On the risk of estimates for block decreasing densities

Abstract

A density f= f( x 1,…, x d ) on [0,∞) d is block decreasing if for each j∈{1,…, d}, it is a decreasing function of x j , when all other components are held fixed. Let us consider the class of all block decreasing densities on [0,1] d bounded by B. We shall study the minimax risk over this class using n i.i.d. observations, the loss being measured by the L 1 distance between the estimate and the true density. We prove that if S=log(1+ B), lower bounds for the risk are of the form C( S d / n) 1/( d+2) , where C is a function of d only. We also prove that a suitable histogram with unequal bin widths as well as a variable kernel estimate achieve the optimal multivariate rate. We present a procedure for choosing all parameters in the kernel estimate automatically without loosing the minimax optimality, even if B and the support of f are unknown.