Math Forum Discussions

Glen Barnett

Posts: 643
Registered: 12/6/04

Re: Maximum Likelihood estimation
Posted: Oct 24, 2006 10:07 PM

Plain Text

perreault.c@gmail.com wrote:
> Hi all:
>
> I'm working on an analytical model of spatial ecology behavior. I use
> this equation, which can apparently be parametrable using Maximum
> Likelihood Regression:
>
> Ji = 1
> ******
> 1+ e
> *****
> Si²Ai^x
>
> Note: Ai^x = Ai power x
>
> I have data for Ji, Si and Ai, and want to estimate the parameter e
> and x.
>
> Maximum likelihood regression seems to be beyond my knowledge of
> statistics, and all the explanation I have found so far assume from
> the
> reader math knowledge I lack.
>
> Can someone help me with this and show me how to resolve this
> practically?
>
> Thanks for your help,
> Charles P.

Do you have a distributional form you're prepared to assume for Ji?
It's a bit hard to do likelihood without a distribution.

Are Si and Ai measured without error? Or is the model a conditional one
relating Ji to observed Si and Ai rather than some underlying notion of
the "correct" value?

What are these quantities?

It's likely that the ML estimates can't be solved for directly and may
require iteration.

To get a starting value for your parameters, you can rearrange things:
(1/ Ji) -1 = e / Si²Ai^x

ln[(1/ Ji) -1] = ln(e) - 2ln(Si) - x ln(Ai)

ln[(1/ Ji) -1] + 2ln(Si) = ln(e) - x ln(Ai)

y = a + b w

So (assuming the numbers are such that you can make those
transformations), you can get starting values to input into your MLE
for e and x by computing:

y = ln[(1/ Ji) -1] + 2ln(Si)
w = ln(Ai)

then fitting a linear regression to estimate a and b. Your initial
estimates of e and x might then be e(0) = exp(a) and x(0) = b.

Once you have your distribution, and from that your likelihood, you may
be able iterate (if necessary) with those starting values.

Glen