Distributed inference for quantile regression processes

Motivation

• Quantile regression targets conditional-distribution features while accommodating heteroskedasticity and avoiding tail conditions that limit mean-focused distributed methods.The framework includes fixed- or growing-dimension linear models and series approximations such as tensor-product B-splines and additive models.Source: §1 · p. 2

Divide-and-conquer limitation

• Averaging quantile-regression estimates across distributed subsamples can invalidate inference when the number of subsamples is too large, with the threshold depending on model properties such as predictor dimension.Figure 1.1 illustrates a rapid coverage decline beyond model-dependent thresholds.Source: §1 · p. 3, Figure 1.1 · p. 3

Quantile projection

• The two-step quantile projection algorithm estimates coefficients at K grid levels, then reconstructs quantiles at arbitrary levels and estimates conditional distribution functions without revisiting the original data.This reduces the cost of estimating a dense quantile grid, while computation depends on the number of subsamples S and quantile levels K.Source: §1 · pp. 3–4

Oracle conditions

• The theory derives sufficient conditions on S and K for oracle-valid inference, with necessity established in some cases up to log N factors and restrictions governed by model complexity.The K lower bound has the form K ≫ N^(1/(2η))∥Z(x)∥^(-1/η), is independent of S, and weakens for larger ∥Z∥.Source: §1 · p. 4

Inference

• The paper proposes normal, t-based, and novel bootstrap procedures that exploit independent subsample estimates, alongside kernel-smoothing extensions for estimating the asymptotic variance.These approaches are designed to address the nonpivotal asymptotic distribution and are intended to require little additional computation.Source: §1 · pp. 4–5

Scope and application

• The algorithms support distributed or locally stored data processing and include conditional distribution estimation as an important application.The procedures are not restricted to large datasets or a particular reason for local data storage.Source: §1 · p. 5

Model framework

• The approximately linear framework covers fixed finite dimension, diverging dimension with local support such as B-splines, and diverging dimension without local support.It includes series approximation models as a special case.Source: §2 · p. 6

Distributed estimation

• The divide-and-conquer algorithm splits the data across S subsamples, estimates quantile-regression coefficients locally at fixed τ, and pools the local estimates.The master aggregates the coefficient estimates transmitted by each local machine.Source: §2 · p. 6

Quantile process construction

• The two-step process computes pooled coefficients on a quantile grid and projects each coefficient component onto a B-spline basis over [τL, τU].The projection uses the empirical grid norm and can be implemented through a closed-form spline-coefficient solution.Source: §2 · pp. 6–7

Conditional distribution inference

• The projected quantile process directly estimates the conditional distribution function by truncating values below τL and above τU, following Chernozhukov et al. (2010) and Volgushev (2013).The estimator is a smooth functional of τ 7→Z(x)⊤β̂(τ), whose process behavior is studied later.Source: §2 · p. 8

3.1. Fixed dimensional linear models.

• Projection adds a bias bounded by O(G^−η), while smoother conditional quantile curves permit smaller grid-size lower bounds and greater computational savings.The projection bias is controlled when τ↦Q(x0;τ) belongs to Λη_c(T), and the lower bound on K decreases as η increases.Source: §3.1 · pp. 9, 11
• The fixed-dimensional upper bounds are sharp up to log N factors, implying that attainable divide-and-conquer precision is limited by the per-machine sample size n=N/S.The paper develops a new Bahadur representation with a sharp remainder expectation bound; prior representations yield the more restrictive (S/N)^(3/4)(log N)^(3/4) bound.Source: §3.1 · pp. 9–10
• Fixed-dimensional models achieve oracle-equivalent inference under sufficient restrictions on sub-samples and quantile-grid size, with necessary conditions matching up to logarithmic factors.For the coefficient process, sufficient conditions include S=o(N^1/2(log N)^−1) and G≫N^(1/(2η)); necessary conditions are S=o(N^1/2) and G≫N^(1/(2η)).Source: §3.1 · p. 11
• Conditional distribution-function estimators inherit the same process convergence as the projected quantile estimator and its oracle counterpart.This follows because the estimated conditional distribution function is a Hadamard differentiable functional of the quantile curve.Source: §3.1 · pp. 11–12

3.2. Approximate linear models with local basis structure.

• In local-basis expansions, coefficient and process oracle rules hold under dimension-, approximation-, smoothness-, and grid-size conditions, with failures established when splitting is too extensive.For spline-type bases, ξm≍√m and local support is satisfied; process convergence also yields conditional distribution-function convergence.Source: §3.2 · pp. 13–15
• For growing local-basis models, sufficient and necessary splitting conditions depend explicitly on model dimension m, becoming more restrictive as complexity increases.More complex models permit less splitting, require larger subsamples, and impose greater computational burden; the framework uses local support and aggregated Bahadur representations.Source: §3.2 · pp. 13–14
• Nonparametric process inference can require a weaker quantile-grid condition, reducing computational burden because oracle convergence is typically slower.The relevant condition is G≫(N/∥Z(x0)∥^2)^(1/(2η)), which can be weaker than G≫N^(1/(2η)); for splines, ∥Z(x0)∥≍m^1/2.Source: §3.2 · p. 16

4. Practical aspects of inference.

• Subsample estimators enable inference with minimal central computation and no additional communication, avoiding the burden of classical bootstrap procedures in massive-data settings.The approach uses the fact that the distributed estimator is an average of approximately normal, independently realized subsample estimators when groups are homogeneous.Source: §4 · pp. 16–17

4.1. Inference utilizing results from subsamples.

• The proposed confidence intervals have asymptotic coverage 1 −α when S ≥2, while the refined t/F-based interval remains valid for fixed S and the normal-based interval requires S →∞.The result assumes the stated corollary conditions and cm(γN) = o(∥Z(x0)∥N−1/2).Source: §4.1 · p. 17
• A subsample-only bootstrap consistently approximates the conditional Gaussian-process limit for uniform inference across quantiles when S →∞.It resamples random weights and recomputes projected quantile-process estimators using only stored subsample estimates, avoiding repeated full-sample estimation unlike Hagemann (2017).Source: §4.1 · pp. 18–19

4.2. Inference based on estimating the asymptotic covariance matrix.

• Averaging distributed covariance components can attain the full-sample convergence rate and asymptotic distribution, but requires careful bandwidth selection and stronger restrictions on S.Under S = o(N2/5(log N)−1), the averaged estimator matches the Powell oracle MSE rate; subsample-optimal h_n ∼n−1/5 instead leads to undersmoothing.Source: §4.2 · pp. 19, 21

5.1. Results for the divide and conquer estimator

• Oracle and t-based intervals maintain nominal coverage for small S, whereas normal and bootstrap intervals under-cover initially; all four converge near S = 20.For m = 32, the oracle rule fails with n = 512 when S > 10, improves at n = 2048, and coverage drops earlier as m increases.Source: §5.1 · p. 22, Figure 5.1 · p. 25
• Coverage deteriorates much faster at tail quantiles τ = 0.1 and 0.9 than at τ = 0.5, plausibly reflecting quantile-specific error-density derivatives in the estimation bias.The paper leaves this explanation for future study.Source: §5.1 · p. 23
• Estimated-variance methods show little coverage difference between n^-1/5 and N^-1/5 bandwidths, while quantreg’s automatic choice performs reasonably well.Once asymptotic variance estimation has at most 5–10% relative error, further accuracy brings little coverage improvement.Source: §5.1 · p. 23, Table 5.1 · p. 24

5.2. Results for the estimator bFY |X(y|x)

• For conditional distribution inference, bootstrap and oracle intervals perform similarly for m = 4 once S ≥20, but m = 32 violates the oracle regime even at S = 2.The bootstrap’s theoretical guarantee applies only when the oracle rule holds; simulations use K = 65 quantile levels and G = 32 spline knots.Source: §5.1 · pp. 23–24, Figure 5.2 · p. 26

5.1. Results for the divide and conquer estimator

• For S ≥20, sub-sample-based inference is reliable; normal intervals are recommended for pointwise inference, bootstrap intervals for process inference, and aggregated variance estimators for S <20.The averaged variance estimator is simple, performs well, and requires no additional implementation.Source: §5.1 · p. 26

APPENDIX A: APPROXIMATE LINEAR MODELS WITHOUT LOCAL BASIS STRUCTURE

• Without local basis structure, oracle rules hold under stronger restrictions on model dimension m and machine count S, including m^4(log N)^10=o(N) and S=o(N^1/2m^-2(log N)^-2) for correctly specified models.The generality requires more stringent growth conditions than local-basis results; when m is fixed, the result reduces to Corollary 3.3.Source: Appendix A · pp. 28–29
• For univariate splines, exploiting local support yields sharper sufficient bounds than the general result, whereas the general corollary applies without Condition (L) and covers broader linear-model settings.The local-support conditions are m^2(log N)^6=o(N) and S=o((Nm^-1(log N)^-4)^1/2 ∧ (Nm^-2(log N)^-10)); the general conditions are more restrictive.Source: Appendix A · p. 29

ONLINE SUPPLEMENTARY MATERIAL: Distributed inference for quantile regression processes

• The supplementary material provides proofs, technical details, and additional simulation results for the distributed quantile-regression framework.It also introduces notation used throughout the supplement, including support-indexed subsets of the unit sphere.Source: ONLINE SUPPLEMENTARY MATERIAL:… · p. 31

APPENDIX S.1: AN ALTERNATIVE ESTIMATOR FOR JP

• An additional communication round estimates J_m(τ) by broadcasting β(τ), aggregating machine-level matrices, and computing the estimator at the master without optimization on each subsample.When β(τ) satisfies the oracle rule, the method achieves covariance estimation at oracle rate N^-2/5 with fairly low additional communication and computational cost.Source: Appendix S.1 · pp. 31–32

S.2.1. Aggregated Bahadur representation for general series approximation model.

• The general-series representation requires conditions such as mξ_m^2log n=o(n), while omitting (log n)^2S^-1=o(1) introduces an additional log n factor in the remainder bound.The representation is formulated for sequences β_N(τ) with controlled approximation error and supplies bounds for distributed estimators.Source: §S.2.1 · pp. 32–33
• For general series models, the aggregated Bahadur representation provides sharper remainder bounds than directly applying the classical representation, improving the sufficient machine-count condition in a fixed-dimensional linear example.The direct approach requires S=o(N^1/3(log N)^-4/3), whereas the aggregated bound permits S=o(N^1/2(log N)^-2) when approximation errors vanish.Source: §S.2.1 · pp. 32–34

S.2.2. Aggregated Bahadur representation for local basis functions.

• For local basis functions, the aggregated Bahadur representation holds under Condition (L), approximation-error restrictions, and index sets containing O(1) consecutive integers; a corollary centers the expansion at the true function under a pointwise bias condition.The local-basis results include a fixed-x0 specialization when c_m(γ_N)=o(∥Z(x0)∥N^-1/2).Source: §S.2.2 · pp. 34–35

• Theorem 3.1 follows from Theorem A.1 because m and ξ_m are constant in N, J_m(τ) is invertible, γ_N ≡ β, and c_m(γ_N) = g_N(γ_N) = 0.Source: §S.3.1 · p. 35
• S = o(N^1/2(log N)^−1) ensures weak convergence of ¯β because ¯β − bβ_or = o_P(N^−1/2), while bβ_or already converges weakly.Source: §S.3.1 · p. 36
• The sufficient part of Corollary 3.4 is proved using Theorem 3.1, spline approximation results, and Lemma 5.1 of Huang (2003).The spline space Θ_G consists of degree-r_τ splines over the grid t_1, ..., t_G on T = [τ_L, τ_U] ⊂ (0, 1).Source: §S.3.1 · p. 36
• This section proves Theorem 3.7 and the sufficient weak-convergence conditions in Corollaries 3.9 and 3.10, while Section S.3.3 contains Theorem 3.8 and the corresponding necessary conditions.Source: §S.3.2 · p. 36

S.3.2.1. Proof of Theorem 3.7

• Theorem 3.7 follows by representing the oracle estimator and showing the remainder terms are O(R_n(S)) under its assumptions.Choosing A = 3 and combining the representation with the remainder bound yields (3.10); the second claim follows by analogous bounds under (A1)–(A3), (L), and (L1).Source: §S.3.2 · p. 37

