Re: On coverage probability of Confidence interval

undiscern wrote:
> Thank you so much Luis & David,
>
> Luis your method is interesting, is it based on resampling of the
> original data? If yes, then that is not what I am looking for at the
> moment. I think my case is much more simple, no need resampling,
> because I know the underlying distribution, therefore just a
> parameteric bootstrapping will do.
>
> This is my case:
>
> I have 10 actual failure time readings, and it follows an exponential
> distribution. In exponential distribution, there is one parameter,
> lambda. From this 10 actual readings I obtained an initial estimate
> of lambda through probability plot. Say its estimate is of value 5.
> Now I am interested in knowing its CI.
>
> Using bootstrapping (Parametric type; since I know that it follows
> Exponential Distribution), I use the estimated value of Lambda (5) to
> get 1000 sets of data, with each set having 10 failure readings.
> Each bootstrap set will give me a new estimate value of lambda, that
> means I have a total of 1000 lambda estimates. Say at 95% CI and
> using the percentile method, I use the largest 25th and smallest 25th
> value of percentile difference to get the Lower and Upper CI.
>
> Say the CI is now [2, 7]. Now back to my original question. how do I
> know if this CI is 'good'? The standard method is to determine the
> coverage probability of [2, 7]. AM I correct to say that it should
> ideally be 95%?

Yes.

Now the issue is HOW to calculate the coverage
> probability of [2, 7] ? I asked earlier if the method is by
> determining how many of the 1000 lambda estimates lies between [2,
> 7]. If 900 of the lambda estimates lies between [2, 7], it means the
> coverage probability is 90%. Is this correct?
>
But from the process of creating the CI, you have defined it so that 95%
of the estimates lie between [2, 7]!

Someone will correct me, but I think here hte likelihood is Gamma distributed. So, from that you can calculate the correct CI (actually, the correct symmetrical CI: as David pointed out, there are many possible CIs).

If you want to check the coverage, you could simulate the data lots of times, using the estimated value above as the 'true' value, and for each simulation use the bootstrap to calculate a confidence limit, and then check whether that limit includes the 'true' value. this should, of course, happen 95% of the time.

(you can even be more elegant by calculating the likely range of the estimated coverage, by treating each simulation as a bernoulli random variable)

Bob

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Bob O'Hara

Dept. of Mathematics and Statistics
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