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ReliaWiki RRY and RRX examples: median-rank regression of six failure times

ToolWeibull Life Analysis
SourceReliaSoft Corporation (2015). "The Weibull Distribution." ReliaWiki (Life Data Analysis Reference). Rank regression on Y and rank regression on X examples on six failures at 16, 34, 53, 75, 93 and 120 hr (copy in References/Weibull/ReliaWiki/).
LevelUndergraduate

Problem

Six units failed at 16, 34, 53, 75, 93 and 120 hours. Assign the exact median ranks, linearize the Weibull CDF and fit the line by least squares, regressing on Y and alternatively on X; report the slope, intercept, correlation coefficient, the Weibull parameters each gives and the reliable life at R = 90 % from the RRX fit.

Inputs to Enter

  • Life data: 16 F, 34 F, 53 F, 75 F, 93 F, 120 F (loaded by default)
  • Primary method: rank regression; median ranks: exact (not Bernard's approximation)
Open the Weibull Life Analysis tool

Answers

  • Exact median ranks 0.1091, 0.2645, 0.4214, 0.5786, 0.7355, 0.8909; Σ ln t = 23.9068, Σ y = −3.0070, Σ (ln t)² = 97.9909, Σ y² = 7.1502, Σ x y = −8.0699
  • RRY: b = 1.4301, a = −6.19935, β = 1.4301, η = 76.318 hr, ρ = 0.9956 (the pdf version prints 1.4302 / 76.317)
  • RRX: b′ = 0.6931, a′ = 4.3318, β = 1.4428, η = 76.0811 hr
  • RRX reliable life at R = 90 %: 15.9933 hr

Notes

Switch the primary method to rank regression and the median ranks to exact; the tool's default Bernard ranks give β within 1 % of these values. Reproduced to 2e-4 on β and 0.01 hr on η (test_weibull.php).

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