Han and Park (2018) introduced kernel smooth-backfitting estimators for nonparametric additive regression in which every covariate is contaminated by classical measurement error, and proved oracle-type one-dimensional convergence rates under two idealized conditions: the measurement-error characteristic functions are known, and the error distributions are ordinary smooth (polynomially decaying characteristic functions) (the optimal one-dimensional rate is established for error-smoothness index β<1/2\beta<1/2, their Corollary 3.5; for β≥1/2\beta\ge1/2 the rates are slower than univariate but faster than multivariate). The open problem is to complete this theory for the realistic regimes the source paper explicitly excluded. First, when the error characteristic functions are unknown and must be estimated (for example from repeated measurements), the plug-in deconvolution kernel no longer satisfies the exact unbiased-scoring identity, and the required convergence rates of the estimated characteristic functions and their propagation through the backfitting projection to bias, variance and projection consistency are all undeveloped. Second, for super-smooth error distributions (characteristic functions decaying exponentially, such as Gaussian errors), the existing smooth-backfitting theory, which relies on compactly supported kernels, does not apply, and a theory based on noncompactly supported deconvolution kernels is missing. Later literature resolved adjacent questions — a two-stage series estimator with one-step backfitting that achieves super-smooth convergence rates and asymptotic normality for a single mismeasured covariate (Dong, Otsu and Taylor 2022), including first-stage rates when the error law is estimated from repeated measurements, and smooth-backfitting extensions to varying-coefficient and partially linear errors-in-variables models — but the smooth-backfitting-specific theory for estimated error laws and super-smooth errors, with multiple mismeasured covariates, remains open.

Background

A nonparametric additive model postulates Y=μ+∑j=1dmj(Xj)+εY=\mu+\sum_{j=1}^{d}m_j(X_j)+\varepsilon with unknown component functions mjm_j, and it is a standard way to evade the curse of dimensionality in multivariate nonparametric regression: each component can be estimated at the one-dimensional nonparametric rate. Backfitting estimates the components by iteratively smoothing the partial residual against each covariate; smooth backfitting, introduced by Mammen, Linton and Nielsen (1999), reformulates this iteration as a projection of the regression function onto the additive function class, which yields existence, uniqueness and oracle asymptotics under weak conditions. Yu, Park and Mammen (2008) extended smooth backfitting to generalized additive models with non-identity link functions.

In an errors-in-variables (measurement error) setting, the true covariates Xj∗X_j^{*} are unobserved; instead one observes Wj=Xj∗+UjW_j=X_j^{*}+U_j where UjU_j is a classical measurement error, independent of Xj∗X_j^{*} and of the regression noise. Naive smoothing on the contaminated WjW_j is biased (attenuated). The classical remedy is deconvolution: a kernel estimator is built whose Fourier transform divides by the error characteristic function ϕUj\phi_{U_j}, so that the contamination is inverted in frequency space. The attainable rates are governed by the tail of ϕU\phi_U (Fan 1991): for ordinary smooth errors, whose characteristic function decays polynomially (e.g. Laplace errors), rates are polynomial in the sample size; for super-smooth errors, whose characteristic function decays exponentially (e.g. Gaussian or Cauchy errors), rates are only logarithmic, and the deconvolution kernel must have a Fourier transform with suitable (e.g. compactly supported) structure, which is incompatible with the kernel itself having compact support. When ϕU\phi_U is unknown it can be identified and estimated from replicated measurements WjW_j and Wj′W_j' of the same latent variable, since Wj−Wj′=Uj−Uj′W_j-W_j'=U_j-U_j' and ∣ϕU(t)∣2=∣Ecos⁡{t(Wj−Wj′)}∣|\phi_U(t)|^2=|E\cos\{t(W_j-W_j')\}| (Delaigle, Hall and Meister 2008); plugging such estimates into deconvolution estimators changes their stochastic structure.

Han and Park (2018) combined these two strands: they extended smooth backfitting to additive models in which all covariates are classically mismeasured, solving a system of integral equations defined through per-covariate deconvolution kernels K~h∗\tilde K_h^{*} and proving that the resulting component estimators achieve oracle-type one-dimensional convergence rates. Their theory, however, assumes that each error characteristic function ϕUj\phi_{U_j} is known and that the error distributions are ordinary smooth, and it develops only convergence rates (not limiting distributions). A companion paper by Han, Müller and Park (2018) treated the different regime of small (negligible) measurement error without deconvolution. The question addressed here concerns exactly the two regimes excluded by Han and Park's theory — estimated error laws and super-smooth errors.

Problem Statement

For the additive errors-in-variables model Y=μ+∑j=1dmj(Xj∗)+εY=\mu+\sum_{j=1}^{d}m_j(X_j^{*})+\varepsilon with classical measurement error Wj=Xj∗+UjW_j=X_j^{*}+U_j on every covariate, develop the asymptotic theory of the smooth-backfitting estimators of Han and Park (2018) beyond their two standing restrictions, in the two directions the source explicitly left open:

  1. Unknown error characteristic functions. Suppose the ϕUj\phi_{U_j} are not known but estimated. Establish the convergence rates that ϕ^Uj\hat\phi_{U_j} must satisfy, and characterize how the estimation error propagates through the deconvolution kernels and the backfitting projection to the bias, variance and projection consistency of the smooth-backfitting component estimators — that is, establish the asymptotic properties of the plug-in estimators as a function of how fast ϕ^Uj\hat\phi_{U_j} converge to ϕUj\phi_{U_j}, as the source formulates the open question. (Limiting-distribution results are currently available neither for the smooth-backfitting estimator nor, for the estimated-error-law case, for any competing estimator; they would strengthen the goal but are not part of the source-stated open scope.)

  2. Super-smooth measurement errors. Suppose some or all UjU_j are super-smooth, i.e. ∣ϕUj(t)∣|\phi_{U_j}(t)| decays exponentially (Gaussian or Cauchy errors). Construct smooth-backfitting estimators that remain well-defined in this regime — which requires deconvolution kernels that need not be compactly supported, outside the current smooth-backfitting framework — and establish their existence and convergence rates for multiple simultaneously mismeasured covariates.

The source presents these as two separate challenging topics for future study (Section 6, pp. 2248–2249); each numbered direction is an independent target, and a treatment of their interaction (estimated error laws under super-smooth errors), including whether the oracle one-dimensional deconvolution rate is retained there, would be a natural extension but is not required. The source paper's further stated model-class extensions — generalized additive models with non-identity link, Berkson-type error models, and prediction under measurement error (Section 6, p. 2249), within the smooth-backfitting errors-in-variables framework — belong to the same research program and are recorded as adjacent open context, not as required deliverables; two of the originally listed extensions (varying-coefficient and partially linear additive models) have since been treated in the literature.

References

  1. Han, K. and Park, B. U. (2018). Smooth backfitting for errors-in-variables additive models. Annals of Statistics 46(5), 2216–2250. DOI: 10.1214/17-aos1617. https://doi.org/10.1214/17-aos1617
  2. Dong, H., Otsu, T. and Taylor, L. (2022). Nonparametric estimation of additive models with errors-in-variables. Econometric Reviews 41(10), 1164–1204. DOI: 10.1080/07474938.2022.2127076. https://doi.org/10.1080/07474938.2022.2127076
  3. Han, K., Lee, Y. K. and Park, B. U. (2020). Smooth backfitting for errors-in-variables varying coefficient regression models. Computational Statistics & Data Analysis 145, 106909. DOI: 10.1016/j.csda.2019.106909. https://doi.org/10.1016/j.csda.2019.106909
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  5. Han, K., Müller, H.-G. and Park, B. U. (2018). Smooth backfitting for additive modeling with small errors-in-variables, with an application to additive functional regression for multiple predictor functions. Bernoulli 24(2), 1233–1265. DOI: 10.3150/16-bej898. https://doi.org/10.3150/16-bej898
  6. Dong, H., Otsu, T. and Taylor, L. (2022). Estimation of varying coefficient models with measurement error. Journal of Econometrics 230(2), 388–415. DOI: 10.1016/j.jeconom.2020.12.013. https://doi.org/10.1016/j.jeconom.2020.12.013
  7. Stoklosa, J., Hwang, W.-H. and Warton, D. I. (2023). A general algorithm for error-in-variables regression modelling using Monte Carlo expectation maximization. PLOS ONE 18(4), e0283798. DOI: 10.1371/journal.pone.0283798. https://doi.org/10.1371/journal.pone.0283798
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