On the Risk of Estimates for Block Decreasing Densities

A density f = f(x1, . . . , xd) on the hypercube [0, 1] d is block decreasing if, for each j ∈ {1, . . . , d}, it is a decreasing function of xj as xj → 1, 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 L1 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(Sd/n)1/(d+2), where C is 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.