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Journal of Probability and Statistics
Volume 2013 (2013), Article ID 432642, 15 pages
On the Generalized Lognormal Distribution
Technological Educational Institute of Athens, Department of Mathematics, Ag. Spyridonos & Palikaridi Street, 12210 Egaleo, Athens, Greece
Received 11 April 2013; Revised 4 June 2013; Accepted 5 June 2013
Academic Editor: Mohammad Fraiwan Al-Saleh
Copyright © 2013 Thomas L. Toulias and Christos 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 -order generalized lognormal distribution (-GLD). Under certain values of the extra shape parameter , the usual lognormal, log-Laplace, and log-uniform distribution, are obtained, as well as the degenerate Dirac distribution. The shape of all the members of the -GLD family is extensively discussed. The cumulative distribution function is evaluated through the generalized error function, while series expansion forms are derived. Moreover, the moments for the -GLD are also studied.
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 . Also, it plays an important role in Astrophysics and Cosmology; see [2–4] among others, while for Lognormal expansions see .
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  attracts attention quite often in environmental sciences, physics, economics, and finance as well as in longitudinal studies . Recently, Log-Laplace distributions have been proposed for modeling growth rates as stock prices  and currency exchange rates .
In this paper a generalized form of Lognormal distribution is introduced, involving a third shape parameter. With this generalization, a family of distributions is emerged, which combines theoretically all the properties of Lognormal, Log-Uniform, and Log-Laplace distributions, depending on the value of this third parameter.
The generalized -order Lognormal distribution (-GLD) is the distribution of a random vector whose logarithm follows the -order normal distribution, an exponential power generalization of the usual normal distribution, introduced by [11, 12]. This family of -dimensional generalized normal distributions, denoted by , is equipped with an extra shape parameter and constructed to play the role of normal distribution for the generalized Fisher’s entropy type of information; see also [13, 14].
The density function of a -variate, -order, normally distributed random variable , with location vector , positive definite scale matrix , and shape parameter , is given by . where is the quadratic form , , while being the normalizing factor
In Section 2, a generalized form of the Lognormal distribution is introduced, which is derived from the univariate family of distributions, denoted by , and includes the Log-Laplace distribution as well as the Log-Uniform distribution. The shape of the members is extensively discussed while it is connected to the tailing behavior of 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 ; see Section 2. The generalized error function denoted by and the generalized complementary error function , , are defined, respectively, as The generalized error function can be expressed (changing to variable), through the lower incomplete gamma function or the upper (complementary) incomplete gamma function , in the form see . Moreover, adopting the series expansion form of the lower incomplete gamma function, a series expansion form of the generalized error function is extracted: Notice that, is the known error function , that is, , while is the function of a straight line through the origin with slope . Applying , the known incomplete gamma function identities such as and are obtained. Moreover, for all and as when .
2. The -Order Lognormal Distribution
The generalized univariate Lognormal distribution is defined, through the univariate generalized -order normal distribution, as follows.
Definition 1. When the logarithm of a random variable follows the univariate -order normal distribution, that is, , then is said to follow the generalized Lognormal distribution, denoted by ; that is, .
The is referred to as the (generalized) -order Lognormal distribution (-GLD). Like the usual Lognormal distribution, the parameter is considered to be log-scaled, while the non log-scaled (i.e. when is assumed log-scaled) is referred to as the location parameter of . Hence, if , then is a -order normally distributed variable; that is, . Therefore, the location parameter of is in fact the mean of ’s natural logarithm, that is, , while see  for details on .
Let with density function as in (1) and . Then, the density function of can be written, through (1), as The probability density function , as in (10), is defined in ; that is, has zero threshold. Therefore, the following definition extends Definition 1.
Definition 2. When the logarithm of a random variable follows the univariate -order normal distribution, that is, , then is said to follow the generalized Lognormal distribution with threshold ; that is, .
It is clear that when , is a -order normally distributed variable, that is, , and thus, is the mean of ’s natural logarithm while is the same as in (8).
Let . The density function of is given by , .
