Reliability HotWire |
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Reliability Basics | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

This test is important because it warns the analyst about the possibility
that the systems, which are assumed to be similar, are in fact different (
The Common Beta Hypothesis test is applicable to the following data types
available in RGA 6:
Multiple Systems (Unknown Equivalent Times), Repairable and Fleet. As shown
by Crow [1], suppose that
(1)
where β_{1} =
β_{2} =...= β_{K}. In other words, the CBH
tests for the hypothesis that the interarrival rate of the failures across
the systems are fairly consistent.
Let denote the conditional maximum likelihood estimate of , which is given by:
(2)
where: -
*K*= 1 -
*M*=_{q}*N*if data on the_{q}*q*^{th}^{ }system is time terminated or*M*if data on the_{q}= (N_{q}- 1)*q*system is failure terminated (^{th }*N*is the number of failures on the_{q}*q*system)^{th } -
*X*is the_{iq}*i*time-to-failure on the^{th}*q*system^{th}
Then for each system, assume that:
are
conditionally distributed as independent Chi-Squared random variables with K = 2, you can test the
null hypothesis, H, using the following statistic:_{o}
(3)
If
F equals
and
has conditionally an F-distribution with (2M)
degrees of freedom. The critical value, _{1}, 2M_{2}F, can then be determined by
referring to the Chi-Squared tables. Now, if K ≥ 2, then the
likelihood ratio procedure [1] can be used to test the hypothesis β
=_{1} β=_{2}
...= β. Consider the following statistic:_{K}
where:
Also, let:
Calculate the statistic
The
statistic D falls between the critical points.
Given that the intensity function for the
test the hypothesis that β
while assuming a significance level equal to 0.05. Calculate
and
using Eqn. 2._{2},
Then:
Using Eqn. 3, calculate the statistic
Since 1.2408 < 2.0980, the hypothesis that β
is accepted at the 5% significance level. Now suppose instead it is desired
to test the hypothesis that _{2 }β=_{1} β=_{2}
β_{3}. Calculate the statistic D using Eqn. 4.
Using the Chi-Squared tables with β=_{2} β_{3} at
the 5% significance level.
This analysis can be replicated in RGA 6. The data is entered in a Multiple Systems (Unknown Equivalent Times) data sheet by making the following selections in the Data Type Expert.
The data set is entered as follows:
The significance level can be specified in the
User Setup (available from
After analyzing the data, the CBH test result (Passed or Failed) will be displayed in the Results area of the Data Folio Control Panel, as shown next.
More detailed CBH results can be found in the
Results window, which can be accessed by selecting
The Results window shown above tells us that
0.1026 < D < 5.9915, therefore the data passed the CBH test (
1. Crow, L.H., "Reliability Analysis for Complex, Repairable Systems in Reliability and Biometry," SIAM, ed. by Proschan and R. J. Serfling, Philadelphia, Pennsylvania, pp. 379-410, 1974. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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