# Re: [R] Density Estimation

From: Pedro Ramirez <pramirez379_at_hotmail.com>
Date: Fri 09 Jun 2006 - 04:31:26 EST

>In mathematical terms the optimal bandwith for density estimation
>decreases at rate n^{-1/5}, while the one for distribution function
>decreases at rate n^{-1/3}, if n is the sample size. In practical terms,
>one must choose an appreciably smaller bandwidth in the second case
>than in the first one.

Thanks a lot for your remark! I was not aware of the fact that the optimal bandwidths for density and distribution do not decrease at the same rate.

>Besides the computational aspect, there is a statistical one:
>the optimal choice of bandwidth for estimating the density function
>is not optimal (and possibly not even jsut sensible) for estimating
>the distribution function, and the stated problem is equivalent to
>estimation of the distribution function.

The given interval "0<x<3" was only an example, in fact I would like to estimate the probability for intervals such as

"0<=x<1" , "1<=x<2" , "2<=x<3" , "3<=x<4" , ....

and compare it with the estimates of a corresponding histogram. In this case the stated problem is not anymore equivalent to the estimation of the distribution function. What do you think, can I go a ahead in this case with the optimal bandwidth for the density? Thanks a lot for your help!

>best wishes,
>
>
>
>PR>
>PR> >
>PR> >--
>PR> >Gregory (Greg) L. Snow Ph.D.
>PR> >Statistical Data Center
>PR> >Intermountain Healthcare
>PR> >greg.snow@intermountainmail.org
>PR> >(801) 408-8111
>PR> >
>PR> >
>PR> >-----Original Message-----
>PR> >From: r-help-bounces@stat.math.ethz.ch
>PR> >[mailto:r-help-bounces@stat.math.ethz.ch] On Behalf Of Pedro
>PR> >Ramirez Sent: Wednesday, June 07, 2006 11:00 AM
>PR> >To: r-help@stat.math.ethz.ch
>PR> >Subject: [R] Density Estimation
>PR> >
>PR> >Dear R-list,
>PR> >
>PR> >I have made a simple kernel density estimation by
>PR> >
>PR> >x <- c(2,1,3,2,3,0,4,5,10,11,12,11,10)
>PR> >kde <- density(x,n=100)
>PR> >
>PR> >Now I would like to know the estimated probability that a new
>PR> >observation falls into the interval 0<x<3.
>PR> >
>PR> >How can I integrate over the corresponding interval?
>PR> >In several R-packages for kernel density estimation I did not
>PR> >found a corresponding function. I could apply Simpson's Rule for
>PR> >integrating, but perhaps somebody knows a better solution.
>PR> >
>PR> >Thanks a lot for help!
>PR> >
>PR> >Pedro
>PR> >
>PR> >_________
>PR> >
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