Maximum Likelihood, Lognormality and Compound Distributions - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 100

This expression is called a compound generating function. Well-known applica-
tions of this theory are the compound Poisson and negative binomial distributions.

Box 3.2: Moments of Compound Poisson and Negative Binomial
Distribution

If K is Poisson, g y ( s ) · g x ( s )]. Then, if X satis￿es the so-called

logarithmic series distribution: PX

¼ exp[ ʻ + ʻ

k
k where

Þ ʱ θ

1
log e 1 θ

ð ¼ k ʱ ¼

Þ ; ð

k 1, Y becomes negative binomial (Feller 1968 ,

p. 291). If X is a continuous random variable, whereas K is discrete, the
characteristic function of Y satis￿es: g y ( u )

¼ 1, 2, ...; 0 < θ <

¼ g k { g x ( u )} with mean E ( Y ) ¼ E

2 ( Y ) 2 ( X )+ 2 ( K )· E 2 ( X ). Suppose that

( K )· E ( X ) and variance ˃ ¼ E ( K )· ˃ ˃

K is Poisson and that X is lognormally distributed, then:

¼ ʻ

2 2

e ʼþ˃

= 2 and 2 Y 2 2

e 2 ʼþ 2 ˃ EY .

If the negative binomial distribution is used for K instead of the Poisson,
whereas X is kept lognormal, the characteristic function of Y becomes: g y u

ðÞ ¼ ʻʱ ¼ ʻ ˃ ðÞ ¼ ʻʱ þ ʲ

ðÞ

h i r .

p
1 qg xðÞ

rq

p e ʼþ˃

2

= 2 and rq

p e 2 ʼþ˃

2

e ˃

2

q
p

¼ with: EY ðÞ ¼ ˃

2 Y

ðÞ ¼ þ

The combination of completely random (Poisson) distribution of mineral
deposits with lognormal value distribution originally was used for exploration
strategy by Allais ( 1957 ). Using the negative binomial instead of the Poisson
represents the exploration strategy problem later conceived by Grif￿ths ( 1966 )
and formalized by Uhler and Bradley ( 1970 ).

3.3.2 Exploration Strategy Example

One of the objectives of economic geology is to design methods for locating places
that are likely to contain hidden mineral deposits. Exploration is expensive mainly
because large deposits are rare events and it is dif￿cult to locate them. Most
deposits that outcrop at the surface of the Earth may already have been found.
Deeper deposits are to be discovered by geological process modeling, drilling and
new, mainly geophysical, exploration methods. The drilling of deep boreholes or
wells remains essential. In later chapters, several techniques developed for the
evaluation of regional mineral potential that provide a starting point for exploration
will be discussed. In this section, equations presented in Box 3.2 will be used to
predict total value of mineral deposits on the Canadian Shield using historical data.
This example originally was presented in Agterberg ( 1974 ).
Slichter ( 1960 ) compiled information on number of valuable mines per unit area,
for 185 units of 1,000 sq. miles each in Ontario. The rocks for the entire area of size
185,000 sq. miles belong to different structural provinces and geological