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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