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 θ 1, θ 2, … that are to be estimated. The probability that a value x i will fall within a narrow interval Δx is approximately f (x i ) Δx. The probability that n values will fall where they do is Δx n · Π n f (x i ). As long as Δx is sufficiently narrow, the choice of it does not matter and one can consider the product Π 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 specific value. If this value is large it can be assumed that choice of parameters is a good one. The likelihood method consists of maximizing Π n f (x i , θ j ) or its logarithm Σ 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.