Maximum likelihood and sufficient statistics

fT(t;B,C) = exp(-t/C)-exp(-t/B) / C-B where our mean is C+B and t>0.

so far i have found my log likelihood functions and differentiated them as follows:

dl/dB = sum[t*exp(t/C) / (B^2(exp(t/c)-exp(t/B)))] +n/(C-B) = 0

i have also found a similar dl/dC.

I have now been asked to comment what you can find in the way of sufficient statistics for estimating these parameters and why there is no simple way of using Maximum Likelihood for estimation in the problem. I am simply unsure as to what to comment upon. Any help would be appreciated. Thanks, Rachel

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this site supports latex, please reformat your question. I am reluctant to do it myself, since the notation is not very clear. –  mpiktas Jan 25 '11 at 13:41

OK, your question isn't perfectly clear but maybe I can help a little.

A statistic $T(X)$ is sufficient for a parameter $\theta$ if

$P(X|T(X), \theta) = P(X|T(X)$

In terms of likelihood functions you can verify that this implies

$f(x;\theta) = h(x)g(T(x); \theta)$

for some $h$ and $g$, which is known by a few different monikers (the factorization theorem/lemme/criteria and sometimes with a name or two attached). This is where @probabilityislogic's comment comes from, although like I said it's just a property of the likelihood function.

There are often a lot of different sufficient statistics (in particular, take $h=1$ and $g=f$, where $T(X)=X$ is just the entire dataset). Since the goal is to find a particular way to reduce the data without losing information, this leads into questions of minimal/complete sufficient statistics, etc. It's not clear what you need for your question, so I'll leave off there.

In terms of the MLE, your notation is a little confusing to me so I'll make a couple general comments. What problems can happen finding the MLE? It might not have a closed form, which is less a problem than a complication. It can fail to be unique, or occur at the edge of the parameter space, be infinite, etc. You need to at least define the parameter space, which you haven't done in your problem statement so far as I can tell.

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