From: juber@uceng.UC.EDU (james uber)
Subject: Re: linear covariance estimate for max likelihood
Keywords: parameter estimation, maximum likelihood, covariance estimate
Organization: College of Engineering, University of Cincinnati
Date: Fri, 21 Aug 1992 19:05:37 GMT

In article <thompson.714414133@daphne.socsci.umn.edu> thompson@atlas.socsci.umn.edu writes:
>juber@uceng.UC.EDU (james uber) writes:
>
>>I obtain parameter estimates via maximum likelihood where
>>my model is in the standard reduced form y = f(p), y are the
>>data and p are the parameters. I assume that the distribution
>>of the model + measurement errors is normal with zero mean
>>and known covariance matrix Ve. Thus i am solving the optimization
>>problem:
>
>>	min Tr(y - f(p))Inv(Ve)(y - f(p))
>>         p
>
>[rest of post deleted]
>
>I do not understand the question.  If y = f(p) (where f is presumably
>a known and fixed function of the parameters) and y is observed then
>there is no measurement error.  Perhaps you meant y = f(p) + e where e
>is a vector of measurement errors.  (This seems implicit in your
>description of the nonlinear least squares procedure.)
>
>But you also refer to "model errors".  What are these and how do they
>fit in?  If the model is really
>
>              y = f(p) + <error>
>
>and f(p) is known (up to the parameters p) and fixed, then
>Var(y) = Var(<error>) regardless of the source of the errors.
>
>Please clarify.

Thanks for replying to my post. That was my fault for being too 
hasty. This is what i really meant (now given a second chance).
The _measurement_ errors are defined as:

        e1 = y - y*

where y are the data and y* are the (unknown) true values. Now the
_model_ errors are defined as:

        y* = f(p*) + e2

where p* are the "true" parameter values. That is, even given the
measurements without error and the true parameters, there still is
some error due to faults in the model theory, inaccuracy in solution
of f(p), and the like. Combining these two equations gets me back
to where i should have started in the beginning:

        y - e1 = f(p*) + e2
        y = f(p*) + e1 + e2
        y = f(p*) + E

where E = e1 + e2 is the combined model + measurment errors. Thus
the relevant distribution to use in estimation of p via maximum 
likelihood is the distribution of E, which i previously assumed
was a normal distribution with known covariance Ve. My
understanding is that, while it is often possible to specify the 
distribution of e1 in a logical way (normal and i.i.d., for example),
the same can not necessarily be said for e2. Hence the significance
of knowing that both exist. I realize that the parameters of the
distribution of E can be estimated, under certain assumptions
about their form.

Now, like you said, if f(p) is known (not random), then the
variability in y comes from the variability in E. This is where my
brain start to hurt, `cause i think to myself, "hey, wait a minute,
we're talking about the variability in the _data_, which are the
_measurement_ errors." Thus i get confused when i look at 
derivations of the covariance matrix of the parameter estimates
that say things like, "the maximum likelihood objective function
depends on the _data_," or "if we vary the _data_ slightly,
replacing y by y + dy, this would cause the minimum (of the 
log likelihood function) to shift from p* to p* + dp*." I just
can't get around thinking of the variability in my data as the
measurement (e1) errors! So, this leads to my original question
of, in the derivation of the p covariance estimate, when we talk
of the variability in p being caused by the variability in y, 
do we mean variability caused by e1 or by E? I am fairly certain 
that it must be E, but applied statistics is a tough business to 
part-timers, and i'm just not sure.

Again, thanks for taking interest in my question. I hope that,
in the least, my ignorance might now be clear.

jim uber
dept. of civil & environmental eng.
univ. of cincinnati
juber@uceng.uc.edu 
-- 
--
james uber
juber@uceng.uc.edu