[S] test whether a density is unimodal

[Siwik, Thomas]

Dear all,

I have got a statistical question concerning a test about the consistency
of a density function.

Consider a real valued time series of prices. I would like to test, whether
the (conditional) density of price changes is unimodal with a maximum at
zero. The density comes from a kernel estimation with a Gaussian kernel.

Has someone an idea how to do the test?

Thank you
Thomas Siwik
WEDIT Deloitte & Touche
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[Michael Axelrod]

There is something called the coefficient of bimodality. It the following
ratio: (1+skewness^2)/(kurtosis+3) where the "kurtosis" vanishes for the
normal.

This coefficient has value .555 for the uniform, and 1 (exactly) for the
Bernoulli distribution. The Bernoulli of course has only two values with
prob p and 1-p, the ultimate bimodal. I don't know any additional
properties. The sampling version of the coefficient is defined slightly
differently to avoid bias for small samples.

Another approach would be to use cross-entropy between the data and a two
parametric forms, one unimodal and the other multimodal. Say a normal with
unknown mean and variance (two free parameters) and the other a mixture of
normals (as many as 5 parameters). Then use the EM algorithm to get the
cross-entropy from the expect log likelihood. The details are a little
tricky. I think in general this is a hard problem.

[ashle...@gmail.com]

Hello,

I realize that this is a dated post now, but I'd be grateful if anyone could please point me to a reference (or original citation in if possible) for the coefficient of bimodality given by Michael Axelrod, coefficient of bimodality = (1+ skewness^2)/(kurtosis + 3)?

Thank you and have a swell day,
Ashley