Distribution-Free Lower Bounds in Density Estimation

Distribution-free Lower Bounds in Density Estimation by Luc Devroye and Clark

fn(x) _ ( nh)-1 E n 1 K((X; x)/h), where K is a bounded even density with compact support, and X1 , • • •, X, are independent random variables with common density f. We treat the problem of placing a lower bound on the Ll error J, = E(f I fn f I) which holds for all f. In particular, we show that there exist A(K) > (9/125)" depending only upon K, and B*(f) > 1 depending only upon f such that (i...

Lower Bounds and Aggregation in Density Estimation

In this paper we prove the optimality of an aggregation procedure. We prove lower bounds for aggregation of model selection type of M density estimators for the Kullback-Leibler divergence (KL), the Hellinger’s distance and the L1-distance. The lower bound, with respect to the KL distance, can be achieved by the on-line type estimate suggested, among others, by Yang (2000a). Combining these res...

Lower Bounds for Estimation

Distribution-Free Testing Lower Bounds for Basic Boolean Functions

In the distribution-free property testing model, the distance between functions is measured with respect to an arbitrary and unknown probability distribution D over the input domain. We consider distribution-free testing of several basic Boolean function classes over {0, 1}, namely monotone conjunctions, general conjunctions, decision lists, and linear threshold functions. We prove that for eac...

Minimax lower bounds for estimation