Maximum likelihood estimator of the following uniform distribution function

by RAHUl JHa   Last Updated January 14, 2018 12:19 PM

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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