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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 << E[z]. In order to mitigate this
discrepancy, we have made the following mapping for z:
z(ξ) = ξ − (μ5 /ξ)µ6 ,
(2)
where μ5 and μ6 are two new parameters. In this mapping, the
domain of ξ is (0, +∞). Furthermore, by properly choosing
the values of μ5 and μ6 so that (μ5 /ξ)µ6 becomes numerically significant only when ξ << E[ξ], we can ensure that
z(ξ) ≈ ξ for all the other scenarios, and hence the advantages
c 2008 IEEE
1089-7798/08$25.00 CHEN et al.: LOGNORMAL SUM APPROXIMATION WITH A VARIANT OF TYPE IV PEARSON DISTRIBUTION
0
−∞
1−(1e−8)
1−(1e−6)
N=40, σ ∈ [4, 5]
i
0.9999
0.999
0.99
CDF GPV(ξ < A)
of the Type IV Pearson approximation described in [7] are
maintained. Based upon the above mapping, a variant of the
Type IV Pearson distribution has been developed, the PDF of
which can be mathematically expressed as follows:
µ μ6 μ5 6
fP [z(ξ)],
gP V (ξ) = z (ξ)fP [z(ξ)] = 1 +
ξ
ξ
(3)
and the CDF of which can be derived as follows:
A
z(A)
gP V (ξ)dξ =
fP (z)dz,
(4)
GP V (ξ < A) =
631
0.9
N=100, σ ∈ [5.5, 6.5]
0.5
N=70, σ ∈ [5, 6]
0.1
N=50, σi ∈ [4.5, 5.5]
i
i
0.01
Empirical PDF, mi=[−.5 .5]
0.001
Type IV Pearson distribution
Type IV Pearson variant
MGF method − head fitting
MGF method − tail fitting
Schwartz−Yeh method
1e−4
µ6
where z(A) = A − (μ5 /A) .
1e−6
1e−8
III. PARAMETER E VALUATIONS FOR THE T YPE IV
P EARSON VARIANT
15
where mi and σi are the parameters of lognormal distributions
in dB, and β = (ln 10)/10. In order to approximate the sum
of these N lognormal RVs with the Type IV Pearson variant,
the mean (MT ), variance (VT ), skewness (SkT ), and kurtosis
(KuT ) of the sum of the lognormal RVs must be derived first
with the method given by [7]. Then the selection parameter
described in [8], denoted as KT here, can be obtained as
follows:
SkT2 (KuT + 3)2
.
4(4KuT − 3SkT2 )(2KuT − 3SkT2 − 6)
(6)
Depending on whether the value of KT is smaller than one
or not, two different methods must be employed respectively
to evaluate the values of μ1 to μ6 in the Type IV Pearson
variant.
3.1. KT < 1
When KT < 1, as it is shown in [7], the lognormal sum distribution can be approximated with a standard Type IV Pearson
distribution. Correspondingly, the values of μ1 , μ2 , μ3 , and μ4
can be derived from the following equations [8]:
μ1 = (r + 2)/2,
−r(r − 2)SkT
,
μ2 = 16(r − 1) − SkT2 (r − 2)2
μ3 = VT [r − 1 − SkT2 (r − 2)2 /16],
μ4 = MT − (r − 2)SkT VT /4,
25
30
35
A(dB)
Let Z1 , · · · , ZN be N independent but not necessarily
identical lognormal RVs with PDFs as follows:
(10 log10 z − mi )2
1
exp −
fZi (z) = √
,
2σi2
2πβσi z
i =1, · · · , N,
(5)
KT =
20
(7)
(8)
(9)
(10)
where r = 6(KuT − SkT2 − 1)/(2KuT − 3SkT2 − 6).
Unfortunately, there is no close-form equation to derive the
values of μ5 and μ6 , so in this paper, we have evaluated the
values of μ5 and μ6 as follows: first, according to the values
of mi , σi , and N , obtain the empirical CDF of the lognormal
sum distribution through Monte Carlo computer simulations,
and draw the empirical CDF on the so-called lognormal paper
[4]; second, use the built-in least square curve fitting function
in MATLAB software, lsqcurvefit, to estimate the optimal
Fig. 1. CDF comparisons of the lognormal sum distributions with KT < 1
and their Type IV Pearson variant approximations.
values of μ5 and μ6 , which achieve the best match between
the empirical CDF of the lognormal sum distribution and the
numerical CDF of the Type IV Pearson variant calculated from
Eq. (4).
Our further investigations on the values of μ5 and μ6 show
that when the values of mi , σi , and N vary within a notvery-wide range, the values of μ5 and μ6 linearly depend on
those of μ1 , μ3 , and μ4 . For example, when mi ∈ [−0.5, 0.5],
N ∈ [40, 100], and σi ∈ [σ − 0.5, σ + 0.5] with σ ∈ [4.5, 6],
the values of μ5 and μ6 can be calculated from the following
empirical equations:
μ5 = −11.8 − 0.07μ1 + 0.3μ3 + 0.95μ4,
μ6 = 3.8 + 0.92μ1 − 0.04μ3 + 0.032μ4.
