The pdf for the Inverse Power Law relationship and the Weibull distribution is given next.
The IPL-Weibull model can be derived by setting η = L(V), yielding the following IPL-Weibull pdf:
This is a three-parameter model. Therefore it is more flexible but it also requires more laborious techniques for parameter estimation. The IPL-Weibull model yields the IPL-exponential model for β = 1.
IPL-Weibull Statistical Properties Summary
Mean or MTTF
The mean, 
, (also called MTTF), of the IPL-Weibull model
is given by:
where 
is the gamma function evaluated at the value
of 
.
Median
The median, 
of the IPL-Weibull model is given by:
(3)
Mode
The mode, 
of the IPL-Weibull model is given by:
(4)
Standard Deviation
The standard deviation, 
of the IPL-Weibull model is given
by:
IPL-Weibull Reliability Function
The IPL-Weibull reliability function is given by:
Conditional Reliability Function
The IPL-Weibull conditional reliability function at a specified stress level is given by:
or:
Reliable Life
For the IPL-Weibull model, the reliable life, TR, of a unit for a specified reliability and starting the mission at age zero is given by:
(5)
IPL-Weibull Failure Rate Function
The IPL-Weibull failure rate function, λ(T), is given by:
Parameter Estimation
Maximum Likelihood Estimation Method
Substituting the inverse power law model into the Weibull log-likelihood function yields:
where:
and:
Fe is the number of groups of exact times-to-failure data points.
Ni is the number of times-to-failure data points in the ith time-to-failure data group.
β is the Weibull shape parameter (unknown, the first of three parameters to be estimated).
K is the IPL parameter (unknown, the second of three parameters to be estimated).
n is the second IPL parameter (unknown, the third of three parameters to be estimated).
Vi is the stress level of the ith group.
Ti is the exact failure time of the ith group.
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 running 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
interval.
The solution (parameter estimates) will be found by solving for β,
K,
n
so that 
= 0, 
= 0 and 
=
0, where:
Example 1
Consider the following times-to-failure data at two different stress levels.
The data set was analyzed jointly and with a complete MLE solution over the entire data set using ReliaSoft's ALTA. The analysis yields:
= 2.61647
= 0.00102241
= 1.32729123
See Also:
Inverse Power Law Relationship
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