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Kurisu, Daisuke (J-TOKYTE-IEC) ; Otsu, Taisuke (4-LSE-EC)
On linearization of nonparametric deconvolution estimators for repeatedmeasurements model. (English summary)J. Multivariate Anal. 189 (2022), Paper No. 104921, 19 pp.

This article is concerned with nonparametric density deconvolution estimation for theclassical measurement error model. Let a scalar variable Y be generated by Y = X +, where X is the error-free latent variable having an unknown density fX ,  is themeasurement error with a density f and E() = 0, and X and  are independent.
Suppose that we are primarily interested in estimating fX nonparametrically, given thenoisy measurement Y . The deconvolution problem has been typically studied either
under the assumption that the distribution of  is known or under the one that priorinformation about its shape is available (e.g., symmetry) even though the distribution
is unknown.
In contrast, leaving both fX and f unspecified, T. Li and Q. H. Vuong [J. Multivariate Anal. 65 (1998), no. 2, 139–165; MR1625869] demonstrated that these densities
can be estimated nonparametrically via repeated measurements. For n pairs of i.i.d.
noisy measurements {(Y1j , Y2j )}nj=1 = {(Xj + 1j , Xj + 2j )}nj=1 , define the empiricalcharacteristic function of (Y1 , Y2 ) and its derivative with respect to the first argument
asnXb 1 , u2 ) = 1
ψ(uexp{i(u1 Y1j + u2 Y2j )} andn j=1nb 1 , u2 )∂ ψ(u1Xbψ1 (u1 , u2 ) =iY1j exp{i(u1 Y1j + u2 Y2j )},
=
∂u1
n j=1
√
respectively, where i = −1. Then, relying on I. I. Kotlarski’s [Pacific J. Math. 20(1967), 69–76; MR0203769] identity, Li and Vuong [op. cit.] proposed to estimate thecharacteristic functions of X and  as
Z ubb u)ψ1 (0, u2 )ψ(0,ϕb X (u) = exp
du2 and ϕb  (u) =,b u2 )ϕb X (u)ψ(0,0respectively. Finally, two densities fX and f can be estimated byZ1bfa (t) =e−iut ϕb a (u)bϕK (hu)du2π RRfor a ∈ {X, }, where ϕb K (u) = R eiux K(x)dx is the Fourier transform of a kernel
function K, and h is the bandwidth.
In this article, it is demonstrated that fbX − fX and fb − f can be represented in an
asymptotic linear form uniformly over a given compact interval. Four results are available, depending on whether the distributions of X and  may be ordinary smooth orsupersmooth. These results are obtained via intermediate Gaussian approximations. Anadvantage of utilizing such approximations is faster uniform convergence rates of fbX
and fb than not only those derived by Li and Vuong [op. cit.] but also those obtainedin yet another article by the authors [Econometric Theory 38 (2022), no. 1, 172–193,
doi:10.1017/S0266466620000572] via maximal inequalities. On the other hand, approx-

imation errors in the sample counterparts of linearization terms have slow convergence
rates. Therefore, the authors recommend using subsample-based bootstrap counterparts of the linearization terms to construct bootstrap confidence bands for fbX andMasayuki Hirukawafb .
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