On minimax estimation in the presence of side information about remote data

We analyze the following model: One person, called "helper" observes an outcome xn=(x1,⋯,xn)∈Xn of the sequence Xn=(X1,⋯,Xn) of i.i.d. RV's and the statistician gets a sample yn=(y1,⋯,yn) of the sequence Yn(θ,xn) of RV's with a density ∏nt=1f(yt∣θ,xt). The helper can give some (side) information about xn to the statistician via an encoding function sn:Xn→N with rate(sn)def=(1/n)log# range(sn)≤R. Based on the knowledge of sn(xn) and yn the statistician tries to estimate θ by an estimator ^θn. For the maximal mean square error en(R)=definf^θninfsn:rate(sn)≤Rsupθ∈ΘEθ|^θn−θ|2 we establish a Cramer-Rao type bound and, in case of a finite X, prove asymptotic achievability of this bound under certain conditions. The proof involves a nonobvious combination of results (some of which are novel) for both coding and estimation.