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 \(x_i<y_i \,\!\)

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:\(\frac{L_{12}}{\sqrt{L_{11}L_{22}}} \,\!\)