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REVIEW 4 major objections 6 minor 36 references

Calibration Prediction Interval for Non-parametric Regression and Neural Networks

T0 review · 4 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A prediction interval built by calibrating conditional CDF estimates at grid points can restore nominal coverage in nonparametric regression.

desk verdict The calibration PI idea is practical and worth engaging with, but the main DNN coverage theorem is not proven as written and the finite-sample guarantee rests on an admitted oracle assumption. read the letter →

arxiv 2509.02735 v1 pith:LEJBHIAA submitted 2025-09-02 stat.ME stat.ML

Kejin Wu , Dimitris N. Politis This is my paper · ORCID
classification stat.MEstat.ML MSC 62G0862G1562G2068T07
keywords predictionintervalcalibrationdeepneuralnetworkskernelestimationuncertaintyquantificationcoverageguaranteeconditionalnonparametricregression
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to establish that prediction intervals in nonparametric regression can be made conditionally valid at the nominal $1-\alpha$ level by a calibration step that explicitly accounts for estimation variability. Rather than assuming normality or linearity, it estimates the conditional distribution function at many fixed thresholds using deep neural networks or kernel estimators, then chooses interval endpoints from those thresholds so that the estimated probability mass inside the interval is at least $1-\alpha$. The paper proves asymptotic coverage for the DNN and kernel versions, a high-probability large-sample coverage guarantee for the kernel version, and a conditional coverage guarantee for the DNN version in finite samples under strong oracle and grid-spacing assumptions. A sympathetic reader would care because plug-in intervals, including the normal-based one, systematically undercover when the estimated mean and variance carry estimation error, and the proposed cPI is a simple, parallelizable correction.

What carries the argument

The central machinery is a grid of threshold indicators $Z_j = \mathbf 1\{Y \le q_j\}$ for $j = 1,\dots,g$, each regressed on $X$ by its own DNN, or, for the kernel variant, obtained by integrating a conditional density estimate. This turns conditional CDF estimation into $g$ binary regression problems whose estimators $\hat H_j(X_f)$ are continuous functions of $x_f$ by Lemma 2.2. The paper's methodological contribution is the subsequent calibration: three monotonicity corrections, $\hat F_{LtoR}$, $\hat F_{RtoL}$, and their average $\hat F_{Avg}$, render the estimated CDF nondecreasing, and an index-selection step picks endpoints with estimated mass at least $1-\alpha$. The adjusted version $PI_{aaa}$ shifts both endpoints outward one grid cell to compensate for worst-case estimation error, which is the mechanism behind the finite-sample coverage claim. The proof engine is a non-asymptotic DNN risk bound, stated as Theorem A.1, that makes $\hat H_j$ consistent, together with a DNN approximation bound, Lemma A.1, expressed through the modulus of continuity.

What would settle it

Train the DNNs of Algorithm 1 on simulated data from Model-1 with $n = 2000$, construct $PI_{aaa}$ at $g = 200$ using standard gradient descent, and compare the empirical conditional coverage over 500 replications with the nominal $1-\alpha$. If coverage systematically falls below nominal, the finite-sample guarantee has no quantitative support because Assumption B4, that the trained $\hat H_j$ matches the oracle $H_j^*$ in $L_\infty$ with error exactly zero, is violated. Directly computing $\|\hat H_j - H_j^*\|_{L_\infty}$ on these trained networks would settle whether the assumed oracle condition holds.

