# Accelerated life testing

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.

## Contents

## Stress dependence of life-time distributions.

The dependence of the cumulative distribution function of a life-time on the applied stress $ S $ was modelled for exponentially distributed life-times, i.e.,

$$ F ( t \mid S ) = 1 - { \mathop{\rm exp} } \left [ - { \frac{t}{\tau ( S ) } } \right ] , \quad t \geq 0, $$

by the dependence of the parameter $ \tau $ on $ S $. For the relationship between $ S $ and the parameter $ \tau $ certain functions

$$ \tau ( S ) = \psi ( S,a,b, \dots ) $$

were used with a known function $ \psi ( \cdot ) $ and unknown constants $ a,b, \dots $ 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 $ F ( \cdot \mid S _ {u} ) $ of the life-time under use stress $ S _ {u} $ and the cumulative distribution function $ F ( \cdot \mid S ) $ of the life-time under accelerating stress $ S \succ S _ {u} $ is given using so-called acceleration functions $ a ( t ) $ by

$$ \tag{a1 } F ( t \mid S ) = F ( a ( t ) \mid S _ {u} ) , \quad t \geq 0. $$

Important forms of acceleration functions are linear acceleration functions

$$ \tag{a2 } a ( t ) = \alpha ( S ) \cdot t, \quad t \geq 0, $$

and power-type acceleration functions

$$ \tag{a3 } a ( t ) = \alpha ( S ) \cdot t ^ {\beta ( S ) } , \quad t \geq 0. $$

The situation of Weibull distributions (cf. Weibull distribution) with cumulative distribution function

$$ F ( t ) = 1 - { \mathop{\rm exp} } \left [ - \left ( { \frac{t} \tau } \right ) ^ \beta \right ] , \quad t \geq 0, $$

and stress-dependent parameters $ \tau ( S ) $ and $ \beta ( S ) $ is covered by power-type acceleration functions.

## Mathematical model.

The model (a1) can be applied for a one-dimensional as well as a $ k $- dimensional stress $ S $.

For linear acceleration functions and one-dimensional stress a differential equation for the acceleration factors $ \alpha ( S ) $ can be derived. It turns out that the function $ \alpha ( S ) $ is determined by the relative acceleration constant $ \alpha _ {1,2 } $ in

$$ F ( t \mid S _ {2} ) = F ( \alpha _ {1,2 } \cdot t \mid S _ {1} ) , \quad t \geq 0, $$

between two accelerating stress levels $ S _ {1} \prec S _ {2} $.

The life-time distribution under usual stress $ S _ {u} $ is related to $ F ( \cdot \mid S ) $ by

$$ F ( t \mid S _ {u} ) = F \left ( \left . { \frac{t}{\alpha ( S ) } } \right | S \right ) , \quad t \geq 0. $$

For power-type acceleration functions, two differential equations for $ \alpha ( S ) $ and $ \beta ( S ) $ are derived, and these functions are determined by relative acceleration constants between two accelerating stress levels. The life-time distribution under use stress $ S _ {u} $ is given here by

$$ F ( t \mid S _ {u} ) = F \left ( \left [ \left . { \frac{t}{\alpha ( S ) } } \right ] ^ { {1 / {\beta ( S ) } } } \right | S \right ) , \quad t \geq 0. $$

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 $ F ( \cdot \mid \theta ( S ) ) $, regression estimators for the constants $ a,b, \dots $ in the stress dependence $ \theta ( S ) = \psi ( S;a,b, \dots ) $ 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]).

#### References

[a1] | R. Viertl, "Statistical methods in accelerated life testing" , Vandenhoeck and Ruprecht (1988) |

[a2] | W. Nelson, "Accelerated testing: statistical models, test plans, and data analyses" , Wiley (1990) |

[a3] | W.Q. Meeker, L.A. Escobar, "A review of recent research and current issues in accelerated testing" Int. Statist. Rev. , 61 : 1 (1993) |

[a4] | R. Viertl, W. Gurker, "Reliability estimation based on fuzzy life time data" T. Onisawa (ed.) J. Kacprzyk (ed.) , Reliability and Safety Analyses under Fuzziness , Physica-Verlag (1995) |

[a5] | R. Viertl, "Statistical methods for non-precise data" , CRC (1996) |

**How to Cite This Entry:**

Accelerated life testing.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Accelerated_life_testing&oldid=45151