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Is it the case that both minimum variance unbiased estimators (MVUEs) and maximum likelihood estimators (MLEs) are always functions of a minimal sufficient statistic?

If so, how do we know? If not, what are some exceptions or what are some ways that exceptions occur?

Glen_b
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Bratt Swan
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    The body of your question was a bit vague; I have attempted to make it more specific. If you want to ask something different or something further, please edit your question. – Glen_b Apr 14 '16 at 00:16
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    If your original post was in fact `self-study` (as described at the tag wiki), then what theorems do you have about the relationships between these things, and what do they tell you? – Glen_b Apr 14 '16 at 01:37
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    What is there is _no_ minimal sufficient statistic? – Xi'an Apr 14 '16 at 08:26

1 Answers1

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One interesting case for MLEs arises for the MLE of the parameters of the Laplace distribution.

The MLE for the location parameter ($\theta$, say) is the median and for the scale ($\tau$, say) is the mean deviation from the median, but I the minimal sufficient statistic is the complete set of order statistics; the minimal sufficient statistic doesn't reduce to only some function of the MLE for $(\theta,\tau)$.

However, the thing stated in the question would still seem to hold (since the MLE of $(\theta,\tau)$ is some function of the MSS), but it seems you could write some functions of the MSS that those two statistics don't capture; say the difference-in-the-extreme-gaps $X_{(n)}-X_{(n-1)}-(X_{(2)}-X_{(1)})$, could not be recovered from them.

Glen_b
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  • Thanks for your reply! This is actually a true or false statement and I want to know the answer. So, will you say "yes" or "no". – Bratt Swan Apr 14 '16 at 01:15
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    @Bratt if you're doing homework/assignments/practice questions/work for a course/textbook exercises/studying old exams/etc etc, then you should mark your question `self-study` and modify it to follow the guidelines at its [tag wiki](http://stats.stackexchange.com/tags/self-study/info) - *in particular* you should show *in detail* your own attempts to answer the question and expect only hints/guidance on your answer. Trying to get people to do you work for you would be grounds for immediate closure of your question. See also the related discussion of homework in the [help/on-topic] – Glen_b Apr 14 '16 at 01:32