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International cooperation and technology transfer
Mussio, Luigi

E[sl] = o * 2 B
var[s 2 b ]=-
The first one simply confirms what has been observed
about (6) and (7); it also shows that our formal develop
ment is correct; the second highlights the dependence of
the variance of in respect of . In other words:
is a correct estimator of the variance of B because its
mean value coincides with , but the goodness of the
estimation depends inversely on
Another well known way of understanding the properties
of an estimator is to study its confidence interval at a
certain confidence level . From (8) it is easily obtain
2 Si U 2
V 2 < â Xn-a 2 — 2 . _ 2 ~ Xn\ 1-
G , +G„
P = 1
where xl a / and xl \-a/ are two real numbers with the
following properties
P [°>X fl 2 ;«/ 2 ]IX n 2 =Y
P [^Xn^lXn =1 —
The confidence intervals for the estimated variance and
standard deviation of B are easily calculated
Y 2 ■
A./»; a/2
—g 2 <5] _ 2 , _ 2
°A + °B „2
_ 2 2
2 °A +Cy B _ 2
v 2 ÎLl
A.n, l-a/2
The role played by the size of the variance of the instru
ment A is also shown in this case: the bigger it is, the
larger the confidence interval becomes. So it confirms the
rule that high values of mean low quality estimation
of the variance of B. In the next section a practical exam
ple will clarify the orders of magnitude of the phenome
For a research job devoted to the determination of the
precision of a DPW at different resolutions, the position
of 25 control points has been measured with an analytical
stereoplotter. The operator has completed not only one
cycle of orientation and point measurements, but twelve
cycles, so to determine the precision of the analytical
measurements. For the X component the standard devia
tion was <3 x =6. cm; having twelve independent meas
urements, mean values have been calculated for each
i 12
12 jS
Their estimated standard deviation of the analytically
measured coordinates is g . =
= 1.9 cm. The
DPW measurements of the X component, at the resolu
tion of 300 dpi, have shown an estimated standard devia
tion s B = 21. cm; this will be assumed as the true value
for the instrument B ( g b = 21. ).
The following picture shows the confidence interval
width of the estimated standard deviation as a func
tion of ; the value of and n are kept fixed; the con
fidence level is a = 0.0 .
Length of the confidence interval of S 0 as a function of CTA
The interval width obviously increases with ; there is
a lower limit, corresponding to the value g a , that can
be improved only by increasing the number of the points,
n\ this limit is unfortunately high in our case, because the
standard deviation estimation improves very slowly, in
respect to the number of the measurements.