Let . Then, the limiting threshold density value of with implies that and therefore that is, the ’s defining domain, for the positive-ordered Lognormal random variable , can be extended to include threshold point by letting .
The generalized Lognormal family of distributions is a wide range family bridging the Log-Uniform , Lognormal , and Log-Laplace distributions, as well as the degenerate Dirac distributions. We have the following.
Theorem 3. The generalized Lognormal distribution , for order values of , is reduced to
Proof. From definition (1) of the order value is a real number outside the closed interval . Let with density function as in (10). We consider the following cases.(i) The limiting case : let such that . Using the gamma function additive identity , , in (10), we have with which is the density function of the Log-Uniform distribution , , with and ; that is, and . Therefore, first-ordered Lognormal distribution is in fact the Log-Uniform distribution, with vanishing threshold density, . For the purposes of statistical application the Log-Uniform moments are not the same as the model parameters; that is, although , .(ii) The “normal” case : it is clear that , as coincides with the Lognormal density function, and therefore the second-ordered Lognormal distribution is in fact the usual Lognormal distribution.(iii) The limiting case : we have with which coincides with the density function of the known Log-Laplace distribution (symmetric log-exponential distribution) with and ; see . Therefore, the infinite-ordered log-normal distribution is in fact the Log-Laplace distribution, with threshold density For the purposes of statistical application, the Log-Laplace moments are not the same as the model parameters; that is, although , .(iv) The limiting case : we have where is the integer value of . For the value , the p.d.f. as in (10) implies through Stirling’s asymptotic formula as . Assuming now , (10), through (18), implies that is, as coincides with the Dirac density function, with the (non-log-scaled) location parameter of being the singular (infinity) point. Therefore, the zero-ordered Lognormal distribution is in fact the degenerate Dirac distribution with pole at the location parameter of (with vanishing threshold density ). From the above limiting cases (i), (iii), and (iv), the defining domain of the order values , used in (1), is safely extended to include the values ; that is, can now be defined outside the open interval . Eventually, the family of the -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 , , can also be extended to include the threshold point by setting for all non-negative-ordered Lognormals, that is, for all , while for the Log-Laplace case of with by setting .
Proposition 4. The mode of the positive-ordered Lognormal random variable , , is given by with corresponding maximum density value,
Proof. Recall the density function of as in (10), and let . Then it holds that ; that is, From , , we have provided that . Otherwise, (23) holds trivially, as (22) implies ; that is, . Moreover, when and thus, is a strictly ascending density function on when and also on when . Similarly, with , is a strictly descending density function on when and also on when . Specifically, for , the point is a nonsmooth point of , as 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).
Proposition 5. The global mode point of the negative-ordered Lognormal random variable , , is (in limit) the threshold which is an infinite (probability) density point. Moreover, the location parameter (i.e., ) of is a nonsmooth (local) mode point for all that corresponds to locally maximum density while is a local minimum (probability) density point with corresponding locally minimum density .
Proof. The negative-ordered Lognormals are formed by density functions admitting threshold 0 (in limit) for their global mode point (of infinite density), as shown in (12). Moreover, from the previously discussed monotonicity of in Proposition 4, all the negative-ordered Lognormals admit also as a local nonsmooth mode point and as a local minimum density point, with densities as in (26) and , respectively; see Figures 1(b1), 1(b2), and 1(b3).
Furthermore, Proposition 5 holds (in limit) for random variables and which provide Log-Uniform and Log-Laplace distributions, respectively. Indeed, for given , with and , we get, through (20) and (21), that with corresponding maximum density (i.e., the maximum value of the density function), Moreover, the nonzero minimum density (i.e., the minimum, but not zero, value of the density function) is obtained at with . These results are in accordance with the Log-Uniform density function in (14).
For , we evaluate, through (20) and (21), that with the corresponding maximum density value being infinite; that is, , provided , and , provided . The same result can also be derived through (26) as . These results are in accordance with the Log-Laplace density function in (15), although for , can be defined, through (15), for any value inside the interval .
The above discussion on behavior of the modes with respect to shape parameter is formed in the following propositions.
