This question is from one of the introductory books in Mathematical Statistics.

Let $X_1,...,X_n$ be a random sample from a pdf $f(x;\theta)=\frac{1}{2\theta+1},0<x<2\theta+1,$ zero elsewhere.

(a) Find the MLE $\hat{\theta}$ of $\theta$

(b) Find a complete and sufficient statistics for $\theta$.

(c) Find the UMVUE of $\theta$

## I think I can solve (a) and a half (b)

For (a):

Let $Y_1<Y_2<...Y_n$ be the order statistics.

$L(\theta)=\frac{1}{(2\theta+1)^{n}} I(0 \leq Y_{1}) I(Y_{n} \leq 2\theta+1)$

$\therefore$ MLE $\hat{\theta}=\frac{1}{2}(Y_{n}-1)$

For (b):

$f(x_1;\theta)f(x_2;\theta)...f(x_n;\theta)=\frac{1}{(2\theta+1)^{n}} I(0 \leq Y_{1}) I(Y_{n} \leq 2\theta+1)$

$\therefore$ By factorization theorem of Neyman, $Y_n=\max(x_i)$ is a sufficient statistic for $\theta$. Therefore, $\frac{1}{2}(Y_{n}-1)$ is also a sufficient statisitc

I was wondering if someone would help me out in showing it is complete and find UMVUE of $\theta$.