The log-likelihood function (without the constant) is composed of three summation portions: (Note: Note that to avoid numerical issues inherent in the Mixed Weibull solution, the mixed Weibull likelihood function used within Weibull++ uses a midpoint approximation in the case of interval data. The likelihood function utilizing this approximation is what is presented in this section.)
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
is the
number of subpopulations
ρk is the proportionality of the kth subpopulation (unknown a priori, the first set of three sets of parameters to be found)
βk is the Weibull shape parameter of the kth subpopulation (unknown a priori, the second set of three sets of parameters to be found)
ηk is the Weibull scale parameter (unknown a priori, the third set of three sets of 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 groups of interval data points
is the
number of intervals in ith group of data intervals
is the
beginning of the ith interval
is the ending of the ith intervalThe solution will be found by solving for a group of parameters:
so that:
See Also:
Appendix C: Distribution Log-Likelihood Equations
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