When each subject in a study is observed only through m≥2m \ge 2 repeated contaminated dd-dimensional measurements (d≥2d \ge 2) and the measurement-error distribution is unknown, its characteristic function can be estimated from within-subject differences, and the density of the unobserved signal can be estimated by a kernel deconvolution estimator involving a full d×dd \times d bandwidth matrix. This estimator is the standard tool in applications such as estimating the joint distribution of long-run dietary intakes from error-prone 24-hour recalls, yet its asymptotic theory is missing: for multivariate panel data there are no convergence rates, no characterization of asymptotically optimal bandwidth matrices, and no data-driven bandwidth-matrix selection rules with guarantees. Later work (Capitao-Miniconi, Gassiat and Lehéricy, Electronic Journal of Statistics 2025) settled the achievable minimax rates for this observation model under coordinate-independent noise using a projection estimator, but left the kernel-estimator theory, correlated-coordinate errors, and bandwidth-matrix selection open. The problem asks for this missing theory: MISE rates for the kernel deconvolution estimator with a panel-estimated error characteristic function, asymptotically optimal (possibly non-diagonal) bandwidth matrices, data-driven selection rules with oracle-type guarantees, and, secondarily, scalable numerical evaluation of the multivariate deconvolution integral.

Background

When a random vector of interest cannot be observed directly but only through contaminated measurements, estimating its density requires undoing the convolution — the density deconvolution problem. In panel data (repeated measurements on each subject), subject ii is observed through m≥2m\ge 2 replicate measurements Yij=Xi+ϵij\boldsymbol{Y}_{ij}=\boldsymbol{X}_i+\boldsymbol{\epsilon}_{ij}, j=1,…,mj=1,\dots,m, in dimension d≥2d\ge 2, where the signal Xi\boldsymbol{X}_i has unknown density fXf_{\boldsymbol{X}} and the replicate errors are independent and identically distributed with unknown distribution, independent of the signal. Replication is what makes the problem identifiable without knowing the error distribution: for a within-subject difference δ=Yi2−Yi1=ϵi2−ϵi1\boldsymbol{\delta}=\boldsymbol{Y}_{i2}-\boldsymbol{Y}_{i1}=\boldsymbol{\epsilon}_{i2}-\boldsymbol{\epsilon}_{i1}, the characteristic function (CF, the Fourier transform of the density) satisfies ϕδ=∣ϕϵ∣2\phi_{\boldsymbol{\delta}}=|\phi_{\boldsymbol{\epsilon}}|^2, and when the error distribution is symmetric with a CF that never vanishes one can recover ϕϵ=ϕδ\phi_{\boldsymbol{\epsilon}}=\sqrt{\phi_{\boldsymbol{\delta}}}. Kernel deconvolution estimators invert the convolution in the frequency domain: with ϕ^Yˉ,n\hat\phi_{\bar{\boldsymbol{Y}},n} the empirical CF of subject averages, K∗K^* the Fourier transform of a smoothing kernel KK, and BnB_n a d×dd\times d positive-definite bandwidth matrix, a plug-in estimator used in practice is

f^X,n(x)=1(2π)d∫e−ıt⋅x  ϕ^Yˉ,n(t) K∗(Bnt)[ϕ^ϵ,n(t/m)]m dt,\hat f_{\boldsymbol{X},n}(\boldsymbol{x})=\frac{1}{(2\pi)^d}\int e^{-\imath\boldsymbol{t}\cdot\boldsymbol{x}}\;\hat\phi_{\bar{\boldsymbol{Y}},n}(\boldsymbol{t})\,\frac{K^*(B_n\boldsymbol{t})}{[\hat\phi_{\boldsymbol{\epsilon},n}(\boldsymbol{t}/m)]^{m}}\,d\boldsymbol{t},

