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MR4101465 62G05 62G08
Chesneau, Christophe (F-CAEN-LM) ; El Kolei, Salima (F-ENSAI-CES) ;Kou, Junke [Kou, Jun Ke] (PRC-GUET-NDM) ; Navarro, Fabien
Nonparametric estimation in a regression model with additive andmultiplicative noise. (English summary)J. Comput. Appl. Math. 380 (2020), 112971, 14 pp.

A nonparametric regression model
Y = f (X)U + V
is investigated, where f : [0, 1]d → R is unknown and (U, V ) are multiplicative andadditive errors so that U is independent of X and V . Applications of this model canbe found, for instance, in stochastic frontier models and signal and image processing.
The focus of this article is on wavelet estimation of the quadratic functional r := f 2
using n random samples (Y1 , X1 ), . . . , (Yn , Xn ). Convergence rates of two estimatorsof r, namely, the linear (r̂nlin ) and nonlinear (r̂nnon ) wavelet estimators, over the mean
integrated squared error are derived. It is demonstrated that if r belongs to the Besovsspace Bp,q([0, 1]d ) with p, q ∈ [1, ∞) and s > d/p and other suitable regularity conditions
hold, then"Z#
 lin
2s02E
r̂n (x) − r(x) dx ≤ cn− 2s0 +d , and[0,1]d"ZE[0,1]d#2{r̂nnon (x) − r(x)} dx2s≤ c(ln n)n− 2s+dwhere c > 0 is a generic constant and s0 = s − d(1/p − 1/2)+ with x+ := max{x, 0}.
Masayuki Hirukawa
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