Goodness-of-Fit Tests for Repairable Systems

It is generally desirable to test the compatibility of a model and data by a statistical goodness-of-fit test. A parametric Cramér-von Mises goodness-of-fit test is used for the multiple system and repairable system Power Law model, as proposed by Crow in [17]. This goodness-of-fit test is appropriate whenever the start time for each system is 0 and the failure data is complete over the continuous interval [0,Tq] with no gaps in the data. The Chi-Squared test is a goodness-of-fit test that can be applied under more general circumstances. In addition, the Common Beta Hypothesis test also can be used to compare the intensity functions of the individual systems by comparing the βq values 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. The Cramér-von Mises and Chi-Squared goodness-of-fit tests are illustrated next.

Cramér-von Mises Test

To illustrate the application of the Cramér-von Mises statistic for multiple system data, suppose that K like systems are under study and you wish to test the hypothesis H1 that their failure times follow a non-homogeneous Poisson process. Suppose information is available for the qth system over the interval [0,Tq], with successive failure times , (q = 1, 2, ... , K). The Cramér-von Mises test can be performed with the following steps:

Step 1: If $x_{N_{q}q}=T_{q}$ (failure terminated) let Mq = Nq -1, and if $x_{N_{q}q}<T$ (time terminated) let Mq = Nq. Then: MATH Step 2: For each system divide each successive failure time by the corresponding end time Tq, i = 1, 2, ... , Mq. Calculate the M values: MATH

Step 3: Next calculate $\overline{\beta }$ , the unbiased estimate of β, from: MATH

Step 4: Treat the Yiq values as one group and order them from smallest to largest. Name these ordered values z1, z2, ..., zM, such that z1 < z2 < ...< zM.

Step 5: Calculate the parametric Cramér-von Mises statistic. MATH

Critical values for the Cramér-von Mises test are presented in Table B.2 of Appendix B.

Step 6: If the calculated $C_{M}^{2}$ is less than the critical value then accept the hypothesis that the failure times for the K systems follow the non-homogeneous Poisson process with intensity function u(t) = λβtβ-1.

Example 2

For the data from Example 1, use the Cramér-von Mises test to examine the compatibility of the model at a significance level α = 0.10.

Solution

Step 1: MATH

Step 2: Calculate Yiq treat the Yiq values as one group and order them from smallest to largest. Name these ordered values z1, z2, ..., zM.

Step 3: Calculate MATH

Step 4: Calculate

Step 5: Find the critical value (CV) from Table B.2 for M = 34 at a significance level α = 0.10. CV = 0.172.

Step 6: Since $C_{M}^{2}<CV$ , accept the hypothesis that the failure times for the K = 3 repairable systems follow the non-homogeneous Poisson process with intensity function u(t) = λβtβ-1.

Chi-Squared Test

The parametric Cramér-von Mises test described above requires that the starting time, Sq, be equal to 0 for each of the K systems. Although not as powerful as the Cramér-von Mises test, the Chi-Squared test can be applied regardless of the starting times. The expected number of failures for a system over its age (a, b) for the Chi-Squared test is estimated by , where $\widehat{\lambda }$ and $\widehat{\beta }$ are the maximum likelihood estimates.

The computed χ2 statistic is: MATH where d is the total number of intervals. The random variable χ2 is approximately Chi-Square distributed with df = d -2 degrees of freedom. There must be at least three intervals and the length of the intervals do not have to be equal. It is common practice to require that the expected number of failures for each interval, θ(j), be at least five. If or if , reject the null hypothesis.