The Gamma Log-Likelihood Functions and their Partials

This log-likelihood function is composed of three summation portions:

or:

where:

Fe is the number of groups of times-to-failure data points
Ni is the number of times-to-failure in the ith time-to-failure data group
μ is the gamma shape parameter (unknown a priori, the first of two parameters to be found)
k is the gamma scale parameter (unknown a priori, the second of two parameters to be found)
Ti is the time of the ith group of time-to-failure data
S is the number of groups of suspension data points
is the number of suspensions in ith group of suspension data points
is the time of the ith suspension data group
FI is the number of interval failure data groups
is the number of intervals in ith group of data intervals
is the beginning of the ith interval
and is the ending of the ith interval

For the purposes of MLE, left censored data will be considered to be intervals with

The solution of the maximum log-likelihood function is found by solving for (, ) so that