5
2
B
— S. —
160
and it will be demonstrated that it depends on the value of
which is an extraction from the estimator rv
2. DISTRIBUTIONAL RESULTS
j n Let’s consider again the estimator . It is not too diffi-
= cult to find how it is distributed. Indeed we have
The (4) means, among other things,
n _ V 1 À?
oi+ci £fc 2 A +oi
(8)
because the terms under the sum are in reality normal
standardised rvs, as it can be easily demonstrated by the
following equalities
which implies, passing from the true quantities to the es
timated ones, that it is possible to give an estimation of
the variance of the instrument B in the following way
Ê
A, 2
I
'A. V
V°A J
C 2 - Ç 2 _
°A
(6)
The equations (5) and (6) are always true, independently
of the value of , and this seems to contradict our
common-sense, which suggests that the estimation of
the precision of the instrument B can be effectively
carried out by comparison with the measurements
given by A only if the latter is better than the former
or, more formally, if the following inequality holds
Now, it is well known that the sum of n independent
normal standardised rvs equals a chi square rv with n de
grees of freedom, and this brings us to (8).
From relation (8) it is immediately deduced that
c 2
10 B ~
> D 2
-ln~
Va<
The point is that (6) is an estimation; it is always true, but
this only means that the mean values of the rvs involved
coincide, that is
£[s 2 ]=£[s;- 0 ; (7)
Moreover the latter equality doesn't guarantee anything
about the concentration of the rv around its mean
value. In other words, it is necessary to investigate the
goodness of the single estimation
which represents the first significant result of our discus
sion: the distribution of depends on the value of the
variance of A. An immediate way to investigate this de
pendence formally and quantitatively is to consider the
two most important descriptors of a probability distribu
tion: mean value and variance. Remembering that for the
chi square rv the following equalities hold
VAR[xiy
whose demonstration can be found in every good statis
tics textbook, we have
