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 Mode 𝑋 < Med 𝑋 < 𝜇𝑋 . The following corollary examines this inequality for the LN𝛾 family of distributions. Corollary 22. For the 𝛾-ordered lognormally distributed 𝑋𝛾 ∼ LN𝛾 (𝜇, 𝜎), it is true that Mode 𝑋𝛾 ≤ Med 𝑋𝛾 = (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 1: from (20) we have 𝛾

Mode 𝑋𝛾 = 𝑒𝜇−𝜎 < 𝑒𝜇 = Med 𝑋𝛾 . (80)

2

𝜅𝑋 = 𝑒4𝜎 + 2𝑒3𝜎 + 3𝑒2𝜎 − 3, (82)

2

(74)

2𝜇

𝜆 𝑋 = − 𝐶𝑉𝑋−3 − 𝐶𝑉𝑋−1 +

:=

4 Γ( 𝜅𝑋 = 𝐶𝑉𝑋−4 + [𝜎𝑋

Substituting 𝑆𝑖 , 𝑖 = 1, 2, 3, from (74), (75), and (77), we obtain (78).

while

2 𝜎𝑋

Substituting 𝑆1 and 𝑆2 from (74) and (75), we obtain (77). Finally, kurtosis 𝜅𝑋 is (by definition) the standardized fourth (4) 4 (central) moment; that is, 𝜅𝑋 := Kurt[𝑋] = 𝜇𝑋 /𝜎𝑋 , which provides, through Theorem 18, that

(84)

For the Log-Laplace case of 𝑋+∞ , it holds Mode 𝑋𝛾 = 𝑒𝜇 = Med 𝑋𝛾 ,

(85)


13

provided that 𝜎 < 1, while for 𝜎 ≥ 1 we have 𝜇

Mode 𝑋𝛾 = 0 < 𝑒 = Med 𝑋𝛾 .

(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 Mode 𝑋𝛾 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 variable 𝑋 ∼ LU(𝑎, 𝑏), 0 < 𝑎 < 𝑏, are given by (𝑡) = 𝜇̃𝑋

𝑏𝑡 − 𝑎𝑡 , 𝑡 log (𝑏/𝑎)

(𝑘) 𝜇𝑋 =

(𝑎 − 𝑏)𝑘 log𝑘 (𝑏/𝑎)

𝑡 ∈ R, +

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 (𝑡) (𝑡) = lim+ 𝜇𝑋 = 𝜇̃𝑋 𝛾 𝛾→1

𝑡 ∈ R, (95) (1) and hence (87) holds. Moreover, 𝜇𝑋 := 𝜇𝑋 = E[𝑋] = (1/2𝜎) 𝜇+𝜎 𝜇−𝜎 (𝑒 − 𝑒 ), and therefore (89) holds. Working similarly, (67) implies 𝑘 𝜇𝑋 𝑛 ∞ [(𝑘 − 𝑛) 𝜎]2𝑚 𝑘 (𝑘) (𝑘) 𝑘𝜇 𝜇𝑋 , = lim+ 𝜇𝑋 = 𝑒 ( ) ∑ ) (− ∑ 𝛾 𝑛 𝛾→1 𝑒𝜇 𝑚=0 (2𝑚 + 1)! 𝑛=0

𝑘 ∈ N. (96)

(87)

1 log (𝑏/𝑎)

Using the exponential odd series expansion, the above expansion becomes

𝑛 𝑘−𝑛 𝑘−𝑛 𝑘−1 𝑘 (𝑎 − 𝑏) (𝑏 − 𝑎 ) , ×∑( ) 𝑛 𝑛 (𝑘 − 𝑛) log (𝑏/𝑎) 𝑛=0

𝑘 ∈ N,

(𝑘) 𝜇𝑋 =

(88) respectively, while the mean, variance, coefficient of variation, skewness, and kurtosis of 𝑋 are given, respectively, by 𝑏−𝑎 , log (𝑏/𝑎)

(89)

(𝑏 − 𝑎)2 (𝑏 − 𝑎) (𝑏 + 𝑎) , + 2 2 log (𝑏/𝑎) log (𝑏/𝑎)

(90)

𝜇𝑋 = 2 𝜎𝑋 =

𝐶𝑉𝑋 = √ 1 + 𝜆𝑋 =

𝑏+𝑎 𝑏 log , 2 (𝑏 − 𝑎) 𝑎

(𝑏 − 𝑎) (𝑏3 − 𝑎3 ) 1 𝑏4 − 𝑎4 [ − 4 4 4 log (𝑏/𝑎) 𝜎𝑋 3 log2 (𝑏/𝑎) (𝑏 − 𝑎)3 (𝑏 + 𝑎) (𝑏 − 𝑎)4 +3 − 3 ]. log3 (𝑏/𝑎) log4 (𝑏/𝑎)

(93)

Corollary 24. The raw and central moments of a Log-Laplace distributed random variable 𝑋 ∼ LL(𝜇, 𝜎, 𝜎) are given by (𝑡) = 𝜇̃𝑋

𝛾−1 + 1) . 𝛾 (94)

𝜇𝑡 𝜎2 > 𝜇𝑡 , 𝜎2 − 𝑡2

𝜎 > 𝑡, 𝑡 ∈ R,

𝑘 𝜎2(𝑛+1) 𝑘 (𝑘) , = 𝜇𝑘 ∑ ( ) 𝜇𝑋 𝑛 (1 − 𝜎2 )𝑛 [𝜎2 − (𝑘 − 𝑛)2 ] 𝑛=0

(98)

𝜎 > 𝑘, 𝑘 ∈ N. (99)

The mean, variance, coefficient of variation, skewness, and kurtosis of 𝑋 are given, respectively, by 𝜇𝑋 = 2 = 𝜎𝑋

∞ 𝛾 2𝑚((𝛾−1)/𝛾) 𝑒𝑡𝜇 (𝑡𝜎)2𝑚+1 ) ( ∑ 𝑡𝜎Γ ((𝛾 − 1) /𝛾 + 1) 𝑚=0 (2𝑚 + 1)! 𝛾 − 1

× Γ ((2𝑚 + 1)

𝑘 ∈ N,

2 := and, through (89), we obtain (88). Moreover, for 𝑘 = 2, 𝜎𝑋 (2) 2 Var[𝑋] = 𝜇𝑋 − 𝜇𝑋 implies (90), and hence (91) also holds. (3) (4) For 𝑘 = 3 and 𝑘 = 4, through 𝜇𝑋 and 𝜇𝑋 , we obtain (92) and (93), respectively.

