# Biasedness of Uniform Distribution MLE

How do I show that the maximum likelihood estimator for uniform distribution on $$[0, \theta]$$ for a random sample of size $$n$$ is biased? I've calculated the MLE as $$\max_i\{X_i\}$$.

Intuitively, we can say that estimator is biased because only the maximum of $$X_i$$ from the sample would be the estimator. How do I derive the distribution of MLE and check for biasedness?

• – StubbornAtom Oct 22 '18 at 12:59
• Also possibly answered here. – StubbornAtom Oct 22 '18 at 13:02

So $$X_1, \dotsc, X_n$$ is iid uniform on $$(0, \theta)$$ with $$\theta > 0$$. Then the maximum likelihood estimator (also sufficient statistic) of $$\theta$$ is $$M=\max_i X_i$$. Now clearly $$M < \theta$$ with probability one, so the expected value of $$M$$ must be smaller than $$\theta$$, so $$M$$ is a biased estimator.
We need to find the distribution of $$M$$. Use that $$P(M \le m)= P(X_1\le m, X_2\le m, \dotsc, X_n\le m)=\left(m/\theta\right)^n$$ and by differentiation you can find the density $$f(m)=n\left(\frac{m}{\theta}\right)^{n-1}\frac1\theta$$, Then integration will yield the expected value as $$\frac{n}{n+1}\theta$$.