Reliability Demonstration

Reliability demonstration is a procedure for testing whether the reliability of a given device (system) at a certain age is sufficiently high. More precisely, a time point $t_0$ and a desired reliability $R_0$ are specified, and we wish to test whether the reliability of the device at age $t_0$, $R(t_0)$, satisfies the requirement that $R(t_0) \geq R_0$. If the life distribution of the device is completely known, including all parameters, there is no problem of reliability demonstration - one computes $R(t_0)$ exactly and determines whether $R(t_0) \geq R_0$. If, as is generally the case, either the life distribution or its parameters are unknown, then the problem of reliability demonstration is that of obtaining suitable data and using them to test the statistical hypothesis that $R(t_0) \geq R_0$ versus the alternative that $R(t_0) < R_0$. Thus, the theory of testing statistical hypotheses provides the tools for reliability demonstration. In the present section we review some of the basic notions of hypothesis testing as they pertain to reliability demonstration.

We develop several tests of interest in reliability demonstration. We remark here that procedures for obtaining confidence intervals for $R(t_0)$, which were discussed in the previous sections, can be used to test hypotheses. Specifically, the procedure involves computing the upper confidence limit of a $(1-2\alpha)$-level confidence interval for $R(t_0)$ and comparing it with the value $R_0$. If the upper confidence limit exceeds $R_0$ then the null hypothesis $H_0 : R(t_0) > R_0$ is accepted, otherwise it is rejected. This test will have a significance level of $\alpha$.

For example, if the specification of the reliability at age $t = t_0$ is $R = .75$ and the confidence interval for $R(t_0)$, at level of confidence $\gamma = .90$, is $(.80,.85)$, the hypothesis $H_0$ can be immediately accepted at a level of significance of $\alpha = (1-\gamma)/2=.05$. There is a duality between procedures for testing hypotheses and for confidence intervals.