S.3.2.2. Proof of (3.13) and the weak convergence of bβor(τ) in Corollary 3.9

• Under (L1), the oracle and distributed estimators differ by o_P(∥Z(x0)∥N^-1/2), and their weak convergence follows from the Lindeberg central limit theorem.The argument adapts finite-dimensional joint convergence to pointwise convergence.Source: §S.3.2 · p. 38

S.3.2.3. Proof of (3.15) and part 2 of Corollary 3.10

• Process convergence of bβor(τ) follows from Theorem 2.4 of Chao et al. (2017), while Theorem 3.7 supplies the uniform approximation needed for (3.15) and Corollary 3.10.The remaining corollary statement is obtained using arguments analogous to Corollary 4.1 of Chao et al. (2017).Source: §S.3.2 · p. 38

S.3.3.2. A special class of data generation processes.

• The special data-generating process uses i.i.d. Y_i with distribution F_a,b and a discrete independent design satisfying P(X_1=e_j/√m)=1/m, yielding explicit quantile-regression solutions.For observations assigned to coordinate j, the smallest-absolute-value solution is β̂_j(τ)=m^-1/2 times the corresponding sample order statistic.Source: §S.3.3.2 · p. 42
• The constructed family demonstrates that the weak convergence in Corollary 3.3 can fail for fixed dimension and for growing dimension under the paper’s boundary regimes.For fixed m, failure is established at u=e_1; for m→∞, the same contradiction applies when m^2(log N)^6=o(N).Source: §S.3.3.2 · pp. 43–44
• The same construction rules out process convergence over a compact quantile interval by selecting a sequence τ_N and showing that the normalized coordinate estimator would otherwise converge to a centered normal limit.The proof then derives a contradiction with the construction’s conditions, and states that the failure of (3.15) follows similarly.Source: §S.3.3.2 · pp. 44–45
• The proofs establish necessity of S=o(N^1/2) for the distributed estimator and G≫N^1/(2η) for spline approximation-based process convergence.Contradictions arise when S≳N^1/2 or when the spline-grid growth condition is violated.Source: §S.3.3.2 · p. 45
• The appendix generalizes the Section A arguments from γ_N to admissible centering sequences β_N, with conditions controlling S, m, c_m(β_N), and approximation errors.Under these conditions, γ_N satisfies the required bounds, including g_N(γ_N)=o(N^-1/2) and m c_m(γ_N) log N=o(1).Source: §S.3.3.2 · pp. 45–47
• The proofs of Theorems A.1 and Corollary A.2 combine oracle-estimator representations, projection-operator bounds for B-splines, spline approximation control, and a Lindeberg central limit argument.The projection operator is uniformly bounded, while the spline approximation error is controlled through the grid modulus of continuity.Source: §S.3.3.2 · pp. 47, 49, 50, 51

APPENDIX S.4: PROOFS FOR SECTION 4

• The Section 4 appendix continues with analogous proofs under the assumptions of Corollaries 3.4, 3.10, and A.5, presenting the Corollary A.5 case as representative.The other cases are stated to follow by similar arguments.Source: §S.4.1 · p. 52

S.4.1. Proof of Theorem 4.1.

• p_N converges to 1−α for fixed S>2, while S→∞ yields t_{S−1,1−α/2}→Φ^−1(1−α/2), completing both parts of Theorem 4.1.The proof treats subsequences with constant S, uses the continuous mapping theorem, and then extends the result to arbitrary sequences S_N.Source: §S.4.1 · pp. 52, 54

S.4.2. Proof of Theorem 4.2.

• Theorem 4.2 is proved by combining finite-dimensional convergence with asymptotic uniform equicontinuity, using a triangular-array Lyapunov CLT and empirical-process bounds.The proof controls increments through metric entropy, Chernoff and Bernstein inequalities, union bounds, and the assumed weight properties.Source: §S.4.2 · pp. 55, 59, 61

S.4.3. Proof of Theorem 4.4.

• The proof of Theorem 4.4 combines empirical-process bounds, Taylor expansion, and uniform remainder control to complete the required approximation.The argument uses leave-one-out empirical processes and invokes Lemma S.6.3 and Theorem S.6.1 for the needed bounds.Source: §S.4.3 · pp. 62–64

APPENDIX S.5: PROOFS FOR APPENDIX S.2

• The appendix introduces a representation from Theorem S.6.1 that supports the subsequent proofs for results in Appendix S.2.This section serves as the transition into the detailed concentration and remainder arguments for Theorems S.2.1 and S.2.4.Source: §S.5.1 · p. 64

S.5.1. Proof of Theorem S.2.1.

