stress strength model – R code, simulations, and modeling

In my previous blog post I demonstrated how to compute bootstrap confidence intervals for stress-strength models.
The following R-code may be used for computing likelihood based confidence intervals for stress-strength models.

Definition of the unreliability in case of a stress-strength model
A component may fail when the stress (or load) exceeds the strength. Accordingly, the unreliability of the component is defined as:

$U_{hat} =\int^{+\infty}_{-\infty} f_l(l)F_s(l) \ dl$

where U_hat is the unreliability of the component, f_l(·) the probability density function of the load, and F_s(·) the cumulative density function of the strength.

Note that both the load and strength random variables never take negative values. Since both these random variables are non-negative, the lower limit of the above integral changes. That is, the lower limit changes from -∞ to 0, since f_l(·) and F_s(·) will be 0 for negative values (i.e., for l<0). As a consequence, the above integral is given by:

$U_{hat} =\int^{+\infty}_{0} f_l(l)F_s(l) \ dl$

The reliability of the component is R_hat=1-U_hat. Continue reading R code for constructing likelihood based confidence intervals for stress-strength models

The following R code may be used for constructing bias-corrected percentile confidence intervals for stress-strength models. These bias-corrected percentile confidence intervals are obtained with a bootstrapping method.

The code focuses at first on a situation in which stress and strength both follow a normal distribution. However, any other combination of distributions may be used in the R code (e.g, having stress follow a lognormal distribution, while strength a Weibull distribution). In fact, it will be demonstrated how to change the code such that the stress and strength both follow a Weibull distribution.

A comprehensive account of bootstrap confidence intervals for stress-strength models is given in Barbiero, A. (2011), Confidence Intervals for Reliability of Stress-Strength Models in the Normal Case, Communications in Statistics – Simulation and Computation, 40, 907-925. Note though that the bootstrap confidence intervals described/reported by Alessandro Barbiero in his 2011 paper are standard (or simple) percentile confidence intervals and not bias-corrected percentile confidence intervals.
Alessandro Barbiero also maintains an R package called StressStrength. However, the package is limited to situations in which stress and strength are both normally distributed.

Continue reading R code for constructing bootstrap confidence intervals for stress-strength models