Minimax Lower Bounds

This thesis deals with lower bounds for the minimax risk in general decision-theoretic problems. Such bounds are useful for assessing the quality of decision rules. After providing a unified treatment of existing techniques, we prove new lower bounds which involve f-divergences, a general class of dissimilarity measures between probability measures. The proofs of our bounds rely on elementary convexity facts and are extremely simple. Special cases and straightforward corollaries of our results include many well-known lower bounds. As applications, we study a covariance matrix estimation problem and the problem of estimation of convex bodies from noisy support function measurements.