Plug-in L2-upper error bounds in deconvolution, for a mixing density estimate in Rd and for its derivatives, via the L1-error for the mixture

Abstract

In deconvolution in Rd, d≥1, with mixing density p(∈P) and kernel h, the mixture density fp(∈Fp) is estimated with MDE fpˆn, having upper L1-error rate, an, in probability or in risk; pˆn∈P. In one application, P consists of L1-separable densities in R with differences changing sign at most J times and h(x−y) Totally Positive. When h is known and p is q˜-smooth, vanishing outside a compact in Rd, plug-in upper bounds are provided for the L2-error rate of pˆn and its [s]-th mixed partial derivative pˆ(s)n, via ∥∥fpˆn−fp∥∥1, with rates (loga−1n)−N1 and aN2n, respectively, for h super-smooth and smooth; q˜∈R+,[s]≤q˜,d≥1, N1>0, N2>0. For an∼(logn)ζ⋅n−δ, the former rate is optimal for any δ>0 and the latter misses the optimal by the factor (logn)ξ when δ=.5; ζ>0,ξ>0. N1 and N2 appear in optimal rates and lower error and risk bounds in the deconvolution literature.