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Extended reading notes

Core claim

The core claim is that the so-called calibration PI (cPI) computed by Algorithms 3 to 7 satisfies $P(Y \in \hat I \mid X_f = x_f) \ge 1-\alpha$ as $n \to \infty$ (Theorem 5.1), and that the finite-sample version $PI_{aaa}$ satisfies the same inequality under oracle DNN training and a sufficiently large grid spacing (Theorem 5.2). The reason a naive interval fails is that it replaces the true conditional mean and variance by plug-in estimates and ignores how much those estimates vary; the cPI instead estimates the conditional CDF $F_{Y|X_f}(q_j)$ at grid points $q_j$, corrects the estimates for monotonicity, and then selects $l$ and $r$ so that $\hat F(q_r) - \hat F(q_l) \ge 1-\alpha$. By construction the interval is widened just enough to absorb the estimation error, and the adjusted version widens it one further grid step. The paper also claims the kernel-based cPI attains coverage with high probability for large samples (Theorem 5.3), a stronger property than plain asymptotic validity.

Load-bearing premise

The finite-sample coverage guarantee in Theorem 5.2 assumes the trained DNN estimates are as good as the best DNN in the class in $L_\infty$ norm, with the estimation error sequence $\delta_n$ treated as exactly zero, an assumption the authors say is possible but may not be practically accomplishable.

Editorial extensions

If this is right

  • Plug-in intervals that ignore estimation variability undercover even when the error is normal; the paper's simulations show the normal-based $PI_b$ stays below the nominal level at $n = 10{,}000$.
  • Any consistent estimator of the conditional CDF can be plugged into the calibration procedure; DNN and kernel estimators are the two instantiations the paper develops.
  • Because the $g+1$ DNNs in Algorithm 1 can be trained in parallel, the calibration step need not add significant wall-clock time over a single point-prediction DNN.
  • The kernel-based cPI gives a high-probability large-sample coverage guarantee, a middle ground between asymptotic validity and the finite-sample guarantee.
  • The finite-sample coverage claim holds only under Assumptions B4 and B5; without oracle training or a wide grid, the guarantee has no quantitative support.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Inference: the same grid-of-thresholds calibration could be applied to any base probabilistic regression model, such as gradient boosting, Gaussian processes, or random forests, so the method's scope is wider than DNN and kernel estimators.
  • Inference: the one-step outward adjustment in $PI_{aaa}$ is a generic conservative device; if a base estimator has a known uniform error bound $\delta_n$, moving each endpoint one grid step should restore coverage whenever the grid spacing exceeds $2\delta_n$, a condition analogous to B5 that could be tested empirically.
  • Inference: a data-driven choice of $g$, for instance the smallest grid whose validation coverage is at least $1-\alpha$, would remove the user-set grid hyperparameter and is a natural extension the paper does not pursue.
  • Inference: if Assumption B4 fails, as the authors concede is likely, a bootstrap estimate of the $L_\infty$ error of $\hat H_j$ could substitute for the oracle condition and yield an adjusted interval whose length reflects actual estimation variability.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper proposes a calibration prediction interval (cPI) for conditional prediction in nonparametric regression. The method estimates the conditional CDF FY|X at fixed grid points using either deep ReLU networks (one network per grid point) or kernel density estimators, applies one of three monotonicity corrections, and then selects interval endpoints so that the estimated CDF mass is at least 1−α. Several variants are proposed (minimal length, symmetric around the estimated mean, asymmetric), including an adjusted version that moves endpoints outward by one grid step. The authors claim asymptotic conditional coverage for both DNN-based and kernel-based cPIs, a high-probability large-sample coverage guarantee for the kernel version, and a finite-sample coverage guarantee for DNN cPIs under oracle-type assumptions. The practical claims are supported by simulations on six nonlinear models and by two wine-quality datasets.

