While using the Crow-AMSAA model
there are four goodness-of-fit tests which, depending on their applicability,
will become available. The
Cramér von Mises goodness-of-fit
test tests the hypothesis that the data follows a nonhomogeneous Poisson
process with failure intensity equal to . This test can be applied when the failure data is complete
over the continuous interval with no gaps
in the data. The Chi-Squared test is a goodness-of-fit test that can be
applied under more general circumstances, particularly when the data is
grouped. In addition, the Common Beta Hypothesis test can also be used
to compare the intensity functions of the individual systems by comparing
the 's of each system. Lastly, the Laplace
Trend test checks for trends within the data. Due to their general application,
the Common Beta Hypothesis test and the Laplace Trend test are both presented
in Appendix
B.
Cramér von Mises
If the individual failure times are
known, a Cramér von Mises statistic is used to test the null hypothesis
that a non-homogeneous Poisson process with failure intensity function
properly describes the reliability growth
of a system. The Cramér-von Mises goodness-of-fit statistic is then given
by the following expression:
where:
The failure times, Ti,
must be ordered so that T1 < T2 < ...< TM.
If the statistic exceeds the critical value corresponding to M for a chosen significance level, then the
null hypothesis that the Crow-AMSAA model adequately fits the data shall
be rejected. Otherwise, the model shall be accepted. Critical values of
are shown in Appendix B,
Table B.2,
where the table is indexed by the total number of observed failures, M.
Chi-Squared
A Chi-Squared goodness-of-fit test
is used to test the null hypothesis that the Crow-AMSAA reliability model
adequately represents a set of grouped data. The expected number of failures
in the interval from Ti-1 to Ti is approximated by:
(56)
For each interval, shall not be less than 5 and, if necessary, adjacent intervals
may have to be combined so that the expected number of failures in any
combined interval is at least 5. Let the number of intervals after this
recombination be d
and let the observed number of failures in the ith new interval be Niand
the expected number of failures in the ith new interval be . Then the following statistic is approximately distributed
as a Chi-Squared random variable with degrees of freedom d - 2.
The null hypothesis is rejected if
the statistic exceeds the critical value
for a chosen significance level. This means that the hypothesis that the
Crow model adequately fits the grouped data shall be rejected. Critical
values for this statistic can be found in tables of the Chi-Squared distribution.
Crow-AMSAA Example 6
An aircraft has scheduled inspections
at intervals of 20 flight hours. The data set from the first 100 hours
of flight testing are given in Table 5.4.
Table
5.4 - Grouped test data
Start Time
End Time
Number Failures
0
20
13
20
40
16
40
60
5
60
80
8
80
100
7
Determine
the maximum likelihood estimators.
Evaluate the goodness-of-fit.
Solution to Crow-AMSAA Example 6
Obtain the estimator of using Eqn.
33. Using RGA the value
of is 0.75285. Now plug this value into
Eqn. 32 and is:
There are a
total of M = 49
observed failures from d
= 5 intervals. Table 5.5 shows that those adjacent intervals do not have
to be combined after applying Eqn. 56 to the original intervals.
Table
5.5 - Observed vs. Expected Number of Failures for Grouped data
Start Time
End Time
Observed Number of Failures
Expected Number of Failures
0
20
13
14.59
20
40
16
9.99
40
60
5
8.77
60
80
8
8.07
80
100
7
7.58
To test the model's goodness-of-fit,
a Chi-Squared statistic of 5.45 is compared to the critical value of 7.8
corresponding to 3 degrees of freedom and a 0.05 significance level. Since
the statistic is less than the critical value, the applicability of the
Crow-AMSAA model is accepted.
. This test can be applied when the failure data is complete
over the continuous interval ![$[0,T_{q}]$](amsaa__466.gif)
with no gaps
in the data. The Chi-Squared test is a goodness-of-fit test that can be
applied under more general circumstances, particularly when the data is
grouped. In addition, the Common Beta Hypothesis test can also be used
to compare the intensity functions of the individual systems by comparing
the 
's of each system. Lastly, the Laplace
Trend test checks for trends within the data. Due to their general application,
the Common Beta Hypothesis test and the Laplace Trend test are both presented
in Appendix
B.
properly describes the reliability growth
of a system. The Cramér-von Mises goodness-of-fit statistic is then given
by the following expression:
exceeds the critical value corresponding to 
are shown in 
(56)
shall not be less than 5 and, if necessary, adjacent intervals
may have to be combined so that the expected number of failures in any
combined interval is at least 5. Let the number of intervals after this
recombination be 
. Then the following statistic is approximately distributed
as a Chi-Squared random variable with degrees of freedom 
statistic exceeds the critical value
for a chosen significance level. This means that the hypothesis that the
Crow model adequately fits the grouped data shall be rejected. Critical
values for this statistic can be found in tables of the Chi-Squared distribution.
using 
is 0.75285. Now plug this value into
is:
