The deconvolution estimator of θ=P(X<Y)\theta=\mathbb{P}(X<Y) constructed by Phuong and Thuy (2022) for the model in which both variables are contaminated by additive measurement errors with unknown distributions — estimated from supplementary pure-error samples under a symmetry assumption — attains the minimax-optimal logarithmic rate only when its cut-off regularization parameter is chosen using unknown supersmooth decay constants of the error characteristic functions, and the paper explicitly defers a data-driven (adaptive) choice to future work. The open problem is to make this estimator fully automatic: construct a data-driven choice of the regularization parameter that provably attains the oracle rates without knowledge of the error-decay or density-smoothness constants.

Background

In reliability engineering one must estimate the probability θ=P(X<Y)\theta=\mathbb{P}(X<Y) that a component's strength YY exceeds the applied stress XX (the stress–strength model); in medical diagnostics the same quantity equals the area under the receiver operating characteristic (ROC) curve when XX and YY are biomarker values of healthy and diseased subjects. Suppose both variables are mismeasured: instead of XX and YY one observes contaminated versions X′=X+ζX'=X+\zeta and Y′=Y+ηY'=Y+\eta, where ζ\zeta and η\eta are random measurement errors independent of XX and YY. The characteristic function (the Fourier transform of a distribution) of an observation factorises as ϕX′=ϕXϕζ\phi_{X'}=\phi_X\phi_\zeta, so recovering any functional of the law of XX from data on X′X' requires dividing by the error characteristic function — the classical deconvolution problem, which is ill-posed whenever the error characteristic function is small in magnitude.

A Fourier inversion identity of Gil–Pelaez type (a classical formula recovering a distribution function from its characteristic function) gives θ=12−1π∫0∞1t Im{ϕX(t)ϕY(−t)} dt\theta=\frac12-\frac1\pi\int_0^\infty\frac1t\,\mathrm{Im}\{\phi_X(t)\phi_Y(-t)\}\,\mathrm{d}t, so when the error characteristic functions are known one can plug in deconvolved empirical characteristic functions and truncate the integral at a frequency TT; this is the estimator studied by Dattner (2013), who obtained logarithmic convergence rates in mean squared error when the errors are supersmooth, meaning their characteristic functions decay roughly like e−d∣t∣γe^{-d|t|^{\gamma}} (for example Gaussian errors, with γ=2\gamma=2). For supersmooth errors the deconvolution problem is severely ill-posed and no estimator can converge faster than a power of 1/log⁡n1/\log n, a barrier established for this very problem by the minimax lower bound of Phuong and Thuy (2022) discussed below.

In most applications the error distributions are unknown. Following the device of Diggle and Hall (1993), Phuong and Thuy (2022) assume that independent pure-error samples ζ1′,…,ζN′\zeta'_1,\dots,\zeta'_N and η1′,…,ηM′\eta'_1,\dots,\eta'_M drawn directly from the error distributions are available (for instance blank calibration measurements), and that the error densities are symmetric about zero, so that their characteristic functions are real and can be unbiasedly estimated by empirical cosine transforms ϕ^ζ(t)=N−1∑pcos⁡(tζp′)\hat\phi_\zeta(t)=N^{-1}\sum_p\cos(t\zeta'_p) and ϕ^η(t)=M−1∑qcos⁡(tηq′)\hat\phi_\eta(t)=M^{-1}\sum_q\cos(t\eta'_q). Dividing by these estimates is unstable where they are close to zero, so the estimator is regularized by indicator truncation together with an overall cut-off TT: θ^T=12−1π∫0T1t Im{ϕ^X′(t)ϕ^Y′(−t)ϕ^ζ(t)ϕ^η(−t)}I(∣ϕ^ζ(t)∣≥N−1/2)I(∣ϕ^η(t)∣≥M−1/2) dt,\hat{\theta}_T=\frac12-\frac1\pi\int_0^T\frac1t\,\mathrm{Im}\left\{\frac{\hat\phi_{X'}(t)\hat\phi_{Y'}(-t)}{\hat\phi_\zeta(t)\hat\phi_\eta(-t)}\right\}\mathbb{I}\big(|\hat\phi_\zeta(t)|\ge N^{-1/2}\big)\mathbb{I}\big(|\hat\phi_\eta(t)|\ge M^{-1/2}\big)\,\mathrm{d}t, where ϕ^X′(t)=n−1∑j=1neitXj′\hat\phi_{X'}(t)=n^{-1}\sum_{j=1}^n e^{itX'_j} and ϕ^Y′(t)=m−1∑k=1meitYk′\hat\phi_{Y'}(t)=m^{-1}\sum_{k=1}^m e^{itY'_k} are the empirical characteristic functions of the contaminated samples. Assuming the error characteristic functions are bounded below by c e−d∣t∣γc\,e^{-d|t|^{\gamma}} (a supersmooth lower decay bound), Phuong and Thuy prove a worst-case mean squared error of logarithmic order, matching a minimax lower bound of Le Cam type (a two-point hypothesis comparison), so their estimator is rate-optimal — provided the auxiliary sample sizes satisfy N≥nN\ge n and M≥mM\ge m, and, crucially, the cut-off is chosen as T=(ln⁡ℓ−[ln⁡(ln⁡ℓ)]22max⁡{dζ,dη})1/max⁡{γζ,γη},ℓ=min⁡{n,m,N,M}.T=\left(\frac{\ln\ell-[\ln(\ln\ell)]^2}{2\max\{d_\zeta,d_\eta\}}\right)^{1/\max\{\gamma_\zeta,\gamma_\eta\}},\qquad \ell=\min\{n,m,N,M\}. This value depends on the decay constants dζ,dη,γζ,γηd_\zeta,d_\eta,\gamma_\zeta,\gamma_\eta of the unknown error distributions, and the rate itself involves the unknown smoothness indices of the densities of XX and YY; the optimal procedure is therefore not available in practice, and the paper defers a data-driven (adaptive) choice of the regularization parameter to future research.

Adaptive deconvolution — choosing the tuning parameter from the data while retaining (near-)optimal rates — is a classical difficulty. For estimating a distribution function with a known error distribution, Dattner and Reiser (2013) gave a data-driven bandwidth; for estimating quantiles (and, as a side result, distribution functions) with an unknown error distribution, Dattner, Reiß and Trabs (2016) constructed an adaptive, minimax-optimal estimator via a data-driven bandwidth; for density deconvolution with an unknown error distribution estimated from an auxiliary noise sample, Kappus and Mabon (2014) constructed an adaptive estimator; in repeated-measurement schemes without any symmetry assumption on the errors, Comte and Kappus (2015) and Capitao-Miniconi, Gassiat and Lehéricy (2025) obtained adaptive procedures, the latter with a full model-selection scheme. None of these covers the functional θ=P(X<Y)\theta=\mathbb{P}(X<Y) in the pure-error-sample scheme, and the subsequent extensions of the P(X<Y) problem by the same research programme (ridge-type regularization — dividing by the estimated error characteristic function shifted by a small constant — allowing error characteristic functions that vanish, repeated observations replacing the pure-error samples, dependent stationary processes) all retain prescribed, non-data-driven tuning parameters.

Problem Statement

Consider the measurement-error model X′=X+ζX'=X+\zeta and Y′=Y+ηY'=Y+\eta, where X,Y,ζ,ηX,Y,\zeta,\eta are mutually independent real random variables, the error densities fζ,fηf_\zeta,f_\eta are unknown and symmetric about zero, and the error characteristic functions satisfy the supersmooth lower bounds ∣ϕζ(t)∣≥cζe−dζ∣t∣γζ|\phi_\zeta(t)|\ge c_\zeta e^{-d_\zeta|t|^{\gamma_\zeta}} and ∣ϕη(t)∣≥cηe−dη∣t∣γη|\phi_\eta(t)|\ge c_\eta e^{-d_\eta|t|^{\gamma_\eta}} for all real tt and some constants cζ,cη,dζ,dη,γζ,γη>0c_\zeta,c_\eta,d_\zeta,d_\eta,\gamma_\zeta,\gamma_\eta>0; here ϕU\phi_U denotes the characteristic function (the Fourier transform of the distribution) of a random variable UU. One observes the contaminated samples X1′,…,Xn′X'_1,\dots,X'_n and Y1′,…,Ym′Y'_1,\dots,Y'_m together with independent pure-error samples ζ1′,…,ζN′\zeta'_1,\dots,\zeta'_N and η1′,…,ηM′\eta'_1,\dots,\eta'_M drawn directly from the error distributions themselves (as with blank calibration measurements). Write ϕ^X′(t)=n−1∑j=1neitXj′\hat\phi_{X'}(t)=n^{-1}\sum_{j=1}^n e^{itX'_j} and ϕ^Y′(t)=m−1∑k=1meitYk′\hat\phi_{Y'}(t)=m^{-1}\sum_{k=1}^m e^{itY'_k} for the empirical characteristic functions of the contaminated samples, and ϕ^ζ(t)=N−1∑p=1Ncos⁡(tζp′)\hat\phi_\zeta(t)=N^{-1}\sum_{p=1}^N\cos(t\zeta'_p), ϕ^η(t)=M−1∑q=1Mcos⁡(tηq′)\hat\phi_\eta(t)=M^{-1}\sum_{q=1}^M\cos(t\eta'_q) for