Maximum Likelihood, Lognormality and Compound Distributions - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 78

Fig. 3.2 Chronogram for Caerfai St. David's boundary example with parabola ￿tted by method of
least squares. E 2

¼ log-likelihood function is plotted in vertical direction creating a basket. Dates
belonging to stages, which are older and younger than boundary, are indicated by o and y ,
respectively. Standard deviation follows from d representing width of parabola for E 2 equal to
its minimum value augmented by 2 (Source: Agterberg 1990 , Fig. 3.13)

by using the method of least squares. If the log-likelihood function is parabolic with
E 2 a + b·t e + c·t e 2 , it follows that the maximum likelihood estimator is normally

distributed with mean m

¼

b /2 c and variance s 2

1/2 c . It will be shown in next
paragraph that, graphically, s can be determined by taking one fourth of the width of
the parabola at the point where E 2 exceeds its minimum value by 2 (see Fig. 3.2 ).
This method differs from the procedure followed by Harland et al. ( 1982 ) who
de￿ned the error of their estimate by taking one-half the age range for which E 2
does not exceed its minimum value by more than 1. This yields a standard deviation
that is

¼ ¼

2 times larger than the one resulting from L a .
According to the theory of mathematical statistics (Kendall and Stuart 1961 ,
pp. 43-44), the likelihood function is asymptotically normal, or:

√

t 2
2

1

e y ¼ p

2

exp

˃

2

˃ ˀ

Here e y represents the standard deviation of this asymp-

totically normal curve that is centered about t

¼ L ( x | t e ) and t ¼ t e ˄ ; ˃

¼ 0. Taking the logarithm at both

t 2 /2

2 where max represents the maximum value

sides gives the parabola y ¼ max ˃

of the log-likelihood function. Setting y . Consequently,

the width of the parabola at two units of y below its maximum value is equal to 4

¼ max 2 then gives t ¼ 2 ˃

˃

.
The parabola shown in Fig. 3.2 (and subsequent illustrations) is assumed to provide
an approximation of the true log-likelihood function. The standard deviation
obtained from the nest-￿tting parabola is written as s . In Fig. 3.2 the Y -axis has