Due to its nature, maximum likelihood offers a very powerful method in estimating the parameters of accelerated testing models, making possible the analysis of very complex models. In the beginning of this Appendix, a graphical method for obtaining the parameters of accelerated testing models was illustrated. It involved estimating the parameters of the life distribution separately for each individual stress level and then plotting the life-stress relationship in a linear manner on a separate life vs. stress plot. In other words, the life distribution and the life-stress relationship were treated separately. However, using the MLE method, the life distribution and the life-stress relationship can be treated as one complete model that describes both. This can be accomplished by including the life-stress relationship into the pdf of the life distribution.
Background Theory
The maximum likelihood for accelerated life testing analysis is formulated in the same way as shown previously, however in this case the stress level of each individual observation is included in the likelihood function. Consider a continuous random variable x(v), where v is the stress. The pdf of the random variable now becomes a function of both x and v.
where 
are k
unknown constant parameters which need to be estimated. Conduct an experiment
and obtain N independent observations,
x1,
x2,
..., xN each at a corresponding
stress, v1,
v2,
..., vN. Then the likelihood
function for complete data is given by:
(28)
The logarithmic likelihood function is given by:
(29)
The maximum likelihood estimators (MLE) of are obtained
by maximizing L or 
.
In this case, are the parameters of the combined model
which includes the parameters of the life distribution and the parameters
of the life-stress relationship. Note that in Eqns. (28) and (29), N
is the total number of observations. This means that the sample size is
no longer broken into the number of observations at each stress level.
In Example 1, the sample size at the stress level of 20V was 4 and 15
at 36V. Using Eqn. (28) or Eqn. (29) however, the test's sample size is
19.
Once the parameters are estimated, they can be substituted back into the life distribution and the life-stress relationship.
Example 2
The following example illustrates the use of the MLE method on accelerated life test data. Consider the inverse power law relationship, given by:
(30)
where:
L represents a quantifiable life measure, V represents the stress level, K is one of the parameters and n is another model parameter.
Assume that the life at each stress follows a Weibull distribution, with a pdf given by:
(31)
where the time-to-failure, t, is a function of stress, V.
A common life measure needs to determined so that it can be easily included in Eqn. (31). In this case, setting η = L(V) (which is the life at 63.2%) in Eqn. (30) and substituting in Eqn. (31), yields the following IPL-Weibull pdf:
The log-likelihood function for the complete data is given by:
Note that β is now the common shape parameter to solve for, along with K and n.
See Also:
Appendix B: Parameter Estimation
Go to Weibull.com
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