17.6.3.2 Algorithms (Weibull Fit)WeibullFit-Algorithm
For /math-e3c0cbf91ece024841ff9fdb38694186.png?v=0 "n\,\!") realizations,/math-6ce07c4566b882f5b04c4fd4ec104550.png?v=0 "y_i \,\!") , from a Weibull distribution, a value /math-b714d71dd0cdb2c881fae49429ac19db.png?v=0 "x_i \,\!") is observed if /math-27744114b47bd9ef390e92b76574d035.png?v=0 "x_i\leq y_i \,\!")
There are two situations:
Exactly specified observations, when /math-e154fe4295c6340907dbb51128201755.png?v=0 "x_i=y_i \,\!")
Right-censored observations, known by a lower bound, when /math-9d9289175fcc3610c2f8f470c38f4780.png?v=0 "x_i<y_i \,\!")
The probability density function of Weibull distribution, and hence the contribution of an exactly specified observation to the likelihood, given by:
/math-0cd981b980541daefd7c750a14da1f4a.png?v=0 "f(x:\theta ,c,\sigma )=\frac c\sigma (\frac{x-\theta }\sigma )^{c-1}\exp (-(\frac{x-\theta }\sigma )^c),x>\theta ,\;for\;c,\sigma >0 \,\!")
While the survival function of Weibull distribution, and hence the contribution of a right-censored observation to the likelihood, is given by:
/math-f5005cef56de6bfe17a790062a747ef2.png?v=0 "S(x;c,\sigma )=\exp (-(\frac{x-\theta }\sigma )^c),x>\theta ,\;for\;c,\sigma >0 \,\!")
Where /math-eda199874d89f6beec2062820f4b61c9.png?v=0 "\theta\,\!") intercept parameter which also be called threshold parameter, /math-79ce6cdd2db95f0db5cefeeadc51b5bc.png?v=0 "c \,\!") is Weibull shape parameter and /math-596589cfba308c784b8a5b3f8919ff44.png?v=0 "\sigma \,\!") is Weibull scale parameter.
If /math-a3fd7097926ade7405b7390b2e5ec4af.png?v=0 "d\,\!") of the observations are exactly specified and indicated by /math-a15b88855e8100619588ea57a7263edf.png?v=0 "i\in D \,\!") . And the remaining /math-a2ed284e0ed3505ce0b4c57148c8cc9e.png?v=0 "(n-d) \,\!") observations are right-censored. Then the likelihood function, /math-3c15a48d811c9ede164e29bbfad740bb.png?v=0 "Like(c,\sigma) \,\!") is given by:
/math-ef581caa6bbddde71d94ccd736703921.png?v=0 "Like(c,\sigma )=(\frac c\sigma )^d(\coprod_{i\in D}(\frac{x_i-\theta }\sigma )^{c-1})\exp (-\sum_{i=1}^n(\frac{x_i-\theta }\sigma )^c) \,\!")
The kernel likelihood function is given:
/math-e52aca194ac7cf0c814164cc05487575.png?v=0 "L(c,\sigma )=d\log (\frac c\sigma )+(c-1)\sum_{i\in D}\log (\frac{x_i-\theta }\sigma )-\sum_{i=1}^n(\frac{x_i-\theta }\sigma )^c \,\!")
If the derivatives /math-cfc693d2ea1d1f2b883f97d834a00d8f.png?v=0 "\frac{\partial L}{\partial c} \,\!") ,/math-12ed5e98f08d91979c6d4010a27847e0.png?v=0 "\frac{\partial L}{\partial \sigma } \,\!") ,/math-d37536fa5f81d9c755553bf553c7715a.png?v=0 "\frac{\partial ^2L}{\partial c^2} \,\!") ,/math-80437f55c93fa9394fd38030ba93265e.png?v=0 "\frac{\partial ^2L}{\partial \sigma \partial c} \,\!") , /math-3d321d44a3f8f24281598bd1e42b5c07.png?v=0 "\frac{\partial ^2L}{\partial \sigma ^2} \,\!") are denoted by /math-c9db0818268102735575605d2e53818c.png?v=0 "L_1 \,\!") ,/math-c21b34a55d751464f855b7b9abca0fd1.png?v=0 "L_2 \,\!") ,/math-f46a681edb663ec30e93cd4964e6e6a3.png?v=0 "L_{11} \,\!") ,/math-01a887bdd494ffb5960adf403c1434e6.png?v=0 "L_{12} \,\!") ,/math-6dd5f68208fb1d4b7385218c441b13c4.png?v=0 "L_{22} \,\!") respectively, then the maximum likelihood estimates, /math-fb5213ebc765e346c619e447a0d85cb5.png?v=0 "\widehat{c} \,\!") and /math-c064f955ca038366c16a6100a3090ea8.png?v=0 "\widehat{\sigma } \,\!") , are the solutions to the equations: /math-4de64a974da3b6ef5335728b39e38af1.png?v=0 "L_1(\widehat{c},\widehat{\sigma })=0 \,\!") and /math-d3b73c0d4da8bfebd3e66284c4d32b55.png?v=0 "L_2(\widehat{c},\widehat{\sigma })=0 \,\!")
Estimates of the asymptotic standard errors of and are given by:
/math-05426c0f1964584836a47c6ebe00580a.png?v=0 "se(\widehat{c})=\sqrt{\frac{-L_{22}}{L_{11}L_{22}-L_{12}^2}} \,\!") and /math-184a54cc28b4a05b7d3de913331ad65e.png?v=0 "se(\widehat{\sigma })=\sqrt{\frac{-L_{11}}{L_{11}L_{22}-L_{12}^2}} \,\!")
An estimate correlation coefficient of /math-34e3a22fc702ec4461ac776139dd07cd.png?v=0 "\widehat{c}\,\!") and /math-daaaa0a22ff76a70ab86e7bf0c2c6b1b.png?v=0 "\widehat{\sigma } \,\!") is given by:/math-c22c7847d807c5f84bf61871a4b1889a.png?v=0 "\frac{L_{12}}{\sqrt{L_{11}L_{22}}} \,\!")
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