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I am wondering if there is a way of generalizing the likelihood function of some parameters $L(\theta | \mathbf x), \theta \in \Theta$ given some data $\mathbf x$ that has been observed coming from a distribution on $\mathbb R^n$, say $\mu$, so that it is defined for distributions that admit neither a density nor mass function. I guess something rooted in measure theory that preserves the relevant properties of likelihood functions would be desired (e.g. under suitable regularity conditions it would be desirable for asymptotic normality and consistency of maximum likelihood estimators to hold).

My only thought is that we may be able to generalize the notion so long as the distribution of the data admits some density with respect to some measure other than Lebesgue or counting measure.

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I think your last paragraph may be the answer to the question in your first.

Sometimes just adding a few measures together will give you one with respect to which all of the measures concerned are absolutely continuous. And probably it won't matter which among such measures you use. There may be a simplest or "smallest" one.

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Well, this is believable. Do you know off-hand if any good results such as $\sqrt n$-consistency of MLE's hold under reasonable regularity conditions? –  guy Jul 16 '11 at 22:08
    
I've forgotten a lot of that sort of stuff, but Serfling's book on asymptotics in statistics should have something on this. –  Michael Hardy Jul 17 '11 at 1:41
    
Excellent. I've hunted it down and will add it to my reading list. –  guy Jul 17 '11 at 2:23

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