Significance. If the theoretical guarantees were valid, the paper would make a practically useful contribution: the calibration idea is simple, avoids normality and linearity assumptions, compensates for estimation variability, and is parallelizable. The numerical study is substantial (six models, normal/t/skew-normal errors, two sample sizes, several DNN widths), and the comparison on the wine datasets is informative. The monotonicity-correction methods are clearly described and appear to work well in the experiments. However, the main DNN asymptotic theorem is not established by the given proof, the kernel lemma contains a bandwidth condition that is inconsistent with consistency, and the finite-sample guarantee rests on an acknowledged oracle assumption for which no supporting result is provided. The theoretical claims in the abstract are therefore currently stronger than what the manuscript establishes.

major comments (4)
  1. [Section 5.1.1, Eq. (11), proof of Theorem 5.1] The proof asserts that ∫(bH_j−H_j)^2 f dx → 0 with probability tending to 1, together with B2 (f(x)>0), implies bH_j(xf) → H_j(xf) for every xf. This implication is false: L2 convergence does not imply pointwise convergence, and positivity of a density on a compact set does not imply a positive lower bound. A sequence of narrow spike functions centered at xf can have L2 distance tending to zero while remaining equal to 1 at xf. Theorem A.1, the only DNN error bound cited in this paper, supplies L2 convergence only; no pointwise or uniform consistency result for bH_j is given. Since Algorithms 3–7 evaluate bH_j at the fixed future point xf and select endpoints from those values, the theorem requires convergence at that xf, and the claim 'for any xf in its domain X' requires simultaneous control over all xf. The headline asymptotic coverage result for DNN cPIs is therefore unsupported by the current proof.
  2. [Lemma 5.1 and Theorem 5.3, Eq. (8)] The condition ln(n)/(nh^{d+1}) → ∞ stated in Lemma 5.1 is inconsistent with the displayed error bound. The variance term (ln(n)/(nh^{d+1}))^{1/2} converges to zero only if ln(n)/(nh^{d+1}) → 0; under the stated condition it diverges, so the claimed uniform consistency of fhat(y|x) does not follow. The same sign error appears in Theorem 5.3. Because the kernel cPI's asymptotic and large-sample coverage results are derived directly from Lemma 5.1, this condition must be corrected before those results can be accepted.
  3. [Section 5.1.2, B4, Remark 5.3, Theorem 5.2] The finite-sample coverage theorem is conditional on B4, which assumes ||bH_j − H*_j||_{L∞(X)} ≤ δ_n with δ_n treated as exactly 0 for all n. Remark 5.3 states that no existing result supports B4 and that achieving it 'may not be practically accomplishable'. The proof of Theorem 5.2 is a non-quantitative description of a diagram rather than a verification that the monotonicity correction and the one-step outward adjustment guarantee eFAvg(q_l~) ≤ α/2 and eFAvg(q_r~) ≥ 1−α/2 under B5. As stated, the theorem does not provide a quantitative finite-sample coverage claim for any realistic estimator, and the abstract's statement that the cPI based on DNN works 'even with finite samples' is not supported.
  4. [Section 5.1.1, Algorithms 3–7] The proof of Theorem 5.1 does not address the discrete index-selection step. Even if bH_j(xf) converged at each grid point, coverage requires that, for the particular corrected estimators eFLtoR, eFRtoL, and eFAvg, the selected indices l and r satisfy the corresponding inequalities with enough slack to imply true coverage. The monotonicity corrections C1–C3 can move estimated values away from H_j at the endpoint grid points, and the arg max/arg min operations in Algorithms 6 and 7 have no defined behavior when no grid point satisfies the required inequality. A rigorous proof would need to control these quantities simultaneously, for all grid points and for all xf in the domain; the current one-paragraph proof does not do so.
minor comments (6)
  1. [Proof of Lemma 2.1] The line '|H0(u) − H0(v)| ≤ ε·M·∞' appears to be a typo; it should be ε·M.
  2. [Section 5.2.1, paragraph after Lemma 5.1] The sentence 'all cPIs with DNN estimators guarantee at least 1−α conditional coverage asymptotically' should refer to kernel estimators, since the surrounding argument concerns the kernel-based estimators bHk_j.
  3. [Algorithm 4, Step 3] 'Fine the index c' should read 'Find the index c'.
  4. [Algorithms 6 and 7] Please define the convention when no grid point satisfies eFAvg(qk) ≤ α/2 or eFAvg(qk) ≥ 1−α/2; otherwise the arg max and arg min are taken over an empty set.
  5. [Theorem 5.1 statement] The statement should specify whether the number of grid points g is fixed or grows with n, and it should specify the probability space for P(Y ∈ bI | Xf = xf), since bI depends on the training sample.
  6. [Table 1] The symbol '√–' used for the DNN-based large-sample coverage entry is not defined in the table note; please explain it in the caption.

Circularity Check

1 steps flagged · score 4.0 of 10

The cPI construction itself is not circular, but the central DNN asymptotic coverage theorem rests on a load-bearing L2 consistency result imported from the authors' own prior work, with an additional unsupported L2-to-pointwise step.

  1. self citation load bearing [Theorem A.1 (Appendix A) and its use in the proof of Theorem 5.1 (Section 5.1.1)]
    "Theorem A.1 (One variant of Theorem 3.1 in Wu and Politis (2024)) Under assumptions A1 to A3 and B1, let FDNN be a class of standard fully connected feedforward DNN functions ... we have that ||bH−H0||2 L2(X) → 0 with probability at least 1−exp(−γ). ... By Theorem A.1, we have E((bHj(X)−Hj(X))2) → 0 ... for j = 0,...,g."

    The asymptotic coverage theorem for the DNN-based cPIs is not proved from first principles in this paper. Its proof invokes Theorem A.1, which is explicitly stated to be 'One variant of Theorem 3.1 in Wu and Politis (2024)' — a result by the same two authors. No external, machine-checked, or independently reproduced derivation is supplied for the uniform-continuity-only L2 consistency bound on which the whole DNN coverage argument depends. The proof then needs pointwise convergence of bHj at every future point xf, but only cites L2 convergence; the load-bearing consistency input is therefore both self-cited and insufficient for the pointwise inference required by Algorithms 3–7.

full rationale

The calibration-PI idea is not circular in the usual sense: the algorithms choose endpoints from corrected estimated conditional CDF values so that the estimated mass is at least 1−α, and the asymptotic coverage claim then reduces to consistency of those CDF estimates. That is a standard plug-in argument, not a self-definitional equivalence. The empirical simulations and real-data comparisons are also self-contained and do not depend on the theoretical claims being true. However, the central DNN asymptotic theorem has a load-bearing self-citation: Theorem 5.1 relies on Theorem A.1, which is a variant of a theorem in the authors' own prior arXiv preprint Wu and Politis (2024), and no independent verification of that uniformity-free L2 bound is given. Moreover, the proof of Theorem 5.1 contains a serious non-circular correctness gap: from Eq. (11) it claims that L2 convergence plus positivity of f(x) implies pointwise convergence bHj(xf) → Hj(xf) for all xf, which is false; L2 convergence does not imply pointwise convergence at a fixed point, and B2 only asserts positivity, not a uniform lower bound. The finite-sample result Theorem 5.2 is explicitly conditional on B4, where the paper states the oracle-error sequence δn is 'treated to be 0 for all n. This is possible but may not be practically accomplishable,' so the finite-sample guarantee is an assumption-dependence limitation rather than an independent derivation. The kernel Lemma 5.1 also states the condition ln(n)/(nh^{d+1}) → ∞, which is the wrong direction for the variance term to vanish; this is another correctness issue, not a circular one. Weighing these, the central claim is not merely a renaming or a fit of its inputs, but it does lean on an unverified same-author consistency theorem, justifying a score of 4 rather than 0.

