Let $x_i$ be independent and identically distributed observations in a sample from a uniform distribution over $[0,θ]$. Now I need to estimate $θ$ based on $N$ observations and I want the estimator to be unbiased.
I thought about simple estimator $\hatθ =\min(x_i)$.
Based on simulation it is not biased, yet I couldn't show it analytically.
Could anyone, please, show how can I get it unbiased?
Best Answer
I propose the $\hat{\theta}_N=\frac{N+1}{N}\max(X_1,\ldots,X_N)$. Indeed, let $Y=\max(X_1,\ldots,X_N)$ then $$\Bbb{P}(Y\leq t)=\left(\frac{t}{\theta}\right)^N,$$ Hence, the density of $Y$ is given by $f_Y(t)=\frac{N}{\theta^N}t^{N-1}{\bf 1}_{[0,\theta]}(t)$ and $$\Bbb{E}(\hat\theta_N)=\frac{N+1}{N}\Bbb{E}(Y)=\frac{N+1}{N}\frac{N}{\theta^N}\int_0^\theta t^Ndt=\theta.$$ and we are done.