the (real-valued, by symmetry) cosine-transform estimators of the error characteristic functions. In this scheme the deconvolution estimator of θ=P(X<Y)\theta=\mathbb{P}(X<Y) proposed by Phuong and Thuy (2022) is θ^T=12−1π∫0T1t Im{ϕ^X′(t)ϕ^Y′(−t)ϕ^ζ(t)ϕ^η(−t)}I(∣ϕ^ζ(t)∣≥N−1/2)I(∣ϕ^η(t)∣≥M−1/2) dt,\hat{\theta}_T=\frac12-\frac1\pi\int_0^T\frac1t\,\mathrm{Im}\left\{\frac{\hat\phi_{X'}(t)\hat\phi_{Y'}(-t)}{\hat\phi_\zeta(t)\hat\phi_\eta(-t)}\right\}\mathbb{I}\big(|\hat\phi_\zeta(t)|\ge N^{-1/2}\big)\mathbb{I}\big(|\hat\phi_\eta(t)|\ge M^{-1/2}\big)\,\mathrm{d}t, where T>0T>0 is a cut-off (regularization) parameter and I(⋅)\mathbb{I}(\cdot) denotes the indicator function. Under these assumptions, when the auxiliary sample sizes satisfy N≥nN\ge n and M≥mM\ge m, this estimator attains the minimax-optimal logarithmic mean-squared-error rate provided the cut-off is chosen as T=(ln⁡ℓ−[ln⁡(ln⁡ℓ)]22max⁡{dζ,dη})1/max⁡{γζ,γη},ℓ=min⁡{n,m,N,M},T=\left(\frac{\ln\ell-[\ln(\ln\ell)]^2}{2\max\{d_\zeta,d_\eta\}}\right)^{1/\max\{\gamma_\zeta,\gamma_\eta\}},\qquad \ell=\min\{n,m,N,M\}, a choice that depends on the unknown decay constants dζ,dη,γζ,γηd_\zeta,d_\eta,\gamma_\zeta,\gamma_\eta, while the rate itself involves the unknown smoothness indices of the densities of XX and YY; as it stands, the optimal procedure is therefore not implementable without knowledge of these constants.

The open problem is to construct a fully data-driven rule T∗=T∗(X1′,…,Xn′, Y1′,…,Ym′, ζ1′,…,ζN′, η1′,…,ηM′)T^*=T^*(X'_1,\dots,X'_n,\,Y'_1,\dots,Y'_m,\,\zeta'_1,\dots,\zeta'_N,\,\eta'_1,\dots,\eta'_M) — together with data-driven thresholds, if the indicator truncation is retained or replaced by another regularization — such that θ^T∗\hat\theta_{T^*} is consistent and achieves the oracle logarithmic rate up to constants or slowly varying factors, uniformly over precisely stated classes of error distributions and of densities of XX and YY within which the decay constants dζ,dη,γζ,γηd_\zeta,d_\eta,\gamma_\zeta,\gamma_\eta and the smoothness indices genuinely vary (so that no fixed, data-independent cut-off can attain the oracle rate across the class), and without knowledge of cζ,cη,dζ,dη,γζ,γηc_\zeta,c_\eta,d_\zeta,d_\eta,\gamma_\zeta,\gamma_\eta or of the smoothness indices.

The adaptive choice of the regularization parameter for estimating θ\theta under unknown measurement error is the problem the source paper explicitly left open. Extending the scheme to settings without any supplementary error information has already been partly addressed through repeated measurements (Phuong and Trang, 2025) and is not required.

