Linear Model Problem

sonicb11

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Nov 15, 2007, 11:42:11 AM11/15/07

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I am working on a linear model problem.

Here is the model:

Wi = a*Zi + b*Zi*Xi + Zi*Ei

Wi is the observed response. a and b are the regression
coefficients. Zi follows a Beta distribution with parameters alpha and
beta (they are both unknown). Xi is a predictor variable, and Ei is
the error with mean zero and variance sigma^2.

I need to estimate a, b, alpha, beta, and sigma^2 using method of
moments. I want formulas for them. I've never seen a linear model like
this before and have no idea how to tackle it. Any help or hints would
be greatly appreciated. Thanks in advance.

sonicb11

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Nov 17, 2007, 8:43:06 AM11/17/07

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Maybe using maximum likelihood, there is a solution? Either method. I
just need some solution to estimate the parameters.

Richard Ulrich

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Nov 17, 2007, 8:43:30 PM11/17/07

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On Thu, 15 Nov 2007 08:42:11 -0800 (PST), sonicb11
<willi...@hotmail.com> wrote:

> I am working on a linear model problem.
>
> Here is the model:
>
> Wi = a*Zi + b*Zi*Xi + Zi*Ei

That's a meaningful model? That has Zi in each term.
Is there a typographical error here?

>
> Wi is the observed response. a and b are the regression
> coefficients. Zi follows a Beta distribution with parameters alpha and
> beta (they are both unknown). Xi is a predictor variable, and Ei is
> the error with mean zero and variance sigma^2.

Does the linear model care how Z is distributed?
That seems to be a separate question, if you care about it.

>
> I need to estimate a, b, alpha, beta, and sigma^2 using method of
> moments. I want formulas for them. I've never seen a linear model like
> this before and have no idea how to tackle it. Any help or hints would
> be greatly appreciated. Thanks in advance.

--
Rich Ulrich, wpi...@pitt.edu
http://www.pitt.edu/~wpilib/index.html

sonicb11

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Nov 17, 2007, 10:34:14 PM11/17/07

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On Nov 17, 8:43 pm, Richard Ulrich <Rich.Ulr...@comcast.net> wrote:
> On Thu, 15 Nov 2007 08:42:11 -0800 (PST), sonicb11
>

Yes, it's a meaningful model. The Zi is there from a previous step,
which I haven't shown. The Zi follow a Beta distribution with unknown
parameters alpha and beta.

Anon.

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Nov 18, 2007, 2:07:52 AM11/18/07

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But is Zi known?

It's not clear to me if Zi is even identifiable. If it is, you might
want to look at an EM algorithm, i.e. fit the standard linear model to
Wi/Zi for estimated values of Zi, then estimate the Zi's, given a and b.

Bob

--
Bob O'Hara
Department of Mathematics and Statistics
P.O. Box 68 (Gustaf Hällströmin katu 2b)
FIN-00014 University of Helsinki
Finland

Telephone: +358-9-191 51479
Mobile: +358 50 599 0540
Fax: +358-9-191 51400
WWW: http://www.RNI.Helsinki.FI/~boh/
Blog: http://deepthoughtsandsilliness.blogspot.com/
Journal of Negative Results - EEB: www.jnr-eeb.org

sonicb11

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Nov 19, 2007, 10:27:01 AM11/19/07

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> Journal of Negative Results - EEB:www.jnr-eeb.org- Hide quoted text -
>
> - Show quoted text -

Thanks. I'm a little confused though. Could you explain that further?
I'm looking for a formula for the estimators, and I don't know if the
EM algorithm will give that. Doesn't matter which method I use, as
long as I get some formulas.