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fore as is called an error probability, the reader will
recognize it as the probability of making a type I error.
If 0 G C2, «2 is the optimal action; it is taken if z G Ö2. If z G Di
we take «1 which is the wrong action in this case, there
fore we call i-ps also an error probability (of making
a type II error).
The risks are
R(Q,s) (i-as) x o xs x n ocsn if 0 GCi
f?(0,s) (i-ps) X rn Ps X o (i-[3s)m if 0 GC2.
The two-state, two-action decision problem has now been com
pletely formulated. We have deliberately used some terms from the
well-known theory of testing hypotheses founded by Neyman and
Pearson to bring out the correspondence; in fact this theory also
specifies the decision problem completely except for the explicit
use of the regrets. It should be noted that in general one cannot
find a test that minimizes the risk for all states of nature. It can
be shown [3] that for testing a simple hypothesis against a simple
alternative, every admissible strategy is a Bayes strategy, and
every Bayes strategy is a likelihood-ratio test.
In geodesy, tests of significance are often used. A null hypothesis
is formulated and a critical region is chosen which has small
probability under the null hypothesis. If the test statistic obtains a
value that belongs to the critical region, the null hypothesis is
rejected: the data are said to contradict the null hypothesis signif
icantly. I In this approach it is, without further consideration, not
clear what critical region should be chosen. Consider the following
example
The angles of a small triangle have been measured. The misclosure
variate is known to have a standard deviation of 10" (normal
distribution). We wish to test the null hypothesis that its mean is
zero. Suppose we take a critical region between 1.0" and 2.3".
Under the null hypothesis, an actually observed misclosure has a
probability of 5% to fall in this region. Consequently, if we observe
a misclosure of, e.g. 1.5" the null hypothesis is rejected at the 5%
level of significance.
Evidently, this is nonsense, and nobody is suspected of using
such a test in a general case, but then the formulation of the testing
problem should be reviewed. The alternative hypothesis and the
regrets were not explicitly taken into account. They should provide
the basis for the choice of the type of the rejection region and the
level of significance.
Of course in many cases it will be extremely difficult to estimate
the losses of utility with any degree of accuracy. This is so for most
types of decision problems. In practice, one often has to content
oneself with a calculation of the action probabilities, and somehow
use one's experience to weigh the consequences of errors.
I
