Maximum Likelihood, Lognormality and Compound Distributions - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 73

Chapter 3
Maximum Likelihood, Lognormality
and Compound Distributions

Abstract Estimators obtained by the classical method of moments, although they
are unbiased, can have the undesirable property of being relatively imprecise, and
more precise estimators can be obtained from the same data by using other methods
of estimation. One such method is based on the principle of maximum likelihood
that is particularly useful in applications to relatively small samples from positively
skewed frequency distributions including the lognormal also to be discussed in this
chapter. The maximum likelihood method was invented by Fisher (1922) who
developed it as an alternative to Bayesian statistics to which he was strongly
opposed. The principle of maximum likelihood is as follows. Suppose that the
frequency distribution of a random variable X contains several parameters

...
that are to be estimated. The probability that a value x i will fall within a narrow
interval

θ 1 , θ 2 ,

Δ x is approximately f ( x i ) Δ x . The probability that n values will fall where

x n ·

they do is x is suf￿ciently narrow, the choice of it does

not matter and one can consider the product

Δ ʠ n f ( x i ). As long as Δ

ʠ n f ( x i ) only. This is the likelihood
function. For any given values of the parameters

θ 1 , θ 2 , ...

, the likelihood function
will assume a speci￿c value. If this value is large it can be assumed that choice of
parameters is a good one. The likelihood method consists of maximizing ʠ n f ( x i , θ j )
or its logarithm ʣ log f ( x i , θ j ) with respect to the θ j . The resulting maximum
likelihood estimators always have the minimum variance property. However, they
can be biased contrary to estimators based on the method of moments. The
maximum likelihood method with its rapid convergence to normality for increased
sample size is helpful in various geoscience applications including the analysis of
age determinations in numerical timescale construction. Other illustrative examples
include applications to chemical element concentration values from small sets of
observations taken from surface rocks or drill-cores, compositions of larger channel
samples from orebodies, thickness measurements of layers in sedimentary rocks,
and occurrences of fossils in stratigraphic sections.