Inverse Power Law (IPL)-Lognormal Model

This example validates the IPL life stress relationship with a lognormal distribution.

The data set is from Example 19.10 on page 504 in book Statistical Methods for Reliability Data by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998.

A Mylar-Polyurethane Insulating structure was tested under several different voltage settings. The test data is given in the table shown next.

The following function is used for the Ln-Mean $$\,\!\mu {}'$$:


 * $$\,\!\mu {}'=\beta _{0}+\beta _{1}\times ln\left ( V \right )$$

where V is the voltage and its natural log transform is used in the above life stress relation.

This function can be written in the following way:


 * $$\,\!e^{\mu {}'}=e^{\alpha _{0}+\alpha _{1}log\left ( V \right )}$$

The above equation is the general log-linear model in ALTA. In ALTA, the coefficients are denoted by $$\,\!\alpha _{i}$$.

In fact, the above model also can be expressed using the traditional IPL (inverse power law) model:


 * $$\,\!e^{\mu {}'}=\frac{1}{K\cdot V^{n}}$$

where $$\,\!K=e^{-\alpha _{0}}$$ and $$\,\!n=-\alpha _{1}$$.

In the book, the following results are provided:


 * ML estimations for the model parameters are: $$\,\!\sigma =1.05$$, $$\,\!\beta _{0}=27.5$$ and $$\,\!\beta _{1}=-4.29$$.


 * The standard deviation of each parameter are: $$\,\!std\left ( \sigma \right )=0.12$$, $$\,\!std\left ( \beta _{0}  \right )=3.0$$ and $$\,\!std\left ( \beta _{1}  \right )=0.6$$.


 * Therefore, their variances are: $$\,\!Var\left ( \sigma \right )=0.0144$$, $$\,\!Var\left ( \beta _{0}  \right )=9$$ and $$\,\!Var\left ( \beta _{1}  \right )=0.36$$.


 * The log-likelihood value is -271.4.


 * The 95% two-sided confidence intervals are: for $$\,\!\sigma$$, it is [0.83, 1.32]; for $$\,\!\beta _{0}$$ , it is [21.6, 33.4]; and for $$\,\!\beta _{1}$$ , it is [-5.46, -3.11].


 * ML estimations for the model parameters are:




 * The variance and covariance matrix for model parameters is:




 * The log-likelihood value is -271.4247.


 * The 95% two-sided confidence intervals are: