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EDF & Laboratoire Jean Kuntzmann (LJK)
Goodness-of-fit tests for the Weibull distribution withcensored data
Florian PRIVÉ1, Olivier GAUDOIN1 & Emmanuel REMY2
1. Univ. Grenoble Alpes, Laboratoire Jean Kuntzmann, France2. EDF R&D, Industrial Risk Management, Chatou, France
2 Definitions and recallsCensoring and testsThe Weibull distribution
3 GOF tests for censored samplesTests based on probability plotsTests based on the empirical distribution functionTests based on the normalized spacingsSimplified likelihood based tests
2 Definitions and recallsCensoring and testsThe Weibull distribution
3 GOF tests for censored samplesTests based on probability plotsTests based on the empirical distribution functionTests based on the normalized spacingsSimplified likelihood based tests
Risk management of industrial facilities, such as EDF’s (major Frenchelectric utility) power plants, needs to accurately predict system reliability:
Building of relevant probabilistic models,Statistical inference of the developed models,Validation of the fitted models using statistical criteria such asgoodness-of-fit tests.
The most usual models for lifetimes are the exponential and Weibulldistributions.
2 Definitions and recallsCensoring and testsThe Weibull distribution
3 GOF tests for censored samplesTests based on probability plotsTests based on the empirical distribution functionTests based on the normalized spacingsSimplified likelihood based tests
Goodness-of-fit testNull hypothesis:H0 : “X1, . . . ,Xn is a sample from the Weibull distribution.”Alternative:H1 : “X1, . . . ,Xn is not a sample from the Weibull distribution.”
Note thatOnly the part X ∗1 , . . . ,X
∗m of X1, . . . ,Xn is observed.
We test the assumption that the sample comes from the family ofWeibull distributions (with unknown parameters), NOT that itcomes from a fully specified Weibull distribution.
2 Definitions and recallsCensoring and testsThe Weibull distribution
3 GOF tests for censored samplesTests based on probability plotsTests based on the empirical distribution functionTests based on the normalized spacingsSimplified likelihood based tests
These tests, adapted from the corresponding ones for complete data (Kritet al, 2016), are based on generalized Weibull distributions (with 3parameters). For instance:
EW(θ, η, β), whose cdf is:
[1− e−(x/η)
β]θ
(10)
then we can test θ = 1 (particular case of Weibull with only 2parameters) vs θ 6= 1.AW(ξ, η, β) whose cdf is:
Shapiro-Wilk type tests, based on the ratio of two linear estimators ofσ = 1/β,Tests based on the Kullback-Leibler information,Others not presented here.
The goal was to be as thorough as possible in order to obtain the bestperforming test statistics.
The powers of a total of 75 goodness-of-fit tests were investigated.
2 Definitions and recallsCensoring and testsThe Weibull distribution
3 GOF tests for censored samplesTests based on probability plotsTests based on the empirical distribution functionTests based on the normalized spacingsSimplified likelihood based tests
We used:500 000 simulations in order to estimate the quantiles of the testsstatistics distribution under H0,100 000 simulations in order to estimate the power of a given teststatistic for a given alternative (16 different ones).simulations for n = 50 and m ∈ {25, 50}.and others for different values of n and m.
We have chosen usual alternatives of the Weibull distribution:the Gamma distribution G(k , θ),the Lognormal distribution LN (µ, σ),the Inverse-Gamma distribution IG(α, β),
but also new ones introduced in Krit et al (2014):several GW distributions: AW(ξ, η, β), EW(θ, η, β) and GG(k, η, β),the distributions I and II of Dhillon: D1(β, b) and D2(λ, b),the Inverse Gaussian distribution IS(µ, λ),Chen’s distribution C(λ, β),
We grouped them according to the shape of their hazard rate:increasing hazard rate (IHR)upside-down bathtub-shaped hazard rate (UBT)decreasing hazard rate (DHR)bathtub-shaped hazard rate (BT)
lifetimes of components in EDF hydropower plants,this sample is censored at more than 95%: it is only the m = 15 firstfailure times of n = 351 components.
If we run a Tiku-Singh test on these values, we get a p-value of 46.4%.
The Weibull assumption can’t be rejected for those data.
2 Definitions and recallsCensoring and testsThe Weibull distribution
3 GOF tests for censored samplesTests based on probability plotsTests based on the empirical distribution functionTests based on the normalized spacingsSimplified likelihood based tests