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Random Censoring scheme in Weibull Distribution

Cross Validated Asked by Soham Bagchi on December 8, 2020

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

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

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

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

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

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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