liyusen007
New member
- Joined
- Dec 30, 2013
- Messages
- 10
what's the question?
I'm not seeing any sort of wondrous cancellations or anything that would lead this to be able to be integrated to a closed form. Each term of the product has an exponential of a polynomial in t.
\(\displaystyle \displaystyle{\prod_{j=1}^n}\left(1-\exp\left[{\displaystyle{-\frac{1}{\lambda^k}\sum_{m=0}^{k-1}}t^m t_j^{k-m}}\right]\right)\)
If you expand the whole thing out it will be a big sum of exponentials of polynomials in t which may be able to be evaluated using the product rule and the generalized error function.
I'm guessing this isn't a homework problem. Is this a real world application? Can you just integrate it numerically?
what I would do is set up the integral in whatever software you plan to do it in and run it using an increasing upper limit of integration. If the integral has any hope of converging you should start to see evidence of that the results of each run start to get closer and closer. Just keep expanding in upper limit until the absolute value of the difference between runs is less than some number you're happy with. A little experience will quickly tell you how far you have to integrate to.
It's a sum of decaying exponentials so it should converge, possible very quickly depending on \(\displaystyle \lambda\)
If you don't see evidence of convergence well then you have to take a much closer look at things. Cross that bridge when you come to it.
I'm looking at this now and I was wrong. It's not a sum of decreasing exponentials.
I'm using linearly spaced \(\displaystyle t_i\)'s and playing with k and \(\displaystyle \lambda\).
Each term in the product tends towards 1 eventually. You end up with the infinite integral of a constant term. That's not going to converge.
I'll keep playing with it and let you know if I find anything of use.
Well thank you.
But I don't have good news. It's pretty clear this thing won't converge.
Can I ask what the actual problem is?
I'm not going to be able to help you out. I'd hit the technical journal index and search for Weibull order statistics. There's some stuff on google but you'll probably need to dig deeper. There's going to have to be something that keeps t from being allowed to go to infinity.
Someone must have calculated the expectation of the max of a set of Weibull distributed rv's before.
another place you might look in in texts on survival analysis.
I'm guessing this problem is related to the remaining lifetime left to a group of components (or people) given that they've survived to various times already.
Good luck!