Normal Log-Likelihood Functions and their Partials

The complete normal likelihood function (without the constant) is composed of three summation portions:

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 mean parameter (unknown a priori, the first of two parameters to be found)
σ is the standard deviation 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 the ith group of suspension data points
is the time of the ith suspension data group
Fi is the number of interval data groups
is the number of intervals in the ith group of data intervals
is the beginning of the ith interval
and is the ending of the ith interval

The solution will be found by solving for a pair of parameters (μ0, σ0) so that and

where:

and:

Complete Data

Note that for the normal distribution, and in the case of complete data only (as was shown in the Statistical Background chapter), there exists a closed-form solution for both of the parameters or:

and: