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I want to get a confidence interval of a function of some parameters. for example, from the data I estimate parameters of Pareto. Now I want to get 95% CI for 90th quantile (it's a function of parameters of Pareto), so I would need standard error.

I know delta method is one option. For simulation method, I am wondering if it is legitimate to simulate 1000 samples of size 50 from Pareto, calculate each of the 90th quantiles and take the standard deviation of the 1000 data. Is the standard deviation I get equivalent to standard error? Thanks for your help!

chl
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Melon
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  • this [question](http://stats.stackexchange.com/questions/5782/variance-of-a-function-of-one-random-variable) and its answers can be of interest. – mpiktas May 13 '11 at 07:23

1 Answers1

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Generating data from a given distribution, then calculating the part of interest then redoing this a bunch of times to get the interval is sometimes called a parametric bootstrap. You might learn more by reading up on this topic.

Why a sample of 50 each time? is the 50 meaningful? if not, then bigger samples are probably better.

One thing that your above method does not take into account is any uncertainties that you have in the parameters of the pareto distribution itself. You may be able to take this into account by doing a 2 stage bootstrap, fit the parameters on a bootstrap sample, then generate your new data from that set of parameters and find the percentile. Then repeat the entire process many times (starting with the bootstrap sample again).

Greg Snow
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  • could you elaborate more on your last point? – Melon May 14 '11 at 15:39
  • You said that you estimated the parameters from the data, so there is uncertainty about the parameters (you could fit a CI to show this). If you only generate data at the values you estimated then it does not take into accont this uncertainty. You can see this by increasing the number of data points you generate, as the number increases your interval estimate will converge towards having 0 width around the value if you knew the exact parameters. To get a realistic estimate you need to include uncertainty about the parameters, bootstrapping to estimate the parameters is one way to do this. – Greg Snow May 16 '11 at 14:31