On the risk of estimates for block decreasing densities. (English) Zbl 1025.62015

Summary: A density \(f=f(x_1,\dots,x_d)\) on \([0,\infty)^d\) is block decreasing if for each \(j\in\{1,\dots,d\}\), 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.


62G07 Density estimation
62H12 Estimation in multivariate analysis
62C20 Minimax procedures in statistical decision theory


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