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}}} \,\!