# 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

## Related Questions

### Why does a class weight fraction improve precision compared to undersampling approach where precision drops?

1  Asked on November 12, 2021

### Is it possible to implement an activation function or layer in Keras that uses two distinct sets of weights?

1  Asked on November 12, 2021

### Expert Knowledge Acquisition and Machine learning

1  Asked on November 12, 2021

### Improving F1 scores using models with good precision and recall

1  Asked on November 12, 2021

### Is ROCR applied to training data or testing data?

1  Asked on November 12, 2021 by fcas80

### Is the plot “White noise”?

0  Asked on November 12, 2021

### Multilabel Tweet Classification

0  Asked on November 12, 2021 by vineet

### Propagating uncertainty through nested random forest models

0  Asked on November 12, 2021

### Identify a contaminated distribution

1  Asked on November 12, 2021 by stephen-clark

### How do we generate the samples of hidden root nodes in the Bayes network (Sigmoid Belief Networks) of a generative model

1  Asked on November 12, 2021 by user6703592

### Which monotone transformations give a very loose confidence interval in transformed space?

0  Asked on November 12, 2021

### Restricted standard deviation of survival time

1  Asked on November 12, 2021 by emma-jean

### What does the term episode mean in meta-learning?

2  Asked on November 12, 2021

### Classification technique to classify categories in two variables when dateset has larger number of numerical variables and few data points

0  Asked on November 12, 2021 by rbeginner

### difference partial dependence and feature weights

1  Asked on November 12, 2021

### Interpreting HRs from stratified cox survival analysis in R

1  Asked on November 12, 2021

### Compare two datasets and whether they agree

2  Asked on November 9, 2021 by jennifer-ruurs

### Handling daily time series data for better accuracy

1  Asked on November 9, 2021 by joy_1379

### Model tuning in the presence of incorrect training labels

1  Asked on November 9, 2021 by astel

### Term for the error in machine learning as a direct result of incorrectly labelled data?

0  Asked on November 9, 2021