# 17.6.3.2 Algorithms (Weibull Fit)

For $n\,\!$ realizations, $y_i \,\!$ , from a Weibull distribution, a value $x_i \,\!$ is observed if $x_i\leq y_i \,\!$

There are two situations:

Exactly specified observations, when $x_i=y_i \,\!$

Right-censored observations, known by a lower bound, when $x_i

The probability density function of Weibull distribution, and hence the contribution of an exactly specified observation to the likelihood, given by: $f(x:\theta ,c,\sigma )=\frac c\sigma (\frac{x-\theta }\sigma )^{c-1}\exp (-(\frac{x-\theta }\sigma )^c),x>\theta ,\;for\;c,\sigma >0 \,\!$

While the survival function of Weibull distribution, and hence the contribution of a right-censored observation to the likelihood, is given by: $S(x;c,\sigma )=\exp (-(\frac{x-\theta }\sigma )^c),x>\theta ,\;for\;c,\sigma >0 \,\!$

Where $\theta\,\!$ intercept parameter which also be called threshold parameter, $c \,\!$ is Weibull shape parameter and $\sigma \,\!$ is Weibull scale parameter.

If $d\,\!$ of the $n\,\!$ observations are exactly specified and indicated by $i\in D \,\!$. And the remaining $(n-d) \,\!$ observations are right-censored. Then the likelihood function, $Like(c,\sigma) \,\!$ is given by: $Like(c,\sigma )=(\frac c\sigma )^d(\coprod_{i\in D}(\frac{x_i-\theta }\sigma )^{c-1})\exp (-\sum_{i=1}^n(\frac{x_i-\theta }\sigma )^c) \,\!$

The kernel likelihood function is given: $L(c,\sigma )=d\log (\frac c\sigma )+(c-1)\sum_{i\in D}\log (\frac{x_i-\theta }\sigma )-\sum_{i=1}^n(\frac{x_i-\theta }\sigma )^c \,\!$

If the derivatives $\frac{\partial L}{\partial c} \,\!$, $\frac{\partial L}{\partial \sigma } \,\!$, $\frac{\partial ^2L}{\partial c^2} \,\!$, $\frac{\partial ^2L}{\partial \sigma \partial c} \,\!$, $\frac{\partial ^2L}{\partial \sigma ^2} \,\!$ are denoted by $L_1 \,\!$, $L_2 \,\!$ , $L_{11} \,\!$ , $L_{12} \,\!$ , $L_{22} \,\!$ respectively, then the maximum likelihood estimates, $\widehat{c} \,\!$ and $\widehat{\sigma } \,\!$, are the solutions to the equations: $L_1(\widehat{c},\widehat{\sigma })=0 \,\!$ and $L_2(\widehat{c},\widehat{\sigma })=0 \,\!$

Estimates of the asymptotic standard errors of $\widehat{c} \,\!$and $\widehat{\sigma } \,\!$ are given by: $se(\widehat{c})=\sqrt{\frac{-L_{22}}{L_{11}L_{22}-L_{12}^2}} \,\!$ and $se(\widehat{\sigma })=\sqrt{\frac{-L_{11}}{L_{11}L_{22}-L_{12}^2}} \,\!$

An estimate correlation coefficient of $\widehat{c}\,\!$ and $\widehat{\sigma } \,\!$ is given by: $\frac{L_{12}}{\sqrt{L_{11}L_{22}}} \,\!$