Timedependend Weibull parameters

[Andrey]

Please somebody answer, very important for me to understand basics !


Suppose we tested a sample of objects and recive a list of records of operation times: times-to-failure and right-suspensions.
Then use Weibull analysis and estimate shape and scale parameters BETA, ETA for Weibul distribution of operation time-to-failure.These parameters are FIXED, i.e. constant values.

But it is known, that real objects have hazard function h(t) described by Bathtub Curve and shape of h(t) changes during operation time:
1-early life (BETA less 1);
2-useful life (BETA is 1);
3-wearout life (BETA more 1).

In our list we SIMULTANEOUSLY have various operation times from ALL these 3 life periods. Hence BETA should be a function of operation time,
but Maximum-likelihood or Regression analysis don't give time-dependend BETA(T).

If it is true, how to solve this problem?

Regards
Andrey

[tarik]

The fact that you have fixed Beta and Eta doesn’t mean that your failure rate is constant (except if Beta=1, then you have a constant failure rate). For example for a Beta1, you have an increasing failure rate.
http://www.weibull.com/LifeDataWeb/the_weibull_distribution.htm
http://www.weibull.com/LifeDataWeb/characteristics_of_the_weibull_distribution.htm


It’s important to note that a single unit/product doesn’t go through all the stages of the bath tub curve. A product only fails once (we are not talking about repairable systems here), which could be during the early life, useful life, or ware out period. The bath tub curve applies to an entire population of products/units which can be modeled with a Mixed Weibull distribution. For example, you could have a portion of your products failing during early life, another portion during useful life, another portion failing during wareout, the entire population would be modeled with a mixed Weibull model made up of 3 Weibull distributions with different parameters.
http://www.weibull.com/LifeDataWeb/the_mixed_weibull_distribution.htm