Thread Subject: Coding Probability Density (that is a large summation)

Hello M-Community:

I am fitting a density to my data. I would therefore like
to estimate parameters of the density function based on my
data. The density, however, is expressed as a summation.
It is not one of the preconfigured densities found in MATLAB.

The summation has 100 terms and cannot be re-expressed in
simpler form. And rather than literally writing every term,
it would be easier to write a loop. But how may I code this
estimation as such in MATLAB?

It can be thought of as an optimization problem for which
the parameters need to be recovered when the x's are
observed. E.g., density function f with unknown parameter a:

f(x) = Sum _k=0 to 100_ { f1(a,k)*f2(x,a,k)}

Both f1 and f2 can be expressed analytically. But clearly
it is unfortunate that f(x) is an infinite summation of the
two functions multiplied together. Since I am interested in
is using my observed 'x''s to estimate 'a'. How can I do
this estimation in MATLAB?

Thanks in advance for your thoughts and suggestions.

OR Stats wrote:

> It can be thought of as an optimization problem for which
> the parameters need to be recovered when the x's are
> observed. E.g., density function f with unknown parameter a:
>
> f(x) = Sum _k=0 to 100_ { f1(a,k)*f2(x,a,k)}
>
> Both f1 and f2 can be expressed analytically. But clearly
> it is unfortunate that f(x) is an infinite summation of the
> two functions multiplied together. Since I am interested in
> is using my observed 'x''s to estimate 'a'. How can I do
> this estimation in MATLAB?

In theory, this is no different than any other maximum likelihood problem, and
could be solved using, for example, FMINSEARCH, FMINCON, MLE (in the Statistics
Toolbox), or others. In practice, who knows what statistical and numeric
properties this likelihood function has.

Hope this helps.

Thanks for your suggestion, Peter. How would I express the
likelihood estimation in MATLAB when the density itself is
100 some terms? Do I have an alternative to coding all 100
terms? f1(a,k)*f2(x,a,k), where k is my index, which is
unrelated to my data points x themselves.

Peter Perkins <Peter.PerkinsRemoveThis@mathworks.com> wrote
in message <g02dh4$l7t$1@fred.mathworks.com>...
> OR Stats wrote:
>
> > It can be thought of as an optimization problem for which
> > the parameters need to be recovered when the x's are
> > observed. E.g., density function f with unknown
parameter a:
> >
> > f(x) = Sum _k=0 to 100_ { f1(a,k)*f2(x,a,k)}
> >
> > Both f1 and f2 can be expressed analytically. But clearly
> > it is unfortunate that f(x) is an infinite summation of the
> > two functions multiplied together. Since I am interested in
> > is using my observed 'x''s to estimate 'a'. How can I do
> > this estimation in MATLAB?
>
> In theory, this is no different than any other maximum
likelihood problem, and
> could be solved using, for example, FMINSEARCH, FMINCON,
MLE (in the Statistics
> Toolbox), or others. In practice, who knows what
statistical and numeric
> properties this likelihood function has.
>
> Hope this helps.

I suppose the confusion is how to express the density when
symbolically it is 100+ terms...

"OR Stats" <stats112@gmail.com> wrote in message
<g02ju0$46k$1@fred.mathworks.com>...
> Thanks for your suggestion, Peter. How would I express the
> likelihood estimation in MATLAB when the density itself is
> 100 some terms? Do I have an alternative to coding all 100
> terms? f1(a,k)*f2(x,a,k), where k is my index, which is
> unrelated to my data points x themselves.
>
> Peter Perkins <Peter.PerkinsRemoveThis@mathworks.com> wrote
> in message <g02dh4$l7t$1@fred.mathworks.com>...
> > OR Stats wrote:
> >
> > > It can be thought of as an optimization problem for which
> > > the parameters need to be recovered when the x's are
> > > observed. E.g., density function f with unknown
> parameter a:
> > >
> > > f(x) = Sum _k=0 to 100_ { f1(a,k)*f2(x,a,k)}
> > >
> > > Both f1 and f2 can be expressed analytically. But clearly
> > > it is unfortunate that f(x) is an infinite summation
of the
> > > two functions multiplied together. Since I am
interested in
> > > is using my observed 'x''s to estimate 'a'. How can I do
> > > this estimation in MATLAB?
> >
> > In theory, this is no different than any other maximum
> likelihood problem, and
> > could be solved using, for example, FMINSEARCH, FMINCON,
> MLE (in the Statistics
> > Toolbox), or others. In practice, who knows what
> statistical and numeric
> > properties this likelihood function has.
> >
> > Hope this helps.
>

OR Stats wrote:
> I suppose the confusion is how to express the density when
> symbolically it is 100+ terms...

OR, I don't have enough information to know the right answer, but if you cannot
write it as a vectorized expression, then you can write a loop.