Maximum Likelihood, Lognormality and Compound Distributions - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 88

maximum likelihood methods is that the relative stratigraphic position of any rock
sample is generalized with respect to stage boundaries that are relatively far apart in
time. The relative stratigraphic position of one sample with respect to others within the
same stage is not considered. A better approach for estimating the age of stage
boundaries currently is to incorporate precisely known stratigraphic positions for
which high-precision age determinations are available. More data of this type have
become available during the past 20 years resulting in two international geologic time
scales (GST2004 and GST2012). Moreover, the establishment of GSSP's
(GSSP¼Global Stratigraphic Section and Point; see Chap. 9 ) isallowingmore
precise estimation of the age from extensive data at or near stage boundaries as in
the following example for the age of Maastrichtian-Paleocene boundary.
Precision of Cenozoic epoch boundaries in GTS2012 (Gradstein et al. eds, 2012 )
has become excellent because of use of the astronomical clock. This improvement
applies to Cenozoic stage boundaries as well. Remaining uncertainty of ages of
stage boundaries is illustrated by the following example. Currently, the best age
estimate of the Maastrichtian-Paleocene boundary is 66.0 0.5 Ma. This 95 %

con￿dence interval extends to the GTS2004 (Gradstein et al. eds, 2004 )estimate
of 65.5

0.3 Ma. The GTS2004 estimate was close to Renne et al.'s ( 1998 ) estimate

of 65.46 1.26 Ma, which was based on back-calculation with external errors of

Swisher et al.'s ( 1993 ) estimate for an ash layer probably coincident with the
Cretaceous-Paleogene boundary. New estimates for this boundary include two ages
recently proposed by Husson et al. ( 2011 ) which are (1) 65.59

0.07 Ma on the basis
of 405 ka eccentricity variation resulting from astronomical solution La2010a, and
(2) 66 0.07 Ma in coherence with radio-isotope datings. However, Renne

et al. ( 2012 )'s preferred estimate of 66.043 0.086 Ma supports the GTS2012

estimate of the Maastrichtian-Paleocene boundary. These authors ascribe earlier
estimates, which are closer to 65.5 Ma as in GTS2004, to previous miscalibration
by two 405 ka eccentricity cycles (Renne et al. 2012 ,p.684).
Undoubtedly, there will be further improvements of the numerical geological
time scale in future. These may include the following: it is likely that the astro-
nomical clock will be extended downward from the Cenozoic into the Mesozoic
and older periods; zone boundary age estimation may become possible (in addition
to further improved stage boundary age estimation); future GSSPs will help to
re￿ne the Mesozoic and Paleozoic time scales; and the Precambrian time scale will
continue to be further improved. Rapidly increasing numbers of high-precision
dates are bound to signi￿cantly improve overall precision in future time scales,
although GTS2012 already is more accurate and precise than its predecessors.
The maximum likelihood method discussed in this chapter may become useful
again in future for ￿ner chronostratigraphic sub-divisions.

3.2 Lognormality and Mixtures of Frequency Distributions

A random variable X has lognormal frequency distribution if the logarithms of its
values are normally distributed. Theory of the lognormal distribution is explained in
detail by Aitchison and Brown ( 1957 ). In general, if X represents a random variable