Maximum likelihood estimation 3

Question

if I have a simple random sample $Y_{1},...,Y_{n}$ of an uniform distribution over interval $(0,2\theta+1)$, how can i compute the maximum likelihood estimation of $\theta$?

Thank you for your time.

Answer 1

The m.l.e. for $2\theta+1$ is the highest order statistic $Y^{(n)}=\max\{Y_i~|~i=1\cdots,n\}$. Now if $T$ is a m.l.e. for parameter $\eta$ then for continuous function $f$, $f(T)$ is the m.l.e. for $f(\eta)$. As $g(x)=\frac{x-1}2$ is a continuous function $g(Y^{(n)})$ is the m.l.e. for $g(2\theta+1)=\theta$.