Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 36 / 3 / c / Solution

For each $n\ge 1$, choose the continuous probability density functionsThe second is at least $1/2$ and integrates to one. Their expected values differ byWriting $g_n/f=1+{\Delta }_n$ givesThe supplied product bound, also obtained from the chi-squared divergence of product measures, impliesFor the final elementary estimate, compare the exponential series with the geometric series: $e^x\le (1-x{)}^{-1}$ for $0\le x<1$. Thus the overlap of the product laws is at least $1/2$. Part (b) now gives the explicit mean-estimation minimax lower bound for continuous densitiesSo one may take $C=1/1152$, uniformly for all $n\ge 1$. The alternatives are allowed to depend on $n$, because the minimax risk takes a supremum over all continuous probability density functions separately at each sample size.