Finney minimum variance unbiased estimator described by Aitcheson and
Brown.24
The Krige method using 4 minimum variance unbiased estimator
is optimal under the assumption of lognormal distribution (Aitchison and
Brown 34 p- 44).
The shift parameter, tT, is calculated using Krige's
quantile formula.??
The tT is involved in the log-transform as log
(x, -T), where X; are measurement values in pCi/gm and may be
negative and below the minimum detectable activity (MDA).
The
param-
eter removes problems of taking logrithms of negative numbers and improves
the approximation to the lognormal probability density function.
computer algorithm calculates the first T.
Figure 8,
The
T = 0.6 pCi/cn,
exhibits an example where the left MDA concentrations are forced more
closely to a straight line than the same data with T = o (Fig. 7) using
the Krige method?? to adjust this parameter about the first approximation to maximize the r-test value.
The means and variances arc
calculated using the Finney fifteen term eapproximetion to 4 mininun
variance estimate (Aitchison and Brown, 24 Eq.
5.37).
A visual inspec-
tion of each data set is done by a log-probability plot (examples on
Figs. 4 to 10).
For comparison,
using the quantile method.2*
the mean and variance was estimated
The minimus variance log-probability-line
was used to find the quantile values.
Numerical approximations to the
cumulative normai distribution used formulas 26.2.23 and 26.2.22 in
Abromowitz and Stegun's handboc::.49
the comparison of methods
Tables 45 through 47 illustrate
for 24) bry which have severa]
and 137¢5 at the same sample location.
For 24. an,
censored values,
the arithmetic
mean frequently underestimates the Krige method (average of 30 percent,
, underestimation range: 0 to -100 percent, 0-15 cm soils), but the
40 -
a
so11114