Past exam of the algorithmic foundations of learning course of the University of Oxford / 2025 / 3 / b / ii

Use the first order statistic and correct its bias:This is a measurable function of the sample. Under $U_{\theta }^n$, write $Z=X_{(1)}-\theta$. It lies in $[0,1]$, and independence givesFor a bounded nonnegative random variable, integrating the identities $Z={\int }_0^1{1}_{\{Z>u\}}du$ and $Z^2={\int }_0^{1}2u{1}_{\{Z>u\}}du$ proves the tail integral formula for moments by Tonelli's theorem. HenceThus $\hat{\theta }$ is an unbiased estimator. Its quadratic risk is the variance of $Z$, independent of $\theta$:Taking the supremum over all real $\theta$ and then the infimum over estimators givesThe final inequality follows from $n^3\le (n+1{)}^2(n+2)$. Together with part (i), this proves that the minimax risk has order $n^{-2}$ for all $n\ge 1$. In the last line of the source, the request for the upper-bound constant calls it $C$; here it is denoted $C^{\prime }$ consistently with the upper-bound inequality.