Maximum Likelihood, Lognormality and Compound Distributions - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 81

Fig. 3.4 Two examples of runs (No. 1 and No. 7) in computer simulation experiment. True dates
(a) were generated ￿rst, classi￿ed and increased (or decreased) by random amount. Younger and
older ages are shown above and below scale (b), respectively (Source: Agterberg 1990 , Fig. 3.15)

standard deviation of this mean depends on the choice of the constant p in exp
(

px 2 ). For example, p

1.0 for f a ( x ) in Fig. 3.1b . Assuming that f ( x ) of Fig. 3.1a
represents the correct weighting function, one can ask for which value of p the
Gaussian function exp (

¼

px 2 ) provides the best approximation to f ( x ) with x 1.

Let u represent the difference between the two curves, so that log e {1 ʦ ( x )} ¼

px 2 + u . Minimizing u 2

ʣ for x i ¼ 0.1 · k ( k ¼ 1. 2. ... , 20) by the method of least

squares gives p opt ¼ 1.13. Because of the large difference between the two curves

near the origin, p opt increases when fewer values x i are used. It decreases when more
values are used. Letting k run to 23 and 24, respectively, yields p opt values equal to
1.0064 and 0.9740, respectively. These results con￿rm the conclusion reached
previously that a Gaussian weighting function with p

¼ 1.0 provides an excellent

approximation to f ( x ).

3.1.5 Computer Simulation Experiments

Computer simulation in geoscience has had a long history of useful applications
(Harbaugh and Bonham-Carter 1970 ). Computer simulation experiments were
performed by Agterberg ( 1988 ) in order to attempt to answer the following ques-
tions: (a) does the theory of the preceding sections remain valid even when the
number of available dates is very small; (b) how do estimates obtained by the
method of ￿tting a parabola to the log-likelihood function compare to estimates
obtained by the method of scoring which is commonly used by statisticians in
maximum likelihood applications (see, e.g., Rao 1973 ); and (c) how do results
derived from the chronograms in Harland et al. ( 1982 ) compare to those obtained by
the maximum likelihood method.
Figure 3.4 and Table 3.1 illustrate the type of computer simulation experiment
performed. Twenty-￿ve random numbers were generated on the interval [0, 10].