Proposition 6. Consider the positive-ordered Lognormal family of distributions with fixed parameters , and . When rises, that is, when one moves from Log-Uniform to Log-Laplace distribution inside the family, the mode points of are(i) strictly increasing from (Log-Uniform case) to (Log-Laplace case) provided that (with their corresponding maximum density values moving smoothly from to ), (ii) fixed at for all (with the corresponding maximum density values moving smoothly from to ), (iii)strictly decreasing from (Log-Uniform case) to threshold (Log-Laplace case) provided that (with their corresponding maximum density values moving smoothly from to ).
Proof. Let , is a smooth monotonous function of for positive-and negative-ordered , as
For we evaluate, through (20) and (21), that
with the corresponding maximum density value being infinite; that is, , provided , and , provided .
Assume that . 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 with , when rises, that is, when one moves from Log-Laplace to degenerate Dirac distribution inside the family, the local minimum (probability) density points of are(i) strictly increasing from threshold 0 (Log-Laplace case) to (Dirac case) provided that ,(ii) fixed at for all ,(iii) strictly decreasing from (Log-Laplace case) to (Dirac case) provided that .
Proof. Assume now that . From (29) we have when and when . Therefore, the local minimum density point (see Proposition 5) for is decreasing from to through (20). When , for all , while for , increases from to through (20).
It is easy to see that for the Log-Laplace case , the local minimum density point of with coincides (in limit) with the local nonsmooth mode point of ; see Figure 1(b3). Also, notice that the local minimum density point , , for the Dirac case , is the limiting point although the (probability) density in case vanishes everywhere except at the infinite pole .
Figure 1 illustrates the probability density functions curves for scale parameters of the positive-ordered lognormally distributed in Figures 1(a1)–1(a3), respectively, while the p.d.f. of negative-ordered lognormally distributed are depicted in Figures 1(b1)–1(b3), respectively. Moreover, the density points on are also depicted (small circles over p.d.f. curves with their corresponding ticks on -axis). According to Proposition 7, in Figures 1(a1)–1(a3), that is, for positive-ordered , these density points represent the mode points on while in Figures 1(b1)–1(b3), that is, for negative-ordered , represent the local minimum density points on 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. of a -order Lognormal random variable is given by
Proof. From density function , as in (10), we have Applying the transformation , , the above c.d.f. is reduced to where is the c.d.f. of the standardized -order normal distribution . Moreover, can be expressed in terms of the generalized error function. In particular, and as is a symmetric density function around zero, we have and thus Substituting the normalizing factor, as in (2), and using (3), we obtain and finally, through (34), we derive (31), which forms (32) through (4).
As the generalized error function is defined in (4), through the upper incomplete gamma function , 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. can be expressed in the series expansion form
Corollary 10. For the negative-ordered lognormally distributed random variable with , , , the finite expansion is obtained as
Example 11. For the -ordered lognormally distributed (i.e., for ), we have while for the -ordered lognormally distributed (i.e., for ), we have
Example 13. For the infinite-ordered Lognormal , setting , we obtain through (41) and the exponential series expansion that and hence which is the c.d.f. of the Log-Laplace distribution as in (15). This is expected as ; 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 , which provides (in limit) the c.d.f. of the infinite-ordered Lognormal distribution.
Example 14. Similarly, for the first-ordered random variable , the expansion (41) can be written as and provided that , we obtain with and . Therefore, coincides with the c.d.f. of the Log-Uniform distribution as in (15). This is expected as ; see Theorem 3.
Table 1 provides the probability values , , for various . Notice that for all values due to the fact that (see Theorem 8); that is, the point 1 coincides with the -invariant median of the family discussed previously. Moreover, the last two columns provide also the 1st and 3rd quartile points and of ; that is, , , for various values. These quartiles are evaluated using the quantile function of r.v. ; that is, for , that is derived through (40). The values of the inverse upper incomplete gamma function were numerically calculated.
Figure 2 illustrates the c.d.f. curves, as in (39), for certain r.v. and for scale parameters in the 3 subfigures, respectively. Moreover, the 1st and 3rd quartile points and are also depicted (small circles over c.d.f. curves with their corresponding ticks on -axis).
