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Let $Y$ be independent and identically distributed normal with mean $2\mu$ and variance $16$.

Show that $\frac{\bar y}{2}$ is the maximum likelihood estimator of $\mu$.

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    $\begingroup$ As a routine textbook type question, this counts as self-study. Please add the self-study tag and review its tag wiki info. With that in mind, what have you tried, and where does your difficulty lie? $\endgroup$
    – Glen_b
    Feb 9, 2014 at 0:40

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You can use invariance property of Maximum Likelihood estimator:

http://en.wikipedia.org/wiki/Maximum_likelihood

If alpha_hat=g(theta_hat) then theta_hat is MLE for alpha_hat.

And notice that 1/2 is jacobian for transformation of alpha=2*theta and average of y is MLE for myy.

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  • $\begingroup$ Why does anyone need to consider the Jacobian? There is no integration over the parameters. $\endgroup$
    – whuber
    Feb 10, 2014 at 17:35

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