# FW: [R] distance between distributions

From: Mike Waters <dr.mike_at_ntlworld.com>
Date: Sat 07 May 2005 - 05:04:13 EST

Sorry, forgot to send this to the list originally.

-----Original Message-----
From: r-help-bounces@stat.math.ethz.ch
[mailto:r-help-bounces@stat.math.ethz.ch] On Behalf Of Campbell Sent: 06 May 2005 11:19
To: vograno; r-help
Subject: Re: [R] distance between distributions

I may have missed the point here but isn't this an obvious case for using the bootstrap. A paper by Mallows, the exact reference escapes me, establishes the conditions under which asymtotics of the marginal distribution imply a well behaved limit. Perhaps a better discussion of the issues can be found and a pair of papers by Bickel and Freedman, see Annals Of Statistics Vol 9, Number 6.

HTH
Phineas Campbell

This is more of a general stat question. I am looking for a easily computable measure of a distance between two empirical distributions. Say I have two samples x and y drawn from X and Y. I want to compute a statistics rho(x,y) which is zero if X = Y and grows as X and Y become less similar.

Kullback-Leibler distance is the most "official" choice, however it needs estimation of the density. The estimation of the density requires one to choose a family of the distributions to fit from or to use some sort of non-parametric estimation. I have no intuition whether the resulting KL distance will be sensitive to the choice of the family of the distribution or of the fitting method.

Any suggestion of an alternative measure or insight into sensitivity of the KL distance will be highly appreciated.

The distributions I deal with are those of stock returns and qualitatively close to the normal dist with much fatter tails. The tails in general should be modeled non-parametrically.

Thanks,

[[alternative HTML version deleted]]

C. L. Mallows. A note on asymptotic joint normality. Annals of Mathematical Statistics,43(2):508-515, 1972

Another reference that might be of relevance is:

Bootstrap Methods for the Nonparametric Assessment of Population Bioequivelance and Similarity of Distributions. Czado C. and Munk A. ; this can be obtained as a postscript document from: http://www-m4.ma.tum.de/Papers/Czado/simrev.ps

HTH Mike

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