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Nelson's diesel generator fan data: Weibull MLE with 58 suspensions and Fisher-matrix bounds

ToolWeibull Life Analysis
SourceNelson, W. (1982). Applied Life Data Analysis. Wiley, New York. p. 317 (70 generator fans, 12 failures, 344,440 hours); ReliaSoft Corporation (2015). "The Weibull Distribution." ReliaWiki (Life Data Analysis Reference). Benchmark examples, Nelson fan data (copy in References/Weibull/).
LevelGraduate

Problem

Seventy diesel generator fans accumulated 344,440 hours of service; twelve failed and the other 58 were still running when the data were collected. Fit the two-parameter Weibull by maximum likelihood with the suspensions in the likelihood and give the two-sided 95 % Fisher-matrix confidence bounds on the shape and scale.

Inputs to Enter

  • Failures: 450 F; 1150 F 2; 1600 F; 2070 F 2; 2080 F; 3100 F; 3450 F; 4600 F; 6100 F; 8750 F
  • Suspensions: 460 S; 1560 S; 1660 S; 1850 S 5; 2030 S 3; 2200 S; 3000 S 4; 3200 S; 3750 S 2; 4150 S 4; 4300 S 4; 4850 S 4; 5000 S 3; 6100 S 3; 6300 S; 6450 S 2; 6700 S; 7450 S; 7800 S 2; 8100 S 2; 8200 S; 8500 S 3; 8750 S 2; 9400 S; 9900 S; 10100 S 3; 11500 S
  • Primary method: maximum likelihood; confidence 95 %
Open the Weibull Life Analysis tool

Answers

  • β̂ = 1.0584, η̂ = 26,297 hr (Nelson prints 26,296)
  • 95 % bounds on β: 0.6441 to 1.7394
  • 95 % bounds on η: 10,552 to 65,532 hr (ReliaWiki prints the lower bound as 10,522; the independent calculation gives 10,552, so 10,522 is taken as a transposition)
  • Exponential alternative: λ = 12 / 344,440 per hour; log-likelihoods Weibull −135.1527, lognormal −134.5496 (μ 10.1432, σ 1.6796)

Notes

Paste the 37 lines above (the trailing integer is a count) and set the confidence to 95 %. The data are the R survival package's genfan set (References/Weibull/Data/). Bounds reproduced to 1e-4 on β and 5 hr on η (test_weibull.php).

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