Question

Ten identical devices are tested until failure at times t_n = 1.7, 3.5, 5.0, 6.5, 8.0, 9.6, 11.0, 13.0, 18.0, 22.0 (X100) h (n=1,...,10). Find the MTTF using: a) moment estimation, b) maximum likelihood estimation. You can assume that the device fails with constant failure rate.

Answer 1

Since the divices has constant failure rate, their pdf is given by

.

The MTTF is . Given the observations , the MLE estimate of is obtained as

loglikelihood is

Differentiating and equating to 0,

Given data 1.7, 3.5, 5.0, 6.5, 8.0, 9.6, 11.0, 13.0, 18.0, 22.0, the MLE estimator of MTTF is

The mean of distribution is .

The first moment is  . So MOM estimator is

So the maximum likelihood and MOM estimators of MTTF are the same.