I would like to ask you, if somebody helps me to solve one statistics problem.
I have $X_1, …, X_n$ a sample of independent random variables with uniform distribution $(\theta, \theta +1)$
I have an estimator $\hat{\theta}_n=\max$ $X_i -1$ for $1 \le i \le n$, and I should find if it is unbiased and consistent, I found, that it is consistent but I think that it is biased. I find density of my estimator:
$f_Y(y)=n(y+1-\theta)^{(n-1)}$, where $y=\max$ $X_i$, but I am not sure, if it is correct, because if I calculate $E(\hat{\theta}_n)$, the result was quite strange.
The last step, which I should do, is following: how should I change $\hat{\theta}_n$ to make it unbiased. I think that I have to substract 1, but then I am not sure.
Please, can somebody check if I calculate the density correctly and how I can find an unbiased estimator (I cannot use MLE)?
Thank you
Best Answer
Your result is correct
Letting $Y=\hat{\theta}_n=\max (X_i) -1$ we have
$$P(Y \le y) = P(\max (X_i) \le y +1)=\prod P(X_i \le y+1) = (y +1 - \theta)^n $$
Hence $$f_Y(y) = n (y+1-\theta)^{n-1} \hspace{1cm } \theta-1\le y \le \theta$$
And $E(Y)=\theta - \frac{1}{n+1}$
Hence the estimator is biased (but also asymptotically unbiased)
(Both results, and the sign of the bias are intuitively obvious : for one thing, note that always $X < \theta+1 \implies \hat{\theta_n} < \theta$. Also, it's easy to see that for large $n$ the maximum will be very near $\theta+1$, hence we should expect $E(\hat{\theta}_n) \to \theta$)
To make it unbiased, you can try some linear transformation $Z=a(Y+b)$