Testing for dependency of failure times in life testing

A method is proposed for testing whether the failure of the first specimen in a test machine that contains multiple specimens shortens the time to failure of the other specimens in that unit. The time to failure is modeled as a two-parameter Weibull distribution. Two estimates of the shape parameter are constructed: an estimate based on the first failures in each of k identical test units, and an estimate based on the ratios of the second to the first failure times within each unit. The supposition is that these ratios will be shortened if the first failure has influenced the second, causing the estimate of the shape parameter constructed from these ratios to tend to be larger than the estimate obtained from the set of first failures. The null distribution of the ratio of the two shape parameter estimates was determined by Monte Carlo sampling for the case of k units containing exactly two specimens each with k = 5(1)20. The power is determined under the assumption that the first failure causes the remaining failures to occur according to a Weibull distribution with the same shape parameter as the original failure distribution but with a scale parameter reduced by the factor c < 1, Numerical results are given for the case where k = 10 and c = .1(.1).9. The article contains a numerical example based on simulated data and a second example based on actual data obtained from fatigue testing of ball bearings. A comparable test is developed for the case where the Weibull shape parameter is assumed to be known.