Statistics of Reliability

A common method of estimating parameters related to component (system) reliability is that of life testing. This consists of selecting a random sample of n components, testing them under specific environmental conditions, and observing the time to failure of each component.

The following examples are drawn from the blue book.

Example 10.7

Assume that the time to failure, X, of a telehone switching system exponentially distributed with a failure rate . We wish to estimate the failure rate from a random sample of n times to failure. Then the likelihood function is:



from which we get the maximum-likelihood estimator of the failure rate to be the reciprocal of the sample mean:



The corresponding maximum-likelihood estimator of the mean life (MTTF) is equal to the sample mean .

Usually, the MTTF (mean time to failure) is so large as to forbid such exhaustive life tests; hence truncated (censored) life tests are common. Such a life test is terminated after the first r failures have occured (sample-truncated, or type II) or after a specific time has elapsed (time-truncated, or type I).

Example 10.9

Consider a sample truncated etst of n components without replacement. Let T₁, T₂, . . ., T_r be the observed times to failure so that T₁ T₂ ... T_r. Specific values of these random variables are denoted by t₁ t₂ ... t_r. Let be the MTTF to be estimated and assume that components follow an exponential failure law.
Since (n-r) componenets have not failed when the test is completed, the likelihood function is defined in the following way. Assume T_r+1, . . . , T_n are the times to failure of the remaining components, whose failures will not actually be observed. Then



and, dividing by the product of h_i's and taking the limit as h_i ->0, we get



Let



be the accumulated life on the test. Differentiating the likelihood function with respect to and setting it equal to zero, we get

Then the maximum-likelihood estimator (MLE) of the mean life is given by

Thus the estimator of the mean life is given by the accumulated life on test, S_n ; r, divided by the number of observed failures.

Example 10.19

Assume that n = 50 chips are placed on a life test without replacement and the test is to be truncated after r = 10 failures have been observed. Observed failure times are t₁ = 80, t₂ = 95, t₃ = 370, t₄ = 415, t₅ = 590, t₆ = 635, t₇ = 835, t₈ = 895, t₉ = 895, t₁₀ = 960 h. Then

S_n;r = (80 + 95 +370 + 415 + 505 + 590 + 635 + 835 + 895 + 960) + (50 - 10)960 = 43,780 h.

The estimated mean life is = 43780/10 = 4378 h, and the estimated failure rate is = 0.0002284 failures per hour. Finally, a 90% confidence interval for mean life is



or 2787 < < 8069 h,

where = 31.410 and = 10.851 values are obtained from a table of chi-square distributions with 20 degrees of freedom.

Recalling that the interevent times of a Poisson process are exponentially distributed, we can obtain a confidence interval for the average rate. Assume that a Poisson process of rate is observed until a fixed number n of events have been counted. Let X_i denote the time between the (i-1)st and the ith event. Then X_i is exponentially distributed with parameter , and the statistic



is n-stage Elrang with parameter . It follows that 2S_n is chi-square distributed with 2n degrees of freedom. Consequently



is a confidence interval for , with confidence coefficient (1-). Sometimes a one-sided confidence interval is sought in place of the two-sided interval given above. For example, an upper one-sided confidence interval of the mean life is denoted by (_L, ) where _L is known as the lower confidence limit. Since 2S_{n ; r} chi-square distributed with 2r degrees of freedom, we have

It follows that



Similarly, a lower one-sided confidence interval of the mean life is denoted by (0,_U), where a value of the upper confidence limit _U is given by



Example 4

We note that for a chi-square distribution with 2r = 20 degrees of freedom, = = 28.41 and = = 12.443. It follows that the 90% lower confidence limit of the mean life is given by



Therefore, with 90% confidence we can assert taht the true mean life is greater than 2082 h. The 90% upper confidence limit is



Therefore, with 90% confidence we can assert that the true mean life is less than 7036 h.

For ultra-high-reliability systems, the mean life may be much larger than the duration of a normal "mission." In this case we are more interested in obtaining a confidence interval for ssytem reliability given a mission time t. We proceed to derive such a confidence interval starting from the 100(1-)% upper one-sided confidence of the mean life . Thus



(since the exponential is a monotonic function)



In other words



is the lower 100(1-)% confidence limit for the reliability for the reliability, given a mission time t. Note that the chi-square distribution here has 2r degrees of freedom, since we are discussing a test, without replacement, truncated after r failures.

---

Many more examples of statistics of reliability can be found in chapters 10 and 11 of the blue book.