IRTG1792DP2018 061
PLUG-IN L2-UPPER ERROR BOUNDS IN DECONVOLUTION, FOR A MIXING DENSITY ESTIMATE IN Rd AND FOR ITS DERIVATIVES
Yannis G. Yatracos
Abstract
In deconvolution in Rd; d 1; with mixing density p(2 P) and kernel h; the mixture
density fp(2 Fp) can always be estimated with f^pn; ^pn 2 P; via Minimum Distance
Estimation approaches proposed herein, with calculation of f^pn's upper L1-error rate, an;
in probability or in risk; h is either known or unknown, an decreases to zero with n: In
applications, an is obtained when P consists either of products of d densities dened on
a compact, or L1 separable densities in R with their dierences changing sign at most J
times; J is either known or unknown. When h is known and p is ~q-smooth, vanishing
outside a compact in Rd; plug-in upper bounds are then also provided for the L2-error
rate of ^pn and its derivatives, respectively, in probability or in risk; ~q 2 R+; d 1: These
L2-upper bounds depend on h's Fourier transform, ~h(6= 0); and have rates (log a??1
n )??N1 and aN2 n , respectively, for h super-smooth and smooth; N1 > 0; N2 > 0: For the typical
an (log n) n??; the former (logarithmic) rate bound is optimal for any > 0 and
the latter misses the optimal rate by the factor (log n) when = :5; > 0; > 0: The
exponents N1 and N2 appear also in optimal rates and lower error and risk bounds in the
deconvolution literature.
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