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Let we have $X_{1},X_{2},..,X_{n}$ as independent and identically distributed random variable from $U(\theta,\theta+1)$. Clearly, the maximum likelihood estimator of such distribution will be $[X_{(n)}-1,X_{(1)}]$. I know that maximum likelihood estimator follows the invariance property. Can any function from these two estimators will be maximum likelihood? Then, why not $\frac{X_{(1)}+X_{(n)}}{2}$ will be maximum likelihood estimator since they are functions of maximum likelihood estimators.

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There is no unique MLE. So perhaps it is a matter of semantics whether the invariance property applies to quantities from the interval $[X_{(n)}-1,X_{(1)}]$ and perhaps difficult to say what invariance would mean.

For $n \ge 2,$ your proposal $T_1 = (X_{(1)}+X_{(n)})/2$ is severely biased; $E(T_1) = \theta + \frac 12.$ The midpoint $T_2$ of the interval is unbiased. (I suspect it may be what you meant to propose.) You might want to explore whether $T_2$ has smaller variance than other unbiased linear functions of the interval endpoints?

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  • $\begingroup$ Perhpas look at this page. $\endgroup$ – BruceET Jan 14 '18 at 20:27

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