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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

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

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