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IEEE COMMUNICATIONS LETTERS, VOL. 12, NO. 9, SEPTEMBER 2008
Lognormal Sum Approximation with a Variant of Type IV Pearson Distribution Shaohua Chen, Hong Nie, Member, IEEE, and Benjamin Ayers-Glassey
Abstract—In this paper, a variant of the Type IV Pearson distribution is proposed to approximate the distribution of the sum of lognormal random variables. Numerical and computer simulations show that independent of the statistical characteristics of the lognormal sum distribution, the Type IV Pearson variant outperforms the standard Type IV Pearson distribution and the normal variant distribution in accurately approximating the lognormal sum distribution for a whole probability range. Index Terms—Lognormal distribution, sum of lognormal random variables, Type IV Pearson variant, curve fitting.
I. I NTRODUCTION
T
HE lognormal distribution has been widely used in wireless communication areas to model and analyze the characteristics of attenuations caused by shadowing fading in wireless channels and power fluctuations due to power control errors [1]. Therefore, frequently the knowledge about the probability density function (PDF) and cumulative distribution function (CDF) of the sum of lognormal random variables (RVs) is required so as to analyze the performance of wireless systems. During the past decades, many research efforts have been devoted to investigate the statistical characteristics of the lognormal sum RVs, and although the closed-form expressions of the PDF and CDF of the lognormal sum RVs are still unknown, various approximation methods have been developed [2-7]. The most widely used approximation method is to represent the lognormal sum RVs still with a lognormal distribution [26]. However, the numerical PDF and CDF of the lognormal sum RVs obtained by [4] have clearly shown that the lognormal sum distribution has considerable discrepancy from a lognormal distribution. In order to eliminate this discrepancy, [7] has proposed to approximate the lognormal sum RVs with the Type IV Pearson distribution, which can accurately approximate the lognormal sum RVs in a wide probability range (from 0.01 to 0.99999). However, the Type IV Pearson approximation proposed by [7] has two limitations. First, the variable domain of the Type IV Pearson distribution is (−∞, +∞), but that of the lognormal sum RVs is (0, +∞). Therefore, the Type IV Pearson approximation leads to a large discrepancy in the head portion (small values) of the lognormal sum distribution. Second, in order to use the Type IV Pearson
Manuscript received April 8, 2008. The associate editor coordinating the review of this letter and approving it for publication was P. Cotae. S. Chen and B. Ayers-Glassey are with the Department of Math, Physics and Geology, Cape Breton University, Sydney, NS, B1P6L2, Canada (e-mail: george
[email protected],
[email protected]). H. Nie is with the Department of Industrial Technology, University of Northern Iowa, Cedar Falls, IA 50614-0178, USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/LCOMM.2008.080553.
approximation, the selection parameter of the lognormal sum RVs must be less than one [8]. Consequently, quite a few number of lognormal sum RVs cannot be approximated with the Type IV Pearson distribution since their selection parameters are larger than one. In order to relieve the above two limitations but still maintain the advantages of the Type IV Pearson approximation, in this paper, we have developed a variant of the Type IV Pearson distribution to approximate the lognormal sum RVs. This newly developed Type IV Pearson variant has a variable domain of (0, +∞), and does not have any preclusive condition on the selection parameter of the lognormal sum RVs to be approximated. Therefore, it can accurately approximate almost all lognormal sum RVs for a whole probability range, from the head portion to the body portion (intermediate values) and the tail portion (large values). It is worth to mention that before the Type IV Pearson variant is proposed, we had investigated other types of Pearson distributions and found none of them can accurately approximate lognormal sum RVs. The paper is organized as follows: a variant of the Type IV Pearson distribution is developed in Section II. Then the parameter evaluations for the newly proposed Type IV Pearson variant is discussed in Section III. Through Numerical and computer simulations, the performance improvement of the Type IV Pearson variant approximation is also described in Section III. Finally, conclusions of the paper are given in Section IV. II. A VARIANT OF T YPE IV P EARSON D ISTRIBUTION As it is shown in [8], the PDF of the Type IV Pearson distribution is given by: −µ1 fP (z) = ν 1 + ((z − μ4 )/μ3 )2 × exp [−μ2 arctan((z − μ4 )/μ3 )] , −∞ < z < +∞, (1) where μ1 to μ4 are four independent +∞ parameters, and the function of ν is to ensure that −∞ fP (z)dz = 1. Clearly, since the domain of z in Eq. (1) is (−∞, +∞) and that of the lognormal sum RVs is (0, +∞), approximating the lognormal sum RVs with the Type IV Pearson distribution leads to a large discrepancy in the head portion of the lognormal sum distribution, i.e. when z