Assumptions & free parameters 4 free parameters · 10 assumptions · 0 invented entities

The central claims rest on standard regularity conditions (A1-A3, B1-B3) plus two strongly customized assumptions: B4 (oracle DNN training, explicitly acknowledged as practically unattainable) and B5 (a grid-spacing condition that basically assumes the discretization error is large enough to absorb estimation error). The DNN consistency result is imported from the authors' own earlier theorem. No new physical entities are introduced.

free parameters (4)
  • number of grid points g = g = 200 in simulations; g = 200, 100, 50, 25, 12, 5 in real data
    The approximation of the conditional CDF and the coverage-length trade-off depend on g. The finite-sample guarantee B5 is a condition on the grid spacing D = (range)/g, so the choice of g directly affects the theoretical coverage claim.
  • DNN width W and depth L = W = 10 to 50 with two hidden layers in simulations; [50] in real data comparison
    The DNN approximation error, and hence the asymptotic coverage proof, depends on the network architecture. In Theorem 5.1, W and L are specified as functions of n, but in practice they are fixed by the user.
  • kernel bandwidth h = chosen by maximum likelihood cross-validation in real data
    The kernel estimator's error bound in Lemma 5.1 depends on h, and the coverage guarantees require h to tend to zero at an appropriate rate. In practice the bandwidth is selected from data.
  • clipping parameter m = m = 20 in simulations
    The DNN parameters are clipped to [-m, m], which is a training detail that can affect estimation accuracy, though it is not central to the theoretical claims.
assumptions (10)
  • domain assumption A1: G(X, eps) is uniformly continuous and Y in L2(P)
    Used in Lemma 2.1 to establish continuity of the conditional mean and in the DNN approximation results.
  • domain assumption A2: X and eps are independent, error density f_eps is bounded
    Needed to express conditional expectations as integrals over the error distribution and to prove continuity.
  • domain assumption A3: F_{Y|X}(y) is continuous in X for all y
    Used in Lemma 2.2 to establish continuity of the conditional CDF as a function of X.
  • domain assumption B1: domains of X and Y are compact
    Used in Theorem 5.1 and Lemma 5.1 to apply uniform approximation and covering arguments.
  • domain assumption B2: marginal density of X is positive and bounded, conditional density f(y|x) is bounded
    Used to convert L2 consistency of DNN estimators into pointwise consistency at all xf.
  • domain assumption B3: f(x) and f(x,y) are twice differentiable with bounded Hessians
    Required for the kernel density error bound in Lemma 5.1, specifically the Q2 term.
  • ad hoc to paper B4: ||bH_j - H*_j||_{L_infty(X)} <= delta_n with delta_n treated as 0
    This is the oracle DNN estimation assumption required for the finite-sample coverage theorem. Remark 5.3 admits that no existing literature supports the asymptotic result and that treating delta_n as 0 'may not be practically accomplishable'.
  • ad hoc to paper B5/B'5: the grid distance D is large enough that adjacent true CDF values differ by more than the estimation error or modulus-of-continuity bound
    This condition directly ensures that the one-step adjustment in PIaaa or PIaaak corrects undercoverage. It is an unverifiable condition on the unknown CDF and the grid, and it essentially encodes the desired conclusion.
  • domain assumption Theorem A.1: the DNN least-squares estimator converges in L2 to the regression function (variant of Theorem 3.1 in Wu and Politis, 2024)
    The asymptotic coverage proof of Theorem 5.1 relies on this self-cited external result for L2 consistency of DNN estimators. It is not proved in this paper and not machine-checked.
  • standard math Lemma A.1: DNNs can approximate uniformly continuous functions (Theorem 4.3 of Shen et al., 2021)
    Used to control the approximation error of the target functions by DNNs in terms of width and depth.

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Pith. "Pith review of Calibration Prediction Interval for Non-parametric Regression and Neural Networks." pith.science (2026). https://pith.science/paper/LEJBHIAA

@misc{pith2026250902735,
  author       = {Pith},
  title        = {Pith review of: Calibration Prediction Interval for Non-parametric Regression and Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LEJBHIAA}},
  note         = {Machine review of arXiv:2509.02735}
}
abstract

Accurate conditional prediction in the regression setting plays an important role in many real-world problems. Typically, a point prediction often falls short since no attempt is made to quantify the prediction accuracy. Classically, under the normality and linearity assumptions, the Prediction Interval (PI) for the response variable can be determined routinely based on the $t$ distribution. Unfortunately, these two assumptions are rarely met in practice. To fully avoid these two conditions, we develop a so-called calibration PI (cPI) which leverages estimations by Deep Neural Networks (DNN) or kernel methods. Moreover, the cPI can be easily adjusted to capture the estimation variability within the prediction procedure, which is a crucial error source often ignored in practice. Under regular assumptions, we verify that our cPI has an asymptotically valid coverage rate. We also demonstrate that cPI based on the kernel method ensures a coverage rate with a high probability when the sample size is large. Besides, with several conditions, the cPI based on DNN works even with finite samples. A comprehensive simulation study supports the usefulness of cPI, and the convincing performance of cPI with a short sample is confirmed with two empirical datasets.

Figures

Figures reproduced from arXiv: 2509.02735 by the authors.

Figure 1
Figure 1. The diagram of one possible location of α/2 quantile of the conditional distribu￾tion and three adjacent grid points for some index i ∈ {1, . . . , g}. 33 [PITH_FULL_IMAGE:figures/full_fig_p033_1.png] view at source ↗
Figure 2
Figure 2. The effects of three monotonicity corrections methods. [PITH_FULL_IMAGE:figures/full_fig_p042_2.png] view at source ↗

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Works this paper leans on

36 extracted references · 20 canonical work pages

  1. [1]

    Nearly-tight vc-dimension and pseudodimension bounds for piecewise linear neural networks

    Peter L Bartlett, Nick Harvey, Christopher Liaw, and Abbas Mehrabian. Nearly-tight vc-dimension and pseudodimension bounds for piecewise linear neural networks. Journal of Machine Learning Research, 20 0 (63): 0 1--17, 2019

    2019

  2. [2]

    On deep learning as a remedy for the curse of dimensionality in nonparametric regression

    Benedikt Bauer and Michael Kohler. On deep learning as a remedy for the curse of dimensionality in nonparametric regression. The Annals of Statistics, 47 0 (4): 0 2261–2285, 2019

    work page 2019

  3. [3]

    Nonparametric estimates of regression quantiles and their local bahadur representation

    Probal Chaudhuri. Nonparametric estimates of regression quantiles and their local bahadur representation. The Annals of statistics, 19 0 (2): 0 760--777, 1991

    work page 1991

  4. [4]

    Modeling wine preferences by data mining from physicochemical properties

    Paulo Cortez, Ant \'o nio Cerdeira, Fernando Almeida, Telmo Matos, and Jos \'e Reis. Modeling wine preferences by data mining from physicochemical properties. Decision support systems, 47 0 (4): 0 547--553, 2009

    2009

  5. [5]

    Nonparametric estimation of the conditional distribution at regression boundary points

    Srinjoy Das and Dimitris N Politis. Nonparametric estimation of the conditional distribution at regression boundary points. The American Statistician, 74 0 (4): 0 233--242, 2020

    work page 2020

  6. [6]

    Deep neural networks for estimation and inference

    Max H Farrell, Tengyuan Liang, and Sanjog Misra. Deep neural networks for estimation and inference. Econometrica, 89 0 (1): 0 181--213, 2021

    2021

  7. [7]

    On calibration of modern neural networks

    Chuan Guo, Geoff Pleiss, Yu Sun, and Kilian Q Weinberger. On calibration of modern neural networks. In International conference on machine learning, pages 1321--1330. PMLR, 2017

    2017

  8. [8]

    Nonparametric estimation of smooth conditional distributions

    Bruce E Hansen. Nonparametric estimation of smooth conditional distributions. Unpublished paper: Department of Economics, University of Wisconsin, 10, 2004

    work page 2004

Show all 36 references
  1. [9]

