Maximum Likelihood, Lognormality and Compound Distributions - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 89

with cumulative distribution F 2 ( x ), and one knows the distribution F 1 [ h ( x )] where
h ( x ) is a function of x , then F 2 ( x ) ¼ F 1 [ h ( x )]. Differentiation of both sides with

regard to x gives:

dF 2 x ðÞ

dx ¼

dF 1 hx ½ ðÞ dh x ðÞ

dx ðÞ dh x

Box 3.1: Moments of Lognormal Distribution

If h ( x ) ¼ log e x , the lognormal density function becomes:

.

2

log e x ʼ
˃

1

2

f 2 x ðÞ ¼ exp ½

p

x ˃ ˀ

are for logarithmi-
cally (base e ) transformed data. If X represents the random variable with
lognormal distribution, its moments can be derived as follows. The moment
generating function of the normal distribution is: m ( u )

In this expression the mean ʼ and standard deviation ˃

2 u 2 /2].
From the equation for the moment generating function of a continuous
random variable,

¼ exp[ ʼ u + ˃

it can be derived that for the lognormal distribution:

0
r ¼

2 r 2 /2]. Consequently,

ʼ exp[ ʼ r + ˃ the moments of the lognormal

distribution constitute the moment generating function of the normal

2 ,

distribution. The mean and variance written as ʱ and ʲ are:

h i

2
2

ʼ þ ˃

2 X 2 2

ʼ ðÞ ¼ X exp ; ˃ ðÞ ¼ exp 2 ½ ʼ þ 2 ˃ exp 2 ½ ʼ þ ˃ .

3.2.1 Estimation of Lognormal Parameters

Examples of application of the method of moments to the data of Tables 2.4 and 2.5
are as follows. The 118 zinc values of Table 2.4 produce the estimates m ¼2.6137

and s 2 ¼ 0.2851 after logarithmic transformation. It follows that m ( x ) ¼ 15.74 and

s 2 ( x ) 81.74 for the untransformed zinc values. However, direct estimation of

mean and variance of the 118 zinc values gives mean and variance equal to 15.61
and 64.13, respectively. The slightly larger value of variance ( s 2 ( x )

¼

81.74)
obtained by the method of moments suggests a slight departure from lognormality
(large-value tail slightly weaker than lognormal) possibly related to the fact that the
largest possible zinc value is signi￿cantly less than 66 % (see Sect. 2.4.1 ). Similar
estimates based on the 61 gold values of Table 2.5 are as follows. Mean and
standard deviation of original data are 907 and 1,213 in comparison with 897 and
1,088 based on logarithmically transformed values. This is not a large difference
indicating that the sample size in this example is suf￿ciently large so that it is not
necessary to apply the following method that is useful for smaller data sets or when
positive skewness is larger.

¼