Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 26K (Principles of Statistics)

Put $r=1-\alpha \in (0,1)$. The variable $Y=X_1-\theta$ has density $r{y}^{r-1}$ on $(0,1)$, so $\mathbb{E}{Y}^j=r/(r+j)$. ThusThe unbiased estimator isTaking expectations verifies that its bias is zero.

The observations have finite mean and variance. The strong law of large numbers gives ${\overline{X}}_n\to \mathbb{E}{X}_1$ almost surely, so ${\tilde{\theta }}_n\to \theta$ almost surely and therefore is a consistent estimator. The central limit theorem gives

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

Set $\boxed{\beta =1/(1-\alpha )=1/r}$. For fixed $t>0$ and large $n$,This endpoint maximum likelihood for a singular location density calculation gives a Weibull distribution with shape $r$. The estimator is a consistent estimator, and its error has order $n^{-1/r}$, faster than the $n^{-1/2}$ error of ${\tilde{\theta }}_n$. Its bias is asymptotic to $\Gamma (1+1/r)n^{-1/r}$ and its mean squared error to $\Gamma (1+2/r)n^{-2/r}$, versus a positive constant times $n^{-1}$ for the sample-mean estimator. Thus the minimum estimator is preferable for its much faster convergence, despite its positive finite-sample bias. This is a nonregular support-dependent model; ordinary regular-model efficiency bounds do not apply.