    Deep nonparametric regression on approximate manifolds: Nonasymptotic error bounds with polynomial prefactors

    Yuling Jiao, Guohao Shen, Yuanyuan Lin, and Jian Huang. Deep nonparametric regression on approximate manifolds: Nonasymptotic error bounds with polynomial prefactors. The Annals of Statistics, 51 0 (2): 0 691--716, 2023

    2023

  2. [10]

    Lower upper bound estimation method for construction of neural network-based prediction intervals

    Abbas Khosravi, Saeid Nahavandi, Doug Creighton, and Amir F Atiya. Lower upper bound estimation method for construction of neural network-based prediction intervals. IEEE transactions on neural networks, 22 0 (3): 0 337--346, 2010

    2010

  3. [11]

    Quantile regression: 40 years on

    Roger Koenker. Quantile regression: 40 years on. Annual review of economics, 9 0 (1): 0 155--176, 2017

    2017

  4. [12]

    Distribution-free prediction bands for non-parametric regression

    Jing Lei and Larry Wasserman. Distribution-free prediction bands for non-parametric regression. Journal of the Royal Statistical Society Series B: Statistical Methodology, 76 0 (1): 0 71--96, 2014

    2014

  5. [13]

    Distribution-free predictive inference for regression

    Jing Lei, Max G’Sell, Alessandro Rinaldo, Ryan J Tibshirani, and Larry Wasserman. Distribution-free predictive inference for regression. Journal of the American Statistical Association, 113 0 (523): 0 1094--1111, 2018

    2018

  6. [14]

    Nonparametric econometrics: theory and practice

    Qi Li and Jeffrey Scott Racine. Nonparametric econometrics: theory and practice. Princeton University Press, 2007

    2007

  7. [15]

    Wasserstein generative learning of conditional distribution

    Shiao Liu, Xingyu Zhou, Yuling Jiao, and Jian Huang. Wasserstein generative learning of conditional distribution. arXiv preprint arXiv:2112.10039, 2021

    2021 arXiv

  8. [16]

    Prediction intervals for deep neural networks

    Tullio Mancini, Hector Calvo-Pardo, and Jose Olmo. Prediction intervals for deep neural networks. arXiv preprint arXiv:2010.04044, 2020

    2010 arXiv

  9. [17]

    Adaptive approximation and generalization of deep neural network with intrinsic dimensionality

    Ryumei Nakada and Masaaki Imaizumi. Adaptive approximation and generalization of deep neural network with intrinsic dimensionality. The Journal of Machine Learning Research, 21 0 (1): 0 7018--7055, 2020

    2020

  10. [18]

    Ospool, 2006

    OSG . Ospool, 2006. URL https://osg-htc.org/services/open_science_pool.html

    2006

  11. [19]

    Open science data federation, 2015

    OSG . Open science data federation, 2015. URL https://osdf.osg-htc.org/

    2015

  12. [20]

    Confidence interval construction and conditional variance estimation with dense relu networks

    Carlos Misael Madrid Padilla, Oscar Hernan Madrid Padilla, Yik Lun Kei, Zhi Zhang, and Yanzhen Chen. Confidence interval construction and conditional variance estimation with dense relu networks. arXiv preprint arXiv:2412.20355, 2024

    2024 arXiv

  13. [21]

    High-quality prediction intervals for deep learning: A distribution-free, ensembled approach

    Tim Pearce, Alexandra Brintrup, Mohamed Zaki, and Andy Neely. High-quality prediction intervals for deep learning: A distribution-free, ensembled approach. In International conference on machine learning, pages 4075--4084. PMLR, 2018

    2018

  14. [22]

    Model-Free Prediction and Regression: A Transformation-Based Approach to Inference

    Dimitris N Politis. Model-Free Prediction and Regression: A Transformation-Based Approach to Inference. Springer, 2015

    2015

  15. [23]

    Scalable subsampling: computation, aggregation and inference

    Dimitris N Politis. Scalable subsampling: computation, aggregation and inference. Biometrika, 111 0 (1): 0 347--354, 2024

    2024

  16. [24]

    The open science grid

    Ruth Pordes, Don Petravick, Bill Kramer, Doug Olson, Miron Livny, Alain Roy, Paul Avery, Kent Blackburn, Torre Wenaus, Frank W \"u rthwein, Ian Foster, Rob Gardner, Mike Wilde, Alan Blatecky, John McGee, and Rob Quick. The open science grid. In J. Phys. Conf. Ser., volume 78 o...

    2007 doi

  17. [25]

    Deep relu network approximation of functions on a manifold

    Johannes Schmidt-Hieber. Deep relu network approximation of functions on a manifold. arXiv preprint arXiv:1908.00695, 2019

    1908 arXiv

  18. [26]

    Nonparametric regression using deep neural networks with relu activation function

    Johannes Schmidt-Hieber. Nonparametric regression using deep neural networks with relu activation function. The Annals of Statistics, 48 0 (4): 0 1875–1897, 2020

    2020

  19. [27]

    The pilot way to grid resources using glideinwms

    Igor Sfiligoi, Daniel C Bradley, Burt Holzman, Parag Mhashilkar, Sanjay Padhi, and Frank Wurthwein. The pilot way to grid resources using glideinwms. In 2009 WRI World Congress on Computer Science and Information Engineering, volume 2 of 2, pages 428--432, 2009. doi:10.1109/CS...

    2009 doi

  20. [28]

    Deep network approximation characterized by number of neurons

    Zuowei Shen, Haizhao Yang, and Shijun Zhang. Deep network approximation characterized by number of neurons. arXiv preprint arXiv:1906.05497, 2021

    1906 arXiv

  21. [29]

    Piven: A deep neural network for prediction intervals with specific value prediction

    Eli Simhayev, Gilad Katz, and Lior Rokach. Piven: A deep neural network for prediction intervals with specific value prediction. arXiv preprint arXiv:2006.05139, 2020

    2006 arXiv

  22. [30]

    Single-model uncertainties for deep learning

    Natasa Tagasovska and David Lopez-Paz. Single-model uncertainties for deep learning. Advances in neural information processing systems, 32, 2019

    2019

  23. [31]

    Nonparametric quantile estimation

    Ichiro Takeuchi, Quoc Le, Timothy Sears, Alexander Smola, et al. Nonparametric quantile estimation. 2006

    2006

  24. [32]

    Methods to compute prediction intervals: A review and new results

    Qinglong Tian, Daniel J Nordman, and William Q Meeker. Methods to compute prediction intervals: A review and new results. Statistical Science, 37 0 (4): 0 580--597, 2022

    2022

  25. [33]

    Model-free bootstrap and conformal prediction in regression: Conditionality, conjecture testing, and pertinent prediction intervals

    Yiren Wang and Dimitris N Politis. Model-free bootstrap and conformal prediction in regression: Conditionality, conjecture testing, and pertinent prediction intervals. arXiv preprint arXiv:2109.12156, 2021

    2021 arXiv

  26. [34]

    Deep limit model-free prediction in regression

    Kejin Wu and Dimitris N Politis. Deep limit model-free prediction in regression. arXiv preprint arXiv:2408.09532, 2024

    2024 arXiv

  27. [35]

    Scalable subsampling inference for deep neural networks

    Kejin Wu and Dimitris N Politis. Scalable subsampling inference for deep neural networks. ACM/IMS Journal of Data Science, 2 0 (1): 0 1--29, 2025

    2025

  28. [36]

    A deep generative approach to conditional sampling

    Xingyu Zhou, Yuling Jiao, Jin Liu, and Jian Huang. A deep generative approach to conditional sampling. Journal of the American Statistical Association, 118 0 (543): 0 1837--1848, 2023

    2023

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.