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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.

MSC:

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

Software:

pyuvdata
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References:

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