Re: Sample Size for Emperical CDF

From: Mark Von Tress (vontressms_at_cs.com)
Date: 10/04/04 

Date: 4 Oct 2004 09:48:10 -0700

petermichaux@yahoo.com (Peter Michaux) wrote in message news:<47af9e6a.0409292058.75e2d401@posting.google.com>...
> I want to use the bootstrap to make confidence intervals for a
> statistic. The quality of a bootstrap's output depends heavily on the
> quality of the empirical cdf. If my sample is of size n, how big does
> n have to be so that the empirical cdf is a good approximation of the
> underlying cdf? I don't know anything about the underlying cdf.
>
> I think the solution might be to watch the change in the empirical cdf
> after I take each new sample. Eventually the empirical cdf won't be
> changing that much after each new sample. At that point I can stop
> sampling. Is this on the right track? I don't know how to quantify
> that change and when the change in that quantity can be considered
> small.
>
> Thanks,
> Peter

Peter,
  Here is a simple way to do this based on the Kolmogorov goodness of
Fit Test. It is based on a 95% confidence interval around the
emperical cdf discussed in Conover, W.J. "Practical Nonparametric
Statistics" John Wiley and Sons. It is in section 6.1 called
Confidence bands for the Population Distribution Function. Conover
discribes a confidence interval about the EDF, say S(x). He adds and
subtracts the Kolmogorov quantile to all points of S(x). He truncates
values above 1 to 1 and values below 0 to 0.

For n>40, the Kolmogorov quantile for a two sided interval is
delta=1.36/sqrt(n). See table 14. Here delta is the desired precision
of your estimate of S(x). So, the sample size formula is
n=(1.36/delta)^2. You have to pick delta.

Mark