Re: Optimal seperability of distribution

In article <1136934132.340004.60780@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Ray Koopman <koopman@xxxxxx> wrote:
>newsgroupie2003@xxxxxxxx wrote:
>> Hi,
>> I have the following problem that I am not sure how to best go about.
>> I have 2 populations (healthy and ill) and I then have a choice of
>> different measurement methods to try and seperate them. The measurement
>> is univariate.
>> What is the best way of characterizing the separability between the
>> healthy and sick populations?

>> I was thinking of ROC curves summarized by the integral of the ROC
>> curve, but i guess the problem is quite general in the sense that what
>> is needed is measure of how well the 2 distributions are seperated.
>> Any input appreciated!

>> Best regards
>> Soren

>Unfortunately, there is no measure of separation that is best for all
>situations. ROC plots preserve all the ordinal information but discard
>all the interval information. The area under an ROC answers the
>question "What is the probability that a randomly chosen score from
>one population will exceed a randomly chosen score from the other
>population?", but that addresses only one aspect of separation.
>Before you choose a measure, you need to decide what you mean by
>"separation".

One way to consider the separation of two distributions
is to consider the sample size needed to distinguish
between them. The asymptotics for this were studied by
Chernoff, and are the Chernoff "distance" for fixed
sample procedures, and the two Kullback-Leibler measures
for sequential procedures. If f and g are the densities
with respect to a measure m, the probabilities of a wrong
decision from a sample of size n can be made less than

(min \int f^t*g^(1-t) dm)^n,

and the exponential rate cannot be improved, so the
negative logarithm of that minimum is the Chernoff distance.


--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

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