Theorem 15. The (non-log-scaled) location parameter is in fact the geometric mean as well as the median for all generalized lognormally distributed . Moreover, this median is also characterized by vanishing median absolute deviation.
Proof. Considering (39) and the fact that , , it holds that . For the geometric mean , we readily obtain as with . A dispersion measure for the median is the so-called median absolute deviation or MAD, defined by . For , we have ; that is, follows the generalized Lognormal distribution with threshold . Furthermore, is the “folded distribution” case of which is distributed through p.d.f. of the form where is the p.d.f. of . For example, see  on the folded normal distribution. However, the density function is defined in due to threshold , while it vanishes elsewhere; that is, Therefore, the c.d.f. of is given by Applying the transformation , , into the first integral of (55) and , , into the other two integrals, we obtain and hence with being the c.d.f. of the standardized r.v. . From (38) and the fact that , , it is clear that (57) implies , for every , and the theorem has been proved.
3. Moments of the -Order Lognormal Distribution
For the evaluation of the moments of the generalized Lognormal distribution, the following holds.
Proposition 16. The th raw moment of a generalized lognormally distributed random variable is given by and coincides with the moment generating function of the -order normally distributed ; that is, .
Proof. From the definition of the th raw moment , we have
and applying the transformation , , we get
Through the exponential series expansion
it is obtained that
Finally, substituting the normalizing factor as in (2) into (62) and utilizing the known integral ,
we obtain (58).
Moreover, for we have , and the proposition has been proved.
Example 17. For the second-ordered lognormally distributed , (58) implies and through the gamma function identity we have which is the th raw moment of the usual lognormally distributed , with mean . This is true as is the known moment-generating function of the normally distributed .
Theorem 18. The th central moment (about the mean) of a generalized lognormally distributed random variable is given by where
Proof. From the definition of the th central moment we have while using the binomial identity we get Applying Proposition 16, (70) implies that while taking the summation index until , we finally obtain (67), and the theorem has been proved.
Example 19. Recall Example 17. Substituting (66) and the mean into (70), the second-ordered lognormally distributed provides while which are the th central moment and the variance, respectively, of the usual lognormally distributed . The same result can be derived directly through (67) for and the use of the known gamma function identity, as in (65).
Theorem 20. The mean , variance , coefficient of variation , skewness and kurtosis of the generalized lognormally distributed are, respectively, given by where the sums , , are given by (68).
Proof. From Proposition 16 we easily obtain (74), as . From Theorem 18 we have Hence, substituting from (74), (75) holds. Moreover, the squared coefficient of variation is readily obtained via (75) and (74). By definition, skewness is the standardized third (central) moment; that is, . Theorem 18 provides that Substituting and from (74) and (75), we obtain (77). Finally, kurtosis is (by definition) the standardized fourth (central) moment; that is, , which provides, through Theorem 18, that Substituting , , from (74), (75), and (77), we obtain (78).
Example 21. For the second-ordered lognormally distributed , utilizing (65) into (68) we get , . Applying this to Theorem 20 we derive (after some algebra) which are the mean, variance, coefficient of variation, skewness, and kurtosis, respectively, of usual lognormally distributed .
For the usual lognormally distributed random variable , it is known that . The following corollary examines this inequality for the family of distributions.
Corollary 22. For the -ordered lognormally distributed , it is true that . The first equality holds for the Log-Laplace distributed with as well as for all the negative-ordered where is considered to be the local (nonsmooth) mode point of . The second equality holds for the degenerate Dirac case of .
Proof. From (74) and Theorem 15 we have
for every . The above inequality becomes equality for the limiting Dirac case of . For the relation between the mode and the median of , the following cases are considered. (i) The positive-ordered Lognormal case : from (20) we have
For the Log-Laplace case of , it holds
provided that , while for we have
For , the inequality (84) clearly holds. (ii) The negative-ordered Lognormal case : from Proposition 4 the inequality as in (86) holds. Moreover, if is considered as the nonsmooth local mode point of the negative-ordered 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 , , are given by respectively, while the mean, variance, coefficient of variation, skewness, and kurtosis of are given, respectively, by