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Meeker and Escobar Example 19.5: Device-A Arrhenius-lognormal fit with a use-temperature cell of no failures
| Tool | Accelerated Life |
| Source | Meeker, W. Q., and Escobar, L. A. (1998). Statistical Methods for Reliability Data. Wiley, New York. Example 19.5 (Device-A). |
| Level | Graduate |
Problem
Device-A units were tested for 5,000 hours at 10, 40, 60 and 80 °C: none of 30 failed at 10 °C, 10 of 100 at 40 °C, 9 of 20 at 60 °C and 14 of 15 at 80 °C. Fit the Arrhenius-lognormal model by maximum likelihood to all four cells and report the log-likelihood, the activation energy and the common slope.
Inputs to Enter
- Model Arrhenius, distribution lognormal, temperatures in K
- 283.15 K: 30 suspended at 5,000 h
- 313.15 K: failures 1298, 1390, 3187, 3241, 3261, 3313, 4501, 4568, 4841, 4982 h; 90 suspended at 5,000 h
- 333.15 K: failures 581, 925, 1432, 1586, 2452, 2734, 2772, 4106, 4674 h; 11 suspended at 5,000 h
- 353.15 K: failures 283, 361, 515, 638, 854, 1024, 1030, 1045, 1767, 1777, 1856, 1951, 1964, 2884 h; 1 suspended at 5,000 h
Answers
- Log-likelihood −321.7 (book and Weibull++)
- ΔH = 0.627853 eV (s.e. 0.08283), ln A = −13.46865, σ = 0.977823 (scipy cross-check of the same fit)
- The 10 °C cell has no own fit (no failures) and is left out of the likelihood-ratio test, which uses the three cells with failures
Notes
Paste the four blocks into the data box with the temperatures in kelvin. Only the log-likelihood is printed by the book; the parameter values are the tool's, confirmed by an independent scipy fit.