Chapter 5.4, Problem 13Q

As an alternative to imposing unbiasedness, an estimator’s distribution can be “centered” by requiring that its median be equal to the unknown parameter $\theta$. If it is, $\hat{\theta }$ is said to be median unbiased. Let $Y_1,Y_2,\mathrm{...},Y_n$ be a random sample of size $n$ from the uniform pdf, $f_Y\left(y;\theta \right)=1\text{/}\theta ,0\le y\le \theta$. For arbitrary $n$, is $\hat{\theta }=\frac{n+1}{n}\cdot Y_{\text{max}}$ median unbiased? Is it median unbiased for any value of $n$?