• Theorem S.2.1 follows from Theorem S.6.1, union bounds, moment controls, and McDiarmid’s inequality applied to the relevant random-function quantities.The proof separately controls remainder components and concludes with an elementary bound involving ℜ3(A log n).Source: §S.5.1 · pp. 65–66

S.5.2. Proof of Theorem S.2.4.

• Theorem S.2.4 is established by combining the approximation representation from Theorem S.6.2 with reverse-triangle and McDiarmid bounds.The proof uses the defining orthogonality condition for γ_N, controls the resulting remainder terms, and concludes the stated bound.Source: §S.5.2 · pp. 67–68

APPENDIX S.6: REFINED BAHADUR REPRESENTATIONS FOR SUB-SAMPLES AND THEIR PROOFS

• Theorem S.6.1 gives refined Bahadur representations for sub-sample estimators and bounds their remainder terms under Conditions (A1)–(A3) and mξ^2_m log n = o(n).The result applies to any β_N(·) satisfying g_N(β_N) = o(ξ^-1_m).Source: Appendix S.6 · p. 69
• The bound in (S.6.8) is new and crucial for the ABR in Theorem S.2.1, whereas bounds in (S.6.2)–(S.6.6) reproduce Theorem 5.1 of Chao et al. (2017).The appendix also defines empirical-process notation, covering and bracketing numbers, and the score function used in the proofs.Source: Appendix S.6 · pp. 69–70
• Under the additional local-basis Condition (L), Theorem S.6.2 provides a refined representation for consecutive index sets and a new remainder bound crucial for Theorem S.2.4.This result assumes Conditions (A1)–(A3), mξ^2_m log n = o(n), and an index set containing at most L consecutive integers.Source: Appendix S.6 · pp. 70–71

S.6.1. Proof of Theorem S.6.1.

• The proof of Theorem S.6.1 uses a leave-one-out estimator to decompose the remainder, separating mean-zero terms from the nonzero expectation of r_n,3(τ).The argument bounds the remaining expectation through Taylor expansion, Cauchy–Schwarz, and the inequality ∥μ(b_1, τ) − μ(b_2, τ)∥ ≤ 2ξ_m.Source: Appendix S.6 · p. 71, §S.6.1 · pp. 72–74
• Lemmas S.6.3–S.6.6 supply the auxiliary empirical-process, function-class, and probabilistic inequalities needed to complete the theorem’s remainder analysis.Their proofs use arguments from Chao et al. (2017), including Lemmas C.1–C.3, while the appendix supplies the needed applications and reductions.Source: §S.6.1 · pp. 74, 76, 77
• Empirical-process and linear-optimization arguments control the relevant events and yield high-probability bounds for the remainder, with probabilities involving e^-κ_n.The proof combines decompositions, auxiliary lemmas, and a choice of δ_n satisfying the resulting inequality.Source: §S.6.1 · pp. 74–76

S.6.2. Proof of Theorem S.6.2.

• Under (A1)–(A3) and (L), the proof uses a decomposition valid when eJ_m(b) is invertible, applies a Taylor expansion, and invokes Lemma S.6.7 to control a remainder.It defines local index sets I(x) and I′(x), and obtains a probability bound containing m(n+1)e^−κ_n.Source: §S.6.2 · pp. 78, 79, 80, 82
• The localization argument defines I(x) and I′(x) through neighborhoods of k_Z(x), with radius r and an additional D log n buffer.These sets support uniform control over the relevant function classes and are used to bound the event Ω̃_2,n(C) by m e^−κ_n log^2 n.Source: §S.6.2 · pp. 80, 82

S.7.3. Auxiliary Lemmas.

