Minimax estimation of norms of a probability density: I. Lower bounds

Minimax lower bounds for estimation

Estimation of a k-monotone density: characterizations, consistency and minimax lower bounds.

The classes of monotone or convex (and necessarily monotone) densities on ℝ(+) can be viewed as special cases of the classes of k-monotone densities on ℝ(+). These classes bridge the gap between the classes of monotone (1-monotone) and convex decreasing (2-monotone) densities for which asymptotic results are known, and the class of completely monotone (∞-monotone) densities on ℝ(+). In this pap...

Minimax lower bounds

Now that we have a good handle on the performance of ERM and its variants, it is time to ask whether we can do better. For example, consider binary classification: we observe n i.i.d. training samples from an unknown joint distribution P on X× {0,1}, where X is some feature space, and for a fixed class F of candidate classifiers f :X→ {0,1} we let f̂n be the ERM solution f̂n = argmin f ∈F 1 n n ∑...

Minimax Lower Bounds

Minimax Lower Bounds Adityanand Guntuboyina 2011 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 measu...

Local Privacy and Minimax Bounds: Sharp Rates for Probability Estimation

We provide a detailed study of the estimation of probability distributions— discrete and continuous—in a stringent setting in which data is kept private even from the statistician. We give sharp minimax rates of convergence for estimation in these locally private settings, exhibiting fundamental trade-offs between privacy and convergence rate, as well as providing tools to allow movement along ...