Maximum Likelihood, Lognormality and Compound Distributions - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 79

Fig. 3.3 Caerfai St. David's boundary example. Age estimated by maximum likelihood method
using L . Standard deviation ( s ) and width of 95 % con￿dence interval are approximated closely by
results shown in Fig. 3.2 (Source: Agterberg 1990 , Fig. 3.14)

E 2 points upwards in order to facilitate comparison with
chronograms in Harland et al. ( 1982 ).
Figure 3.3 shows estimates based on L . The resulting best-￿tting parabola is
almost equal to the one in Fig. 3.2 that was based on L a instead of L . Consequently,
the estimated ages of the Caerfai-St David's boundary and their standard deviations
obtained from L a and L also are similar. This conclusion will be corroborated by a
more detailed comparison of the weighting functions for L and L a at the end of this
section, and by computer simulation experiments to be described in the next
section. However, it will be shown that L a does not provide a good approximation
when inconsistent data are missing. When n is small, L also produces better results
than L a because a parabolic chronogram is more readily obtained when the consis-
tent ages are used together with the inconsistent ages as will be illustrated by the
following example. An age estimate based on Harland et al.'s ( 1982 , Fig. 3.4h,
p. 54) chronogram for the Norian-Rhaetian boundary is 213 Ma. The corresponding
standard error reported by Harland et al. ( 1982 ) is 9 Ma. The L -based maximum
likelihood method using the same data set of only six dates gives an estimated age
of 215.5 Ma with standard error of 4.2 Ma.

been inverted so that - y ¼

3.1.4 The Chronogram Interpreted as an Inverted
Log-Likelihood Function

The approach taken in this section differs slightly from the one originally taken by
Cox and Dalrymple ( 1967 ). The basic assumptions that the dates are uniformly
distributed through time and subject to measurement errors are made in both