Re: Sample Size for Emperical CDF

From: David Jones (dajxxx_at_ceh.ac.uk)
Date: 10/01/04 

Date: Fri, 1 Oct 2004 12:44:35 +0100

Peter Michaux wrote:
> Thanks for the reply, David.
>
>> You can do something "in advance" by the usual simple argument ...
>
> The problem for me is I don't know that this is usual. Can you
direct
> me to a reference where this is discussed in more detail? Perhaps a
> standard textbook.
>
> Thanks,
> Peter

Difficult to find, perhaps everybody thinks its so obvious.
However see Eq (4.2.3) in Chapter 4 (Direct Simulation) of
"Monte Carlo Methods" JM Hammersley & DC Handscomb, Methuen 1964.

Clearly the above book is a little old now (possibly later editions?)
and you may find some later books on stochastic simulation that will
go further in answering your questions. However, confidence intervals
from bootstrapping is a special topic: you should see any advice in
books on this topic, such as Section 5.2.3 of
"Bootstrap Methods and their Application" AC Davison & DV Hinkley,
Cambridge UP, 1997.
In particular, bootstrap methods are only an approximate solution to
any real problem (even if an infinite number of samples are used), so
that using an extremely large number of samples would be pointless.
You may be better to concentrate on studying repeated simulations of
something close to the "real data", from each of which you extract a
confidence interval by bootstrapping, in order to examine the
properties of the confidence intervals (possibly as they relate to the
number of bootstrap samples).

In a sightly different context (you need to be able to see that use of
the empirical df is equivalent to estimating a proportion), you could
see Equation (2.40) of Section 2.13 (Choosing Sample Sizes) of
"Statistics for Envronmental Science and Management" BFJ Manly,
Chapman&Hall/CRC,2001.

David Jones

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