Monte Carlo confidence intervals

[Geoff Lee]

[Originally Posted: 5/14/00--Transferred by ReliaSoft Moderator]

Can someone please help to clarify how Monte Carlo simulation is used to determine confidence intervals?

I had assumed, for example, that one would use the measured sample data to estimate (by MLE or other method) the population parameters (e.g. Beta, Eta etc). A large number of random (Monte Carlo) samples would then be taken, assuming a population having those estimated parameters. The population parameters would be recalculated for each of the samples. The confidence interval for the calculated parameter would be the interval which contains the desired percentage of the corresponding parameter estimates from the Monte Carlo data.

However, such intervals would not be dependent on the goodness of fit of the model to the initial sample data. By comparison with ML intervals, a good fit (e.g. well defined peak on MLE contour) should give a narrower confidence interval than a poor fit.

So, I assume my understanding of Monte Carlo confidence intervals is incorrect. Can someone please enlighten me? Thanks for your assistance.

[Dr. Dave Olwell]

[Originally Posted: 5/23/00-- Transferred by ReliaSoft Moderator]

What you have described is known as a parametric bootstrap and you have described it correctly!

It can not be used to assess goodness of fit, because you assume that the model you resample from is correct when you draw the samples.

In other words, say I have ten data points and I fit a Weibull(beta,eta) model to the data, and get beta = 2 and eta = 1000. If I do 10,000 Monte Carlo samples from a Weibull(2,1000) distribution, I should see good agreement between the resampled data and a Weibull(2,1000). That good agreement tells me nothing about the agreement between the original data and the Weibull(2,1000). It does allow me to see how variable the beta and eta estimates are, which I can use to infer how precise the original beta = 2 and eta = 1 estimates are.

Best wishes,

Dr. Dave

[Geoff]

[Originally Posted: 5/23/00-- Transferred by ReliaSoft Moderator]

Thanks very much Dr. Dave.

Geoff.

[Jeff Lischer]

[Originally Posted: 9/20/00--Transferred by ReliaSoft Moderator]

What about the use of a straight bootstrap technique here? That would involve generating the random Monte Carlo samples from the original data points (with replacement). Then the eta,beta would be calculated for each of these samples and the distribution of the eta,beta's would be used to determine your confidence intervals?

Would that work too, or is the parametric bootstrap better in this application?