Instantaneous Failure Intensity

[zubinpepsi]

Hi,
I am tryiny to apply Crow-AmSAA model to estimate the reliability growth for grouped data.
Im stuck up in the calculation of instantaneosu failure intensity.
I learnt from the other thread how to calculate instantaneous MTBF and failure intensity for each failure time instances when you have failure times and no.of failures.
But how do you calculate instantaenous MBTF/ failure intensity for future failure times ( some random) to see how the trend goes... Can I apply this formula
http://www.weibull.com/RelGrowthWeb/amsaa__14.gif.
When can i apply this formula?

[David]

Hi zubinpepsi,

Yes, you can use the formula you mentioned to calculate the instantaneous failure intensity for grouped data. As always, I would just be careful as to how far out you extrapolate. While a continuous line can be shown, in general, the failure intensity vs. time plot is displayed using intervals. The intervals represent the constant failure intensity during each stage and cannot be extrapolated to future times.

I hope this helps.

[Arai.M]

Just to expand a little on the previous message...

There are two ways to calculate the instantaneous failure intensity.
One is a non-parametric method where the failure intensity is calculated at each failure time. Denote T(i) as the ith failure time and N(i) as the cumulative number of failures at T(i) for a single system. The instantaneous failure intensity at time T(i) can be calculated using: (N(i)-N(i-1))/(T(i)-T(i-1)).
For multiple systems and for grouped interval data, this equation can also be used.

However, the above method cannot be used for prediction. In order to predict the future failure intensity, you need to use the equation in your message.

In that equation, T can be any time. The parameters in the equation are obtained by fitting the equation to the observed failure intensities. Therefore, at each failure time, the failure intensity calculated from the equation should be close to the observed value. For times after the observed values, you don’t have data so the value from the equation is the expected failure intensity at a future time, T.

As Dave mentions, any extrapolation is risky. You are assuming future failures will have the same trend as past failures. You will also get very wide confidence bounds if the future time is too far from the current time.

Arai

[zubinpepsi]

Thank you very much David and Arai for your replies. :)
Now I have another problem.
I am trying to calculate the confidence bound for cumulative no. of failures.
The Crow confidence bounds on cumulative number of failures are:
http://www.weibull.com/RelGrowthWeb/amsaa__440.gif
when i tried to appply this formula, i get the bounds higher than the oberved N.
Example, my N for the 1st interval is 12. but I get the bounds 23.9 - 36.7..
Is this formula wrong?
Pls help me.

[David]

First, I can say that the formula is correct. Keep in mind that it is dependent on the upper and lower confidence bounds on the instantaneous failure intensity which are the inverse of the instantaneous MTBF confidence bounds. So, it all depends on how these values are determined. Information regarding confidence bounds on the instantaneous MTBF for grouped data can be found at http://www.weibull.com/RelGrowthWeb/Grouped_Data_Confidence_Bounds_on_MTBF_for_the_Cro w-AMSAA_Model.htm.