Rank-Based Estimation of U-Shaped Biomarker Risk Curves and Critical Points for Time-to-Event Outcomes

Mei-Cheng Wang; Yu Du; Yuxin Zhu; Zhirui Fu

REVIEW 3 major objections 5 minor 12 references

The paper proposes a semiparametric survival model that writes the U-shaped risk curve as a piecewise-linear index inside an unknown transformation, so the biomarker value of minimum risk (the critical point) is an explicit function of para

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2026-08-01 12:52 UTC pith:PJ25UWSZ

load-bearing objection Potentially useful rank-based method for U-shaped survival risk, but the UK Biobank application fails an internal consistency check: Table 2's parameters cannot produce the reported critical points. the 3 major comments →

arxiv 2607.19280 v1 pith:PJ25UWSZ submitted 2026-07-21 stat.ME

Rank-Based Estimation of U-Shaped Biomarker Risk Curves and Critical Points for Time-to-Event Outcomes

Zhirui Fu , Mei-Cheng Wang , Yu Du , Yuxin Zhu This is my paper

classification stat.ME MSC 62N0162N0262G0562G20

keywords U-shaped riskcritical pointmaximum C-index estimatorsemiparametric transformation modelsurvival outcomescensoringbootstrap inferenceBMI

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tackles a gap: even though U-shaped biomarker–risk relationships are common in medicine, no existing survival method could estimate the biomarker value of lowest risk (the critical point) and attach valid uncertainty to it. The authors propose a semiparametric transformation model in which the critical point is explicitly parameterized by four parameter blocks, and a rank-based maximum C-index estimator that recovers these parameters without specifying the transformation or error distribution. They prove the estimator is √n-consistent and asymptotically normal, use a smoothed Kaplan–Meier estimator for the nonparametric component, and show in simulations that Cox-style alternatives yield systematically biased critical points whereas this method achieves near-nominal coverage. Applied to UK Biobank BMI and all-cause mortality, the method yields subgroup-specific critical points with confidence intervals, suggesting different 'optimal' BMI values by age and sex.

Core claim

The central claim is that model T=G(-max(-X+Z^T alpha1, beta0+beta1 X+Z^T alpha2)+epsilon), with G and epsilon distribution unspecified, places the critical point Xc=(1+beta1)^-1(Z^T(alpha1-alpha2)-beta0) inside the parametric component. Because the two linear branches cross at Xc, maximizing the concordance index over the four parameter blocks identifies the critical point directly; the MCE is √n-consistent and asymptotically normal with variance V^-1 U V^-1, and bootstrap/delta-method intervals attain nominal coverage at moderate sample sizes. The same framework estimates the whole risk curve through a kernel-weighted Kaplan–Meier estimator, and the UK Biobank analysis finds age- and sex-d

What carries the argument

The engine is the piecewise-linear risk index H(X,Z;theta)=max(-X+Z^T alpha1, beta0+beta1 X+Z^T alpha2) placed inside an unspecified strictly increasing transformation G. The two affine pieces, left slope -1 and right slope beta1>0, intersect exactly at the critical point Xc; rank-based comparison of H across comparable pairs, through a Harrell-type C-index, identifies theta without specifying G or the error distribution. Asymptotic normality flows from U-statistic and empirical-process arguments for maximum rank correlation estimators, adapted to the nondifferentiable index.

Load-bearing premise

Identification hinges on the biomarker X having a strictly positive Lebesgue density conditional on covariates, so that the ordering induced by the risk index uniquely determines the parameters; with coarsely recorded biomarkers such as BMI recorded on a grid, this is only approximately true and ties may flatten the objective.

What would settle it

Simulate from the model with X drