Robust density estimation over star-shaped density classes

Abstract

We establish a novel criterion for comparing the performance of two densities, g_1 and g_2, within the context of corrupted data. Utilizing this criterion, we propose an algorithm to construct a density estimator within a star-shaped density class, F, under conditions of data corruption. We proceed to derive the minimax upper and lower bounds for density estimation across this star-shaped density class, characterized by densities that are uniformly bounded above and below (in the sup norm), in the presence of adversarially corrupted data. Specifically, we assume that a fraction 13 of the N observations are arbitrarily corrupted. We obtain the minimax upper bound \ _J^2, \ d^2. Under certain conditions, we obtain the minimax risk, up to proportionality constants, under the squared L_2 loss as where ^* := \ : N^2 M_F^loc(, c) \ for a sufficiently large constant c. Here, M_F^loc(, c) denotes the local entropy of the set F, and d is the L_2 diameter of F.

Tasks

Reproductions