(11)
(12)
Therefore, when KT < 1, for most cases the values of μ1
to μ6 can be easily evaluated from the close-form analytical
equations and the empirical equations.
In order to demonstrate the approximation improvement of
the newly developed Type IV Pearson variant, in Fig. 1 we
have plotted the CDFs of i) the lognormal sum distribution
with 4 × 108 random samples, ii) the standard Type IV
Pearson approximation [7], iii) the Type IV Pearson variant
approximation, iv) the MGF lognormal approximation [5], and
v) the S-Y lognormal approximation [3] for four different sets
of N and σi . Since on the lognormal paper the CDF of the
lognormal sum distribution is a convex curve and that of a
lognormal distribution is a straight line, as shown in Fig. 1,
the MGF approximation is good at either the head portion or
the tail portion but not the both, and the S-Y approximation is
good only at the body portion. Furthermore, from Fig. 1 it is
clear that the standard Type IV Pearson approximation leads to
a large discrepancy in the head portions of the lognormal sum
distribution. Only the Type IV Pearson variant shows excellent
approximation for a whole probability range, from the head
portion, the body portion, to the tail portion.
3.2. KT ≥ 1
When KT ≥ 1, the standard Type IV Pearson distribution
can no longer be used to approximate the lognormal sum
distribution. Correspondingly, there is no close-form analytical
632
IEEE COMMUNICATIONS LETTERS, VOL. 12, NO. 9, SEPTEMBER 2008
TABLE I
T HE PARAMETERS OF THE T YPE IV P EARSON VARIANT FOR σ = 12
μ2
-6.47
-4.88
-2.17
μ3
62.2
647
3393
μ4
-38
403
4836
μ5
641
1287
7985
0.9999
μ6
0.95
4.57
8.43
equation to derive the values of μ1 to μ4 either. In order to
approximate the lognormal sum distribution with the Type IV
Pearson variant, we have evaluated all the six parameters, μ1
to μ6 , with the curve fitting method listed in the previous
subsection. In previous publications, [9] also used the curve
fitting method to approximate the lognormal sum distribution.
However, a variant of normal distribution, instead of the Type
IV Pearson variant, is employed by [9] to approximate the
lognormal sum distribution.
In order to compare the performance of the Type IV
Pearson variant approximation with that of the normal variant
approximation proposed in [9], we have evaluated the values
of μ1 to μ6 with the same parameters as those in [9], i.e.,
mi = m = 0 and σi = σ = 12. The obtained optimal values
for μ1 to μ6 when N = 20, 100, 300 are listed in Table 1.
According to the values of μ1 to μ6 listed in Table 1,
in Fig. 2 the numerical CDFs of the Type IV Pearson
variant approximation and the normal variant approximation
are plotted on the lognormal paper together with the empirical CDFs of the lognormal sum distribution obtained from
computer simulations. As shown in Fig. 2, when the value
of N is small, e.g. N = 20, the Type IV Pearson variant
approximation has almost the same performance as the normal
variant approximation. However, when the value of N is
large, e.g. N = 100 and 300, the Type IV Pearson variant
approximation outperforms the normal variant approximation
in almost all probability ranges, especially in the tail portion of
the lognormal sum distribution. Furthermore, fitting curves in
the lognormal paper puts much more emphasis on minimizing
the approximation discrepancy in the head and tail portions
rather than in the body portion. In order to accurately illustrate
the approximation discrepancy in the body portion, in Fig. 3
for N = 100 we have plotted the PDFs of the Type IV Pearson
variant approximation, the normal variant approximation, and
the lognormal sum distribution in a linear-scale paper. Clearly,
when the value of N is large, the normal variant approximation
has to sacrifice its accuracy in the body portion of the
lognormal sum distribution so as to approximate the head and
tail portions of the lognormal sum distribution. However, the
Type IV Pearson variant can still accurately approximate all
portions of the lognormal sum distribution.
IV. C ONCLUSIONS
In this paper, we have developed a variant of the Type IV
Pearson distribution with a variable domain of (0, +∞) to
approximate the lognormal sum RVs. This newly developed
Type IV Pearson variant approximation maintains all the
advantages of the standard Type IV Pearson approximation as
well as relieves its two limitations. Therefore, independent of
the statistical characteristics of the lognormal sum distribution,
the Type IV Pearson variant can accurately approximate the
lognormal sum distributions for a whole probability range.
0.999
0.99
0.9
CDF GPV(ξ < A)
μ1
1.23
1.37
1.41
N=100
N=20
0.5
N=300
0.1
0.01
0.001
Empirical CDF
Type IV Pearson variant
Normal variant distribution
1e−4
1e−6
10
20
30
40
50
60
70
A(dB)
CDF comparisons for σ = 12 in the lognormal paper.
Fig. 2.
−4
x 10
Empirical PDF
Type IV Pearson variant
Normal variant distribution
3
Probability Density Function
N
20
100
300
1−(1e−6)
2
1
0
24
26
28
30
32
34
36
38
40
42
44
A(dB)
Fig. 3.
paper.
PDF comparisons for σ = 12 and N = 100 in the linear-scale
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