• Lemma S.7.1 models U_τ conditional on binomial W_n with a Beta distribution and derives bounds for quadratic expectations and conditional variances when n/m ≥ 1.The proof uses Bernstein tail bounds and P(W_n > 0) ≥ 1/2 for n ≥ 2 and n/m ≥ 1.Source: §S.7.3 · pp. 89–90
• Lemma S.7.2 establishes localization sequences τ_N in T for both lim sup_N→∞ n/m < ∞ and lim sup_N→∞ n/m = ∞.The result assumes N ≥ m, G ≫ (N/m)^α, S ≫ (N/m)^1/2, and r_τ ≥ 2.Source: §S.7.3 · p. 91
• The auxiliary lemmas prove uniform bounds by combining local constancy, Bernstein’s inequality, and spline basis support properties under K ≥ G.The proof controls remainder terms uniformly in τ and completes the target bounds using Lemma S.7.3, binomial concentration, and consecutive-support index arguments.Source: §S.7.3 · pp. 93–96

APPENDIX S.8: TECHNICAL RESULTS FROM EMPIRICAL PROCESS THEORY

• The empirical-process appendix records symmetrization and refined Talagrand concentration inequalities for uniformly bounded function classes under variance and entropy conditions.It also states an empirical-process result for U_N(τ) when m(log n)^2 = o(N), citing Koltchinskii (2006) and Massart (2000).Source: Appendix S.8 · pp. 96–97
• A polynomial lemma supplies interval-to-grid inequalities for degree-q polynomials, proved by rescaling intervals, bounding partition errors, and using finite-dimensional operator boundedness.The proof controls interior and boundary partition terms through a bounded linear operator on polynomial spaces.Source: Appendix S.8 · pp. 97–98

S.9.1

• The supplement adds simulation details and examines confidence-interval coverage, including the case m = 16 and large values of S.Section S.9.1 covers the simulation settings and additional figures for FY |X(y|x0).Source: Appendix S.9 · p. 98

S.9.3

• Additional linear-model simulations illustrate oracle properties for the divide-and-conquer estimator β(τ) and distribution-function estimator bFY |X(y|x).Section S.9.3 uses the same linear model as Section 5 and emphasizes the oracle theory.Source: Appendix S.9 · p. 98

S.9.4

• Nonlinear-model simulations likewise assess oracle properties of β(τ) and bFY |X(y|x).Section S.9.4 extends the illustration to a non-linear model.Source: Appendix S.9 · p. 98

S.9.5

• Section S.9.5 summarizes practical conclusions from the simulation studies.Source: Appendix S.9 · p. 98

Implementation details

• The pooled covariance estimator averages sub-sample covariance estimates computed with quantreg’s se='ker' and covariance=TRUE options.The procedure applies the rq function separately to each sub-sample before averaging.Source: §S.9.1 · p. 98
• The tuning constant c*(τ) uses Kato’s theoretical formula, with a corrected α(τ), is symmetric around τ = 0.5, and is approximated across m using values computed at m = 4.An independent simulated data set supplies c*(τ), whose value changes little with m under the stated Z design; Table S.9.1 reports values based on (S.9.1).Source: §S.9.1 · pp. 98–99, Table S.9.1 · p. 99

Additional simulation results

• Additional simulations evaluate Q(x0; τ) confidence-interval coverage for m = 16 under the homoskedastic model, varying S, τ, and n.Figure S.9.1 compares asymptotic oracle, t-based, normal-based, and bootstrap intervals at nominal coverage 0.95, with x0 = (1, ..., 1)/m^1/2.Source: §S.9.1 · p. 99, Figure S.9.1 · p. 99
• Further simulations assess coverage for F_Y|X(y|x0) under the Section 5 settings, including large S for m = 4 and 32 and m = 16.Figure S.9.2 uses n = 512 and nominal coverage 0.95 to compare oracle asymptotic and bootstrap intervals with B = 500; Figure S.9.3 reports the m = 16 case.Source: §S.9.1 · pp. 99–100, Figure S.9.2 · p. 100

S.9.2. Heteroskedastic model.

