On the risk of estimates for block decreasing densities

A density f=f(x1,…,xd) on [0,∞)d is block decreasing if for each j∈{1,…,d}, it is a decreasing function of xj, 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 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.