Convolution and Minimax Theorems

Abstract

Let H be a linear subspace of a Hilbert space with inner product (·, ·) and norm I · II. For each n ∈ N and h ∈ H, let P _n,h be a probability measure on a measurable space (X _n , A _n ). Consider the problem of estimating a “parameter” k _n (h) given an “observation” X _n with law P _n,h . The convolution theorem and the minimax theorem give a lower bound on how well k _n (h) can be estimated asymptotically as ↦ ∞. Suppose the sequence of statistical experiments (X _n, A _n , P _n,h : h ∈ H) is “asymptotically normal” and the sequence of parameters is “regular”. Then the limit distribution of every “regular” estimator sequence is the convolution of a certain Gaussian distribution and a noise factor. Furthermore, the maximum risk of any estimator sequence is bounded below by the “risk” of this Gaussian distribution. These concepts are defined as follows.