• In the heteroskedastic model, confidence-interval coverage follows homoskedastic patterns but starts dropping earlier for m=4, 16, and 32.The study compares several 95% intervals across sub-sample counts S, using oracle asymptotic intervals as benchmarks.Source: §S.9.2 · pp. 100–101
• Estimating the asymptotic variance produces coverage patterns similar to Table 5.1, but probabilities begin declining earlier in the heteroskedastic model.The analysis replaces homoskedastic constants with c∗_g(τ) and evaluates multiple covariance-estimation methods.Source: §S.9.2 · pp. 101–102, Table S.9.3 · p. 102
• For bF_Y|X(y|x), bootstrap and oracle confidence intervals likewise show overall coverage beginning to decline earlier under heteroskedasticity.The simulations use K=65 quantile-grid points and G=32 spline knots, selected to preserve nominal oracle coverage.Source: §S.9.2 · p. 104

S.9.3. Additional simulations for the linear model.

• The additional linear-model simulations target sufficient and necessary thresholds for S∗ and K∗ that preserve oracle properties of β(τ) and bF_Y|X(y|x).They examine normal and exponential errors, τ=0.1, 0.5, 0.9, m=4, 16, 32, and T=[0.05,0.95].Source: §S.9.2 · p. 107

S.9.3.1. Oracle rule for

• Coverage reaches 95% for S<S∗ but drops quickly for S>S∗; as N increases, S∗ approaches N^1/2, while larger m lowers S∗.For exponential errors, coverage becomes asymmetric in τ, and small τ requires larger N to attain comparable S∗.Source: §S.9.3.1 · p. 108
• For the distribution-function estimator, increasing basis dimension q improves coverage at fixed m and S, while larger m permits smaller q.The required growth condition is q≫N^(1/(2η))∥Z(x0)∥^−η, and q and K have similar effects when other parameters are fixed.Source: §S.9.3.1 · p. 108
• Under exponential errors with rate 0.8, distribution-function coverage is better at small τ than under the normal-error simulations.The simulations vary S, m, q, and K while fixing N=2^14 and evaluating coverage of the oracle interval.Source: §S.9.3.1 · p. 108

Model specification

• The quantile model uses cubic B-splines and has Q(x; τ) = 2.5 + sin(2x) + 2 exp(−16x2) + 0.7Φ−1(τ).The basis is defined by m + 4 knots partitioning [−1, 1], with boundary repetitions.Source: §S.9.4 · p. 109

Simulation design

• Confidence-interval coverage is evaluated for Q(x; τ) at fixed τ and for FY |X(y|x), with N = 216 in both subsections.The quantile and distribution-function intervals follow Corollaries 3.9 and 3.10, respectively.Source: §S.9.4 · p. 109

S.9.4.1. Oracle rule for β(τ)

• m = 13 at x0 = 0 produces the worst coverage, because large curvature creates bias; increasing m reduces bias but also decreases S*.These results are shown in Figure S.9.14.Source: §S.9.4 · p. 109

S.9.4.2. Oracle rule for bFY |X(y|x)

• Larger m improves coverage by reducing estimation bias under fixed S, q, and K, while its effects on S, q, and K otherwise resemble the linear normal-error case.Figure S.9.15 reports these effects for τ = 0.1; τ = 0.5, 0.9 are omitted because they add no insights.Source: §S.9.4 · p. 109

Practical recommendations

• Practical guidance recommends choosing S as small and K as large as possible, while selecting S carefully for models with more predictors.The coverage threshold depends on the quantile and the unknown conditional response density, so universal defaults are difficult.Source: §S.9.5 · p. 110

S.9.5. Practical conclusions from simulations.

• Use an effective subsample size n/m of at least 50 and choose S no larger than n/m; rapidly changing conditional-quantile spread requires extra caution.Coverage drops earlier for smaller quantiles under exponential errors, where the conditional quantile function has larger second derivative.Source: §S.9.5 · p. 110, Figure S.9.10 · p. 111
• For spline approximation, K > N^1/4 worked across simulations, while cubic splines with G = K/2 provided a reasonable choice.Although the theorem requires K/G →∞, the proof indicates that K/G > c for a constant c is sufficient for cubic splines.Source: §S.9.5 · p. 110
• The simulations assess 95% confidence-interval coverage for conditional quantiles and conditional distribution functions under both linear and nonlinear models.The linear experiments include normal and exponential errors, while the nonlinear experiments use the model (S.9.7).Source: Figure S.9.9 · p. 111, Figure S.9.11 · p. 112, Figure S.9.12 · p. 113