Re: Question regarding validity of markov parameters calculated from dodgy data

From: David Jones (dajxxx_at_ceh.ac.uk)
Date: 02/21/05 

Date: Mon, 21 Feb 2005 11:24:04 -0000

Ben wrote:
> "David Jones" <dajxxx@ceh.ac.uk> wrote in
> news:4214714a$1@news.nwl.ac.uk:
>
>> Ben wrote:
>>>
>>> My question is this. If I ignore the fact that the data does not
>>> precisely fit the assumptions, what are the consequences if I go
>>> ahead anyway and calculate the maximum likelihoods estimators of
the
>>> markov parameters? It seems to me the values in any case say
>>> something meaningful about my data. So why shouldn't I use them?
>>>
>>
>> (i) This is probably OK, particularly if you can see by putting the
>> Max.Lik. expressions into estimating-equation form, that the
>> estimates are estimating something sensible.
>>
>> (ii) However, do not rely on the usual Max.Lik. way of estimating
>> parameter unceratinty if you know that the model isn't adequate.
>>
>> David Jones
>>
>>
>>
>
> Thanks for the advice. Do you know of any references which might
> cover this issue?
>

Not really, partly because answer (i) isn't true in general. I think
you need to think about what happens in large samples and what the
maximum likelihood estimators would converge to if the model is wrong.
For example with a simple Markov model, the Max.Lik. estimates for a
true Markov modelwould converge to something describing
  prob(X(t+1) | X(t), X(t-1),...) (1)
but which actually estimates something associated with
  prob(X(t+1) | X(t)) (2)
(2) and (1) are the same because of the Markov assumption.
If the real world process isn't Markov, then you are just estimating
(2), not (1), which may be what you want.

However:
(a) you might like to see my paper:
 Jones DA (1983) Statistical analysis of empirical models fitted by
optimisation. Biometrika 70 (1), 67-88
 .... this may help on the subject of estimating parameter uncertainty
for incorrect/unknown models.

(b) In the above paper I reference some papers on associated topics,
of which the following now seem relevant to you.

Klimko LA & Nelson PI (1978) On conditional least squares estimation
for stochastic processes. Ann. Statist. 6, 629-42

Ogata Y (1980) Maximum likelihood estimates of incorrect Markov models
for time series and the derivation of AIC. J. Appl. Prob. 17, 59-72.

Shibata R (1976) Selection of the order of an incorrect Markov model
by Akaike's information criterion. Biometrika 63,117-26

All the above possibly concentrates too much on the mathematical
theory of some specialist concerns which might not really be of use to
what you really want to do. You should consider finding someone in the
Stats Department at Oxford to discuss things with ... possibly someone
within your own department will already have good links there.

David Jones

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