Proof. Recall Proposition 16 with 𝑋𝛾 ∼ LN𝛾 (𝜇, 𝜎). Through the gamma function additive identity (58) can be written as (𝑡) = 𝜇̃𝑋 𝛾

𝜇 𝑛 𝑒(𝑘−𝑛)𝜎 − 𝑒−(𝑘−𝑛)𝜎 𝑒𝑘𝜇 𝑘 𝑘 , ) ∑ ( ) (− 𝑋 2𝜎 𝑛=0 𝑛 𝑒𝜇 (𝑘 − 𝑛)

(97)

(91)

𝑏3 − 𝑎3 1 (𝑏 − 𝑎)2 (𝑏 + 𝑎) (𝑏 − 𝑎)3 [ ], −3 +2 3 3 2 𝜎𝑋 3 log (𝑏/𝑎) 2 log (𝑏/𝑎) log (𝑏/𝑎) (92)

𝜅𝑋 =

𝑒𝑡𝜇 ∞ (𝑡𝜎)2𝑚+1 𝑒𝑡(𝜇+𝜎) − 𝑒𝑡(𝜇−𝜎) , = ∑ 𝑡𝜎 𝑚=0 (2𝑚 + 1)! 2𝑡𝜎

𝜇2 𝜎2 (2𝜎2 + 1) (𝜎2 − 4) (𝜎2 − 1)

𝐶𝑉𝑋 = 𝜆𝑋 =

𝜇𝜎2 > 𝜇, 𝜎2 − 1

1 √ 2𝜎2 + 1 , 𝜎 𝜎2 − 4

2 (15𝜎4 + 7𝜎2 + 2) 𝜎 (𝜎2 − 9)

√

2

𝜎 > 1,

(100)

,

(101)

𝜎 > 2,

(102)

𝜎 > 2,

𝜎2 − 4 3

(2𝜎2 + 1)

,

𝜎 > 3,

(103)

14


𝜅𝑋 =

3 (8𝜎8 + 212𝜎6 + 95𝜎4 + 33𝜎2 + 12) (𝜎2 − 4) (𝜎2

−

16) (𝜎2

−

9) (2𝜎2

2

+ 1)

,

(104)

𝜎 > 4. Proof. Let 𝑋𝛾 ∼ LL𝛾 (𝜇, 𝜎, 𝜎) = LN𝛾 (log 𝜇, 1/𝜎, 1/𝜎). For 𝛾 = ±∞, that is, 𝛾/(𝛾 − 1) = 1, the raw moments as in (58) provide ∞ 𝑡 2𝑘 (𝑡) (𝑡) 𝑡 = 𝜇̃𝑋 = 𝜇 ( ) , 𝜇̃𝑋 ∑ ±∞ 𝜎 𝑘=0

Acknowledgment 𝑡 ∈ R,

(105)

as 𝑋 = 𝑋±∞ , while through the even geometric series expansion, it is 1 𝑡 ∞ 𝑡 𝑘 ∞ 𝑡 𝑘 (𝑡) = [ ( + (− 𝜇 ) ) ] 𝜇̃𝑋 ∑ ∑ ±∞ 2 𝜎 𝜎 𝑘=0 𝑘=0

(106)

𝜎 1 𝜎 = 𝜇𝑡 ( + ), 2 𝜎−𝑡 𝜎+𝑡 provided that 𝜎 > 𝑡, and hence (98) holds. Moreover, 𝜇𝑋 := (1) = E[𝑋], and hence (100) holds. 𝜇̃𝑋 Working similarly, (67) implies 𝑛

𝑘 (−𝜇𝑋 /𝜇) 𝑘 (𝑘) = 𝜇𝑘 𝜎2 ∑ ( ) , 𝜇𝑋 2 𝑛 𝜎 − (𝑘 − 𝑛)2 𝑛=0

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.

𝑘 ∈ N,

(107)

provided 𝜎 > 𝑘, and hence, through (100), the central moments (99) are obtained. (2) 2 2 := Var[𝑋] = 𝜇̃𝑋 − 𝜇𝑋 , Moreover, for 𝑘 = 2 and due to 𝜎𝑋 (101) holds true, while for 𝑘 = 3 and 𝑘 = 4 we derive, through (3) (4) and 𝜇𝑋 , (103) and (104), respectively. 𝜇𝑋 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, (𝑡) = (𝑒𝑡𝑏 − 𝑒𝑡𝑎 )(1/𝑡(𝑏 − 𝑎)). 𝑀𝑈(𝑡) := E[𝑒𝑡𝑈] = 𝜇̃LU 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 momentgenerating function of the Laplace distributed 𝐿 ∼ L(𝜇, 𝜎); (𝑡) = 𝑒𝑡𝜎 (1 − 𝑡2 𝜎2 )−1 . that is, 𝑀𝐿 (𝑡) := E[𝑒𝑡𝐿 ] = 𝜇̃LL

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

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

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