Global Positioning System Reference
In-Depth Information
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A.5.3 Hypothesis Tests
A
hypothesis is a statement about the parameters of a distribution. A test of a hypoth-
es
is is a rule that, based on the sample values, leads to a decision to accept or reject the
nu
ll hypothesis. A test statistic is computed from the sample values (the observations)
an
d from the specifications of the null hypothesis. If the test statistic falls within a
cr
itical region, the null hypothesis is rejected. For example,
v
T
P v
is a test statistic
ha
ving a chi-square distribution. The computed test statistic is
v
T
Pv
. The specifica-
tio
n of the zero hypothesis could be that the a posteriori variance of unit weight has
a
certain numerical value that, in turn, specifies the variance-covariance matrix of
th
e observations (which is a parameter of the multivariate normal distribution of the
ob
servations).
Because the sample statistic is computed from sample values (observations), the
co
mputed value may fall inside the critical region even though the null hypothesis
H
0
is true. There is a probability
[36
that this can happen. One speaks of a type-I error
if
the hypothesis
H
0
is rejected although it is true; the probability of a type-I error
is
α
α
, which, incidentally, is also the significance level of the test. However, there
is
a probability that the sample statistics fall in the critical region when
H
0
is false
(a
nd hence
H
1
is true). That probability is denoted by 1
Lin
—
1.3
——
Nor
PgE
− β
in Figure A.6. If the
sa
mple statistic does not fall in the critical region, but the alternative hypothesis
H
1
is
true, one would mistakenly accept
H
0
and commit a type-II error. The probability
of
committing a type-II error is
.
Figure A.6 displays the probability density functions of the test statistics under
th
e specifications of the null hypothesis
H
0
and the alternative hypothesis
H
1
. The
fig
ure also shows the critical region for which the null hypothesis is rejected, and
β
[36
Figure A.6
Example of probability distributions of test statistics and critical region.