I'm not certain I understand your terminology. "Finite Mixture
distributions" are distributions that are, for example, 20% N(0,
sigma^2) and 80% beta(shape1, shape2). Your use of the term "mixture"
does not seem to fit this standard.
I still don't understand your problem, but the following might help.
Suppose X ~ N(0, sigma^2), E ~ beta(shape1, shape2), with sigma,
shap1, shap2 known. If we observe Y = X + E, then the likelihood =
probability (density) of what we observe is as follows:
c*exp(-0.5*(x/sigma)^2)*((y-x)^shape1)*((1-y+x)^shape2)
where c = 1/(sigma*sqrt(2*pi)*beta(shape1, shape2).
Given any observation y, you can plot this likelhood as a function of x.
Differentiate it with respect to x and set the result to 0, and you
get a cubic, which you can then solve for x.
If X ~ beta(shape1, shape2) and E ~ N(0, sigma^2), you get a similar
likelihood; the method still works.
If this does not solve your problem, please explain why you think the
distribution is a mixture of normal(0, sigma^2) and beta(shape1, shape2).
hth, spencer graves
############################
Xao Ping wrote:
> Spenser:
> Thanks again. Distribution is a mixture of normal(0, sigma^2) and
> beta(shape1, shape2).
> Sigma and shapes are KNOWN. Again, my question is NOT how to estimate
> them,
> but HOW to PREDICT X's knowing Y's, that is how to extract signal
> knowing signal plus noise and statistical characteristics of both signal
> and noise.
> Xao
>
>
> Spencer Graves <spencer.graves@PDF.COM> wrote:
>
> What do your two distributions look like?
>
> If there really is structure, then the observations are not
> independence. What do normal probability plots look like? If it's a
> straight line or a gentle curve, then it is one distribution. If it
> looks like two or three straight lines possibly with gaps in between,
> that is the fingerprint of a mixture of normals. See, e.g.,
>
> Titterington, D. M.. (1985) Statistical analysis of finite mixture
> distributions (New York : Wiley)
>
> or Geoffrey McLachlan and David Peel (2000) Finite Mixture Models
> (Wiley)
>
> For information on software for this, check the R archives at
> "www.r-project.org" -> search -> R site search. If you find nothing
> there, send another query to r-help. Sundar Dorai-Raj, a regular
> contributor to r-help, showed me something on this almost a year
ago. I
> don't have it with me now, but I believe there should be something
> available.
>
> hope this helps. spencer graves
>
> Xao Ping wrote:
> > Spenser, thank you for immediate response, Yes, X and E are
> (assumed)
> > independent.
> > Let me shed a little bit more light on the problem. Empirical
> > distribution of Y shows
> > (in many samples of Y-type) a kind of tricky structure which
> suggests
> > the idea that PDF of Y is a convolution of two distributions. I've
> > devised an additive model which reasonably well reproduces the
> artefact.
> > Using method of moments I've found the unknown parameters. This
> > parameters are found reasonably stable from sample to sample. Now
> the
> > question: so what? Does all this knowledge help to reduce noise
> in data?
> > If yes than what is the appropriate framework for such an
> analysis. I
> > realize of course that
> > the solution is in no way unique. However, what worries me more
> is the
> > question: so what? Suppose that I have the model, does it help to
> > extract signal and to reduce noise.
> > Thanks again
> > Simon
> >
########################################################
If you can provide more structure, then we might be able to do
something. For example, are all the x[i]'s and e[i]'s independent of
each other? If you assume some correlation structure among either the
x[i]'s or the e[i]'s, we might be able to make some progress.
Otherwise, in a sample of N, all I see right now are N equations and 2N
unknowns.
hth. spencer graves
Xao Ping wrote:
> Dear All:
> Suppose that I have a sample Y. Suppose also that it is known that Y=X+E
> where X is considered as a signal and E as noise. The PDFs of X and E
> are known: F(y, theta) and
> G(e, xi). Parameters theta and xi are also a priori known. Given all
> this knowledge,
> is that possible to estimate signal X? Just to be precise, I need to
> substitute each data point in Y by the predicted Y' in such a way as it
> would be, in a sense, closer to X than in the original sample Y.
> Thank you
> Xao
>
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