References

  1. Cao Xuan Phuong and Le Thi Hong Thuy (2022), 'Deconvolution of P(X<Y) with unknown error distributions', Communications in Statistics – Theory and Methods 51(17): 5889–5912. DOI: 10.1080/03610926.2020.1849722, https://doi.org/10.1080/03610926.2020.1849722 (source paper)
  2. I. Dattner (2013), 'Deconvolution of P(X<Y) with supersmooth error distributions', Statistics & Probability Letters 83(8): 1880–1887. DOI: 10.1016/j.spl.2013.04.024, https://doi.org/10.1016/j.spl.2013.04.024
  3. Peter J. Diggle and Peter Hall (1993), 'A Fourier Approach to Nonparametric Deconvolution of a Density Estimate', Journal of the Royal Statistical Society, Series B 55(2): 523–531. DOI: 10.1111/j.2517-6161.1993.tb01920.x, https://doi.org/10.1111/j.2517-6161.1993.tb01920.x
  4. I. Dattner and B. Reiser (2013), 'Estimation of distribution functions in measurement error models', Journal of Statistical Planning and Inference 143(3): 479–493. DOI: 10.1016/j.jspi.2012.09.004, https://doi.org/10.1016/j.jspi.2012.09.004
  5. Cao Xuan Phuong (2020), 'Deconvolution of Cumulative Distribution Function with Unknown Noise Distribution', Acta Applicandae Mathematicae 170(1): 483–514. DOI: 10.1007/s10440-020-00343-9, https://doi.org/10.1007/s10440-020-00343-9
  6. Cao Xuan Phuong and Le Thi Hong Thuy (2024), 'Nonparametric estimation of P(X<Y) from noisy data samples with non-standard error distributions', Metrika 87(8): 973–1006. DOI: 10.1007/s00184-023-00941-1, https://doi.org/10.1007/s00184-023-00941-1
  7. Bui Thuy Trang, Le Thi Hong Thuy and Cao Xuan Phuong (2024), 'Non parametric deconvolution of cumulative distribution function from repeated observations with unknown noise distribution', Communications in Statistics – Theory and Methods 53(24): 8787–8818. DOI: 10.1080/03610926.2023.2298896, https://doi.org/10.1080/03610926.2023.2298896
  8. Cao Xuan Phuong and Bui Thuy Trang (2025), 'Deconvolution of P(X<Y) from repeated data samples with unknown noise distributions', Communications in Statistics – Theory and Methods 54(22): 7011–7051. DOI: 10.1080/03610926.2025.2465646, https://doi.org/10.1080/03610926.2025.2465646
  9. Dang Duc Trong and Thai Phuc Hung (2024), 'Deconvolution of P(X_t < Y_t) for stationary processes with supersmooth error distributions', Statistics 58(6): 1463–1487. DOI: 10.1080/02331888.2024.2407913, https://doi.org/10.1080/02331888.2024.2407913
  10. Thai Phuc Hung and Nguyen Cao Phong (2025), 'Estimating the Probability P(X_t<Y_t) for m-dependent Processes from Noisy Data with Mixtures of Normal Errors', Journal of Science Technology and Engineering Mien Tay Construction University. DOI: 10.66195/mtu.2025.12.058, https://doi.org/10.66195/mtu.2025.12.058
  11. Johanna Kappus and Gwennaëlle Mabon (2014), 'Adaptive density estimation in deconvolution problems with unknown error distribution', Electronic Journal of Statistics 8(2). DOI: 10.1214/14-EJS976, https://doi.org/10.1214/14-EJS976
  12. Fabienne Comte and Johanna Kappus (2015), 'Density deconvolution from repeated measurements without symmetry assumption on the errors', Journal of Multivariate Analysis 140: 31–46. DOI: 10.1016/j.jmva.2015.04.004, https://doi.org/10.1016/j.jmva.2015.04.004
  13. Jérémie Capitao-Miniconi, Elisabeth Gassiat and Luc Lehéricy (2025), 'Deconvolution of repeated measurements corrupted by unknown noise', Electronic Journal of Statistics 19(2). DOI: 10.1214/25-EJS2415, https://doi.org/10.1214/25-EJS2415 (preprint: arXiv:2409.02014, https://arxiv.org/abs/2409.02014)
  14. Itai Dattner, Markus Reiß and Mathias Trabs (2016), 'Adaptive quantile estimation in deconvolution with unknown error distribution', Bernoulli 22(1). DOI: 10.3150/14-bej626, https://doi.org/10.3150/14-bej626