Questioning the if this estimator $\min\{\ Y_1,0\}$ is unbiased and consistent, where $Y_1=\min X_k, X_k: U(0, \theta)$

Question

Questioning the if this estimator (which I found using the method of maximum likelihood )$$\hat\theta_n=\min\{ Y_1,0\}$$ is unbiased and consistent, where $$Y_1=\min X_k; \qquad X_k : U( \theta,0), \quad \theta<0.$$

I've done this with other ones, such as $\overline X_n$ for example, but I have no clue how to go about solving this one. $E\hat\theta_n=\theta\ \ ?$

Answer 1

With probability $1$, $Y_1<0$. Hence $\widehat{\theta}_n=Y_1$ a.s.

Thus you need to find $E(Y_1)$ which is simple: just find the pdf of $Y_1$.

Note, for any $t\in (\theta,0)$, $P(Y_1>t)=P(X_k>t\forall k=1,2,\ldots,n) = P(X>t)^n$ (using independence and identical distribution of the $X_k$) $=\left(\dfrac{-t}{-\theta}\right)^n=\dfrac{t^n}{\theta^n}$.

So pdf of $Y_1$ evaluated at $t$ is $\dfrac{-nt^{n-1}}{\theta^n}$ for $t\in (\theta,0)$.

Now simply find $E(Y_1)$ and check consistency!

Answer 2

Since $\Pr(\widehat\theta_n > \theta)=1$ and $\Pr(\widehat\theta_n = \theta) = 0$, you must have $\operatorname{E}(\widehat\theta)>\theta$, so this is biased.