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D. Statistical test criterions for parameter testing in photogrammetric point
determination problems
Recently photogrammetry uses more and more the methods of statistical interval
estimation and hypotheses testing. Thereby statistical test criterions are some-
times applied in a little careless manner, what may often lead to wrong decisions
concerning the acceptance or rejection of a hypothesis.
So the problem of statistical hypotheses testing is treated in a more general way
and some test criterions are derived which are often useful for the analysis of
results obtained by methods of photogrammetric point determination.
At any rate one should always keep in mind that a statistical test cannot be an
end in itself, it cannot serve as an inevitable standard, but must only be regar-
ded as an aid for decision.
For hypotheses testing we suppose the linear model of self-calibrating bundle
adjustment, derived from (1) |
E(l) s"Ax""; ;
Ee) 7 9. , "D(e) « D(1) «op (28)
with the general linear null-hypothesis of full rank
HH: Bx = WW ', (29)
Oo
where B isa bxu- matrix with rank(B) = b<u.
The minimum variance unbiased estimators for x and o * from (28) are with r = n-u
-1
= {A PA} API (30a)
1
T
^
^2. ^
O75 i P(AK-1) . (30b)
For statistical analysis the distribution of 1 is assumed as TaN (AX, 0° or ly,
(AX-1)
i.e. a n-dimensional normal distribution.
Testing H, requires the derivation of a test function T such that the distribution
of T is known when Bx = w (i.e. the null-hypothesis is true).
Functions with optimal properties are the Likelihood ratio functions. Their appli-
cation leads to the test criterion
R
Ts 31
587 634)
with -1 -]
R edBA-W)(B(ATPA) BI) (BR-w) . | (32)
For the derivation of T see Graybill /9/, Kooh /13/.
If Ho is true, then T is distributed as Snedecor's F(b,r) .
To obtain the power of the test i, against an alternative hypothesis Ha the
distribution of T must also be known, if the alternative hypothesis Hp: Bx # w
(B'x = w') is true. Then we find T distributed as the noncentral F' (b,r,\) with
the noncentrality parameter
Ten “nl
1 PA) !B 3
x
À = (Bx-w) (B(A
g
-l(Bx-w) : (33)
on
The power g(r) of the test may be computed from Tang's tables of the noncentral
beta distribution (Graybill /9/).
