Accelerated life tests are used to obtain advance information on the distribution
of the
life-time,
also called
time to failure,
of engineering systems. Test units are subjected to higher than usual levels of stress
or stresses like temperature, voltage, pressure, humidity, etc. The results
from these tests, i.e., the observed life-time data, are used to make predictions
about product life under more moderate conditions of use, called
use stress.
In the following the basic statistical concepts in accelerated life testing are explained.
Stress dependence of life-time distributions.
The dependence of the cumulative
distribution function
of a life-time on the applied stress
was modelled for exponentially distributed life-times, i.e.,
by the dependence of the parameter
on
.
For the relationship between
and the parameter
certain functions
were used with a known function
and unknown constants
which have to be estimated from observed life-time data.
It turned out that the assumption of exponential distribution is not always justified.
Therefore more general methods were needed and a non-parametric approach
was developed. The formal relationship between the cumulative distribution function
of the life-time under use stress
and the cumulative distribution function
of the life-time under accelerating stress
is given using so-called
acceleration functions
by
Important forms of acceleration functions are
linear acceleration functions
and
power-type acceleration functions
The situation of Weibull distributions (cf.
Weibull distribution)
with cumulative distribution function
and stress-dependent parameters
and
is covered by power-type acceleration functions.
Mathematical model.
The model
(a1)
can be applied for a one-dimensional as well as a
-dimensional
stress
.
For linear acceleration functions and one-dimensional stress a differential
equation for the acceleration factors
can be derived. It turns out that the function
is determined by the
relative acceleration constant
in
between two accelerating stress levels
.
The life-time distribution under usual stress
is related to
by
For power-type acceleration functions, two differential equations for
and
are derived, and these functions are determined by relative acceleration constants
between two accelerating stress levels. The life-time distribution under
use stress
is given here by
For parametric, as well as semi-parametric, models, Bayesian methods are
also applied.
Statistical inference.
Based on life-time observations on different accelerating stress levels,
the following inference methods are used.
For parametric models
,
regression estimators for the constants
in the stress dependence
of the parameter, as well as least-squares estimators, were developed.
Also, Bayesian approaches (cf.
Bayesian approach)
for estimating the life-time distribution under use stress are possible.
Special methods for parametric, as well as for semi-parametric, accelerated
life-testing models are available (compare
[a1]).
Often, observations of life-times are not precise real numbers, but more
or less non-precise. This imprecision is different from errors. Recently
(1996),
models for describing non-precise life-times and
methods for accelerated life testing based on this kind of data were published (compare
[a4],
[a5]).