Example of GED with Complete data

Suppose we have a two component system connected in series. The failure distribution for component 1 and component 2 are given by and respectively. We assume that the scale parameter () for both the components as same and the shape parameters are  for component 1 and  for component 2 respectively. Consider we have n such systems which are identical to each other. We assume that  are independent random variables for i=1, 2,…,n and j=1,2 and they are distributed according to Generalized Exponential distribution with parameters ().  and are the time to failure for the ith system due to component 1 and component 2 respectively and  denote the cause of failure for the i^th system where =1, 2. Then, the time to failure for the ith system will be the minimum of the two, i.e., . Then the expected lifetime of the system due to cause 1 is , where  is evaluated using the method of moments  and is given by = , where  is the digamma function. The relative risk rate () due to cause 1 is  an similarly .Suppose that out of the n systems  fail due to cause 1 and  fail due to cause 2. Then the likelihood function for the observed data is given by

(12.7)

and the log Likelihood Function () is given by

(12.8)

The corresponding likelihood equations can be obtained by taking the partial derivatives with respect to the parameters  and remembering the fact that for evaluating the MLEs of we have to simultaneously solve the set of equations (12.7) and (12.8). As they do not have a closed form solution, hence we use the multidimensional Newton Raphson Method to numerically solve the Likelihood equations to get the MLE estimates .