Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 26K / c

The support restricts $\theta$ to $\theta >0$, $\theta \le X_{(1)}$ and $\theta \ge X_{(n)}-1$. Within its feasible interior, the likelihood function is proportional to ${\prod }_i(X_i-\theta {)}^{-\alpha }$ and is strictly increasing in $\theta$, diverging as $\theta \uparrow X_{(1)}$. Under the printed infinite-endpoint-density, or extended-likelihood convention, the maximum-likelihood estimator isA density is defined only almost everywhere. If its value at the singular endpoint is reassigned a finite value, a finite maximum need not exist; the sample-minimum answer uses the stated endpoint/supremum convention.

Independence and $\mathbb{P}(Y\le t)=t^r$ on $[0,1]$ giveThe error is positive almost surely, so the estimator is not an unbiased estimator. More explicitly its bias is