Maximum Likelihood, Lognormality and Compound Distributions - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 80

methods of approach. Cox and Dalrymple ( 1967 , see their Fig. 4 on p. 2608)
demonstrated that, under these conditions, the inconsistent dates for younger
rocks have probability of occurrence P Iy with:

˄
˃ m 2

t

p

P Iy t ðÞ ¼ ½ erfc

where erfc denotes complementary error function and ˄ is true age of the chronostra-

tigraphic boundary ( boundary between geomagnetic polarity epochs in Cox and

Dalrymple's original paper). The standard deviation for measurement errors is
written as

¼

˃ m . Setting ˄ ¼ 0 and using the relationship ½ erfc ( z / √ 2) ¼ 1 ʦ ( z )it

follows that:

¼

t
˃ m

t
˃ m

P Iy t ðÞ ¼ 1 ʦ f

˃ m is replaced by x , the weighting function shown in Fig. 3.1 is obtained.
Consequently, this weighting function can be interpreted as the probability that an
inconsistent age t a is measured for the younger rocks. Likewise, P Io ( t ) ¼

If t /

t / ˃ m )
can be interpreted as the probability that an inconsistent age t a can be de￿ned for
older rocks.
Cox and Dalrymple ( 1967 ) next introduced the trial boundary age t e and de￿ned
a measure of dispersion of all inconsistent dates t a satisfying:

f (

Z 1

D 2

2 P I t

ð t a t e Þ ¼ ð t t e Þ ðÞ dt

1

where P I ( t ) , this quantity is a

minimum (see Cox and Dalrymple 1967 , Fig. 5 on p. 2608). A normalized version
of E 2 can be directly compared to the theoretical error for D 2 ( t a

¼ P Iy ( t )if t 0; and P I ¼ P Io ( t )if t < 0. For t e ¼ ˄

t e ) when the
number of inconsistent dates is large. This normalization consists of dividing E 2 by
average number of dates per unit time interval. It is noted that P I ( t ) does not
represent a probability density function because it can be shown that
Z 1

p

2

P I t ðÞ dt =ˀ : <

1if t e ¼ ˄

¼ ¼ 0 798

1

In this section, E 2 is not interpreted as a quantity that is approximately propor-
tional to D 2 ( t a

t e ). Instead of this, it is regarded as the inverse of a log-likelihood
function with Gaussian weighting function. For very large samples, good estimates
can be obtained using the inconsistent dates only. For small samples, however,
signi￿cantly better results are obtained by using the consistent dates together with
the inconsistent dates by replacing the Gaussian weighting function by f ( x ).
All Gaussian weighting functions provide the same mean age for a chronostra-
tigraphic boundary when the maximum likelihood method is used. However, the