Lognormal Log-Likelihood Functions and their Partials

The general log-likelihood function (without the constant) for the lognormal distribution 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

• is the mean of the natural logarithms of the times-to-failure (unknown a priori, the first of two parameters to be found)

• is the standard deviation of the natural logarithms of the times-to-failure (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 (, ) so that and :

where:

and:

See Also:
Appendix C: Distribution Log-Likelihood Equations

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