# Random Censoring scheme in Weibull Distribution

I'm trying to derive the estimators of the parameters using maximum likelihood method for Weibull distribution in random censoring scheme

$$f(t)=\alpha\lambda(\lambda t)^{\alpha -1}e^{{-\lambda t}^{\alpha}}$$,$$\alpha >0$$,$$\lambda>0$$

Now i reparametrize $$\gamma=\lambda^{\alpha}$$ then $$f(t)=\gamma\alpha t^{\alpha-1} e^{-\gamma t^{\alpha}}$$

here $$n_{u}$$=number of uncensored observations

So the survival function $$S(t)=e^{-\gamma t^{\alpha}}$$

L=$$(\gamma \alpha)^{n_{u}}$$ $$(\prod_{u} t_{i} ^{\alpha-1})$$ exp{$$-\gamma \sum_{u}t_{i} ^{\alpha}$$} exp{$$-\gamma \sum_{c} c_{i} ^{\alpha}$$},

=$$(\gamma \alpha)^{n_{u}}$$($$\prod_{u} t_{i} ^{\alpha -1}$$)exp{$$-\gamma \sum_{i=1}^n y_{i} ^{\alpha}$$}

Then log L=$$n_{u}log \gamma$$ + $$n_{u} log \alpha$$ + $$(\alpha -1) \sum_{u} log t_{i}$$ + $$\gamma \sum_{i=1}^n y_{i} ^{\alpha}$$

So, $$\frac{\partial}{\partial \gamma}log L=\frac{n_{u}}{\gamma}-\sum_{i=1}^n y_{i} ^{\alpha}$$

$$\frac{\partial}{\partial \alpha}log L=\frac{n_{u}}{\alpha}+\sum_{u} log t_{i}- \gamma\sum_{i=1}^n y_{i} ^{\alpha} log y_{i}$$

Now to write a program in R and to simulate the results,I need to obtain equations for $$\hat{\alpha}$$ and $$\hat{\gamma}$$ in the form to solve NR method.

Suggest me a way to proceed

• You have to solve them numerically. – whuber Jan 22 at 14:16