This log-likelihood function 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
β is the Weibull shape parameter (unknown a priori, the first of two parameters to be found)
η is the Weibull 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
is the ending of the ith intervalFor the purposes of MLE, left censored data will be considered to be
intervals with 
The solution will be found by solving for a pair of parameters (
, 
) so that 
and 
It should be noted that other methods can also be used, such
as direct maximization of the likelihood function, without having to compute
the derivatives.
This log-likelihood function is again 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
β is the Weibull shape parameter (unknown a priori, the first of three parameters to be found)
η is the Weibull scale parameter (unknown a priori, the second of three parameters to be found)
Ti is the time of the ith group of time-to-failure data
γ is the Weibull location parameter (unknown a priori, the third of three parameters to be found)
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 data groups
is the
number of intervals in the ith group of data intervals
is the
beginning of the ith interval
is the ending of the ith intervalThe solution is found by solving for (, , 
) so that 
and
It should be pointed out that the solution to the three-parameter Weibull via MLE is not always stable and can collapse if β ~ 1. In estimating the true MLE of the three-parameter Weibull distribution, two difficulties arise. The first is a problem of "non-regularity" and the second is the "parameter divergence problem" [14].
Non-regularity occurs when
β 
2. In general, there are no MLE solutions in
the region of 0
<
β < 1. When 1 <
β < 2,
MLE solutions exist but are not asymptotically normal [14].
In the case of non-regularity, the solution is treated anomalously.
Weibull++ attempts to find a solution in all
of the regions using a variety of methods, but the user should be forewarned
that not all possible data can be addressed. Thus, some solutions using
MLE for the three-parameter Weibull will fail when the algorithm has reached
predefined limits or fails to converge. In these cases, the user can change
to the non-true MLE approach (in Weibull++ User Setup), where
γ
is estimated using non-linear regression. Once
γ is obtained,
the MLE estimates of and are computed
using the transformation 
See Also:
Appendix C: Distribution Log-Likelihood Equations
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