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probably meaningful. This becomes a practical estimation problem in that, if
lognormality is accepted, the risk is taken of the model yielding biased estimates
because of a departure from lognormality. The Uhler-Bradley model discussed in
the preceding section is fairly sensitive to departures from lognormality. Assuming
lognormality, the preceding statistics give an estimated value of 82.18 · 10
6
dollars.
An unbiased variance estimate can be obtained by using 0.985 ·
s
2
¼
3.096 instead
of
s
2
3.103. This results in an estimated value of 80.34 · 10
6
dollars.
Inasmuch as the original estimates of mean (
¼
60.5 · 10
6
) and logarithmic mean
(¼2.858) are based on 254 data, both are fairly good estimates. The preceding two
estimates of logarithmic variance may be too large because of positive skewness of
the value histogram. For this reason, it may be preferable to base the estimate of
logarithmic variance on the estimates of mean and logarithmic mean. This gives
s
2
¼
3.096 and 3.103).
The number of orebodies in favorable environments outside the control area is
estimated at (219,220/100)
¼
2.490, which is less than the preceding two estimates (
¼
1,478. With average expected value of
60.5 · 10
6
, their total value would be 1,478
0.508
¼
60.5 · 10
6
89.4 · 10
9
dollars. The
¼
corresponding average value per unit area of 10
miles size is
E
(
Y
)
60.5 · 10
6
30.73 · 10
6
. Our estimate of the corresponding variance
2
(
Y
)
0.508
¼
˃
can be based on the negative binomial distribution model for which
E
(
K
)
¼
rq
/
2
(
K
)
rq
/
p
2
p
0.508 and
˃
¼
2.016. With
p
+
q
¼
1, this yields the estimated
2
values
p
0
¼
0.2520,
q
0
¼
0.7480,
r
0
¼
0.1711. With
ʼ
2.858 and
˃
2.490, it
167 · 10
6
dollars.
It seems that an analytical expression for the random variable
Y
does not exist.
However, it can be assumed that its frequency distribution is positively skew and
multimodal. Because of the central-limit theorem, the sum of
n
random variables
Y
that can be written as
Z
2
(
Y
)
27,922 · 10
12
and
follows that
˃
˃
(
Y
)
¼
ʣ
Y
i
(
i
¼
1, 2,
...
,
n
) converges to normal form when
12.30 · 10
9
n
increases. For example, if
n
¼
400, we have
E
(
Z
)
and
˃
(
Z
)
3.34 · 10
9
. The coefcient of variation
ʳ
0.27 then is quite small. In that
situation it is permissible to determine the 95 % condence interval that becomes
6.5 · 10
9
on the predicted total value of 12.30 · 10
9
dollars. It should be kept in mind
that the preceding calculations were based on orebodies located in a control area
that had been discovered in 1968, and only the uppermost part of the Earth's crust
had been scrutinized for mineral occurrences.
(Z)
References
Agterberg FP (1965) Frequency distribution of trace elements in the Muskox layered intrusion. In:
Dotson JC, Peters WC (eds) Short course and symposium on computers and computer
applications in mining and exploration. University of Arizona, Tucson, pp G1-G33
Agterberg FP (1974) Geomathematics. Elsevier, Amsterdam
Agterberg FP (1988) Quality of time scales - a statistical appraisal. In: Merriam DF (ed) Current
trends in geomathematics. Plenum, New York, pp 57-103
Agterberg FP (1990) Automated stratigraphic correlation. Elsevier, Amsterdam
Agterberg FP (1994) Estimation of the Mesozoic geological time scale. Math Geol 26:857-876
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