Maximum Likelihood, Lognormality and Compound Distributions - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 86

Fig. 3.6 Maximum
likelihood methods of
Fig. 3.4 used for estimating
mean age of stage boundary
in Run 7. Log-likelihood
function E 2 does not
provide a good result in this
application. (a) All dates;
(b) Inconsistent dates only
(Source: Agterberg 1990 ,
Fig. 3.18)

a m − s m m + s

− 1

− 2

− 3

y y y

o o

− 4

4.0 4.5 5.0 5.5 6.0 6.5 7.0

Simulated geologic time

b

yyy

o o

1

m

m + s = 9.69

+ +

+ ++++++++++++++++++

0

− 1

4.0 4.5 5.0 5.5 6.0 6.5 7.0

Simulated geologic time

difference between successive approximations became negligibly small, Then the
standard deviation of the ￿nal estimate is given by SD 1/| D 2 |.

For L , the scoring method generally yields estimates of SD which are slightly
greater than those resulting from the parabola method. However, the difference is
negligible small (Table 3.3 ). For L a the scoring method provides an answer only in
six of the experiments of Table 3.3 . Similar results were obtained in a second type
of computer simulation experiment using variable measurement error (see
Agterberg 1988 , for details). In total, 50 runs were made for each of the two
types of experiments. For constant variance of measurement errors, the parabola
method for L gave an overall mean of 4.9287 and standard deviation 0.4979 as
calculated from 50 means. The corresponding numbers for the second type of
experiment were 4.9442 and 0.5160. The Gaussian weighting scheme provided
overall means equal to 4.9213 and 4.9414 in the two types of experiments, and
corresponding standard deviations equal to 0.5790 and 0.6541, respectively. If the
parabola did not provide a good ￿t to the function E 2 , because of zero values around
its minimum, the mean was approximated by the mid-point of the range of zero
values in these calculations. The results of the 50 runs for the two types of
experiments con￿rm the earlier results described in this section. Additionally,
they show that the Gaussian weighting function (using L a ) provides results that
are almost as good as the method of maximum likelihood, unless there are no
inconsistent dates in the data set.

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