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MLE is an algorithm to find a reasonable estimator (personally, it really woos my mind - simple and yet so beautiful.).

Now, well - Life is hard. People has devised ways to check if an esimator is good or not - why will they care I like it or not.

So, they have developed Small Sample Properties and Large Sample Properties to do the quality control of MLE.

This post tests the flamboyancy of MLE is terms of the idea of "Sufficiency".

We discover a rich relationship between the two. Again, MLE wins my heart. Does it win yours? Check with the hints and the solution.

Hint

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Food For Thought

- Use Neymann Factorization Theorem to prove that MLE is a function of a Sufficient Statistic
T . - Write the Likelihood of the data is terms of
T and progress towards the proof. - Now, take a sample from Uniform (\(\theta,\theta +1\)).
- Now, observe that MLE is not unique and find a MLE which is not a function of
T - a small but creative step. - Look out for the Food for Thought.

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- Is MLE a sufficient statistic?
- Is MLE consistent?
- IS MLE a complete sufficient statistic?
- Is MLE asymptotically normal?
- Is MLE UMVUE?
- What if the underlying distribution is Exponential Family?

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