where ϕ^ϵ,n\hat\phi_{\boldsymbol{\epsilon},n} is a difference-based estimator of the error CF (an empirical CF of differences or the CF of a kernel density estimate of them). Common kernel choices include the sinc kernel, whose Fourier transform is the indicator function of a centered rectangle, and flat-top kernels, whose Fourier transform equals 1 on a neighborhood of the origin and then tapers. Two features dominate the difficulty. First, the factor 1/ϕϵ1/\phi_{\boldsymbol{\epsilon}} amplifies frequencies where the error CF is small; error CFs are classified as ordinarily smooth (polynomial decay) or supersmooth (exponential decay, e.g., Gaussian errors), and this classification controls the attainable convergence rates. Second, the error CF in the denominator is itself estimated, so its random fluctuations near small values must be stabilized, typically by ridge terms (adding a small positive constant to keep the denominator away from zero) or truncation (capping the integration domain). The relevant prior theory splits into two strands, neither of which covers multivariate panel data. For multivariate observations with a known error distribution, rates of convergence of kernel deconvolution estimators go back to Masry (1991); bandwidth selection was addressed by Youndjé and Wells (2008), but restricted to a single bandwidth parameter, that is, a diagonal bandwidth matrix with equal diagonal entries; adaptive anisotropic theory (different smoothing in different directions) was developed by Comte and Lacour (2013) for kernel estimators and by Dussap (2022) with Laguerre-basis projections on the positive orthant, still with a known error distribution. For univariate panel data with unknown error distribution, consistency, rates and data-driven procedures were established by Neumann (2007), Delaigle, Hall and Meister (2008), Comte and Kappus (2015, dropping the symmetry assumption via Kotlarski's identity, which identifies the signal CF from repeated observations), and Kurisu and Otsu (2022a, 2022b). Basulto-Elias, Carriquiry, De Brabanter and Nordman (2021) studied the bivariate panel case by simulation — comparing kernel types, panel pre-processing schemes, and error-CF estimators, under symmetric errors with nonvanishing CF and possibly correlated error coordinates — and provided numerical FFT-based integration recipes, but no asymptotic theory; their Discussion states the missing theory for multivariate panel deconvolution (rates of convergence of the kernel density estimators, optimal bandwidth choices, and data-driven bandwidth selection) as the principal open problem.

Problem Statement

Develop the asymptotic theory for kernel deconvolution density estimation from multivariate panel data with an unknown error distribution, as posed in the Discussion of Basulto-Elias et al. (2021).

Setting: independent subjects i=1,…,ni=1,\dots,n, each observed through m≥2m\ge 2 repeated measurements

Yij=Xi+ϵij,j=1,…,m,\boldsymbol{Y}_{ij}=\boldsymbol{X}_i+\boldsymbol{\epsilon}_{ij},\qquad j=1,\dots,m,

in dimension d≥2d\ge 2, where the signals Xi\boldsymbol{X}_i are i.i.d. with unknown density fXf_{\boldsymbol{X}} and the replicate errors are i.i.d. across repetitions with unknown distribution, independent of the signal; in the source framework the error distribution is symmetric with a characteristic function that vanishes nowhere (error coordinates possibly correlated). The error characteristic function is estimated from within-subject differences, and the plug-in kernel deconvolution estimator is

f^X,n(x)=1(2π)d∫e−ıt⋅x  ϕ^Yˉ,n(t) K∗(Bnt)[ϕ^ϵ,n(t/m)]m dt,\hat f_{\boldsymbol{X},n}(\boldsymbol{x})=\frac{1}{(2\pi)^d}\int e^{-\imath\boldsymbol{t}\cdot\boldsymbol{x}}\;\hat\phi_{\bar{\boldsymbol{Y}},n}(\boldsymbol{t})\,\frac{K^*(B_n\boldsymbol{t})}{[\hat\phi_{\boldsymbol{\epsilon},n}(\boldsymbol{t}/m)]^{m}}\,d\boldsymbol{t},

with kernel KK (for example sinc or flat-top), a full d×dd\times d positive-definite bandwidth matrix BnB_n, and stabilization (ridge or truncation) of the estimated error CF in the denominator where needed.

The problem asks for:

  1. Convergence rates of these estimators (for the mean integrated squared error or pointwise mean squared error) over standard smoothness classes of fXf_{\boldsymbol{X}} and for both ordinarily smooth and supersmooth error characteristic functions, quantifying how estimating the error characteristic function from panel differences affects the rates (including the role of the number of repetitions mm and of any ridge/truncation).
  2. The characterization of asymptotically optimal bandwidth matrices BnB_n — not restricted to diagonal or equal-diagonal matrices — for this panel setting, and how they depend on the directional smoothness of fXf_{\boldsymbol{X}} and the directional decay of the error CF.
  3. Data-driven bandwidth-matrix selection rules (plug-in, cross-validation, penalization, or comparable) together with theoretical guarantees of their validity (for example, oracle inequalities, meaning risk bounds relative to the best bandwidth matrix in the class, or asymptotic optimality) when the error distribution is estimated from the panel. As a secondary practical objective from the same source discussion, scalable and numerically reliable evaluation of the dd-dimensional deconvolution Fourier integral, since grid-based (fast Fourier transform / Riemann-sum) approximations become computationally prohibitive beyond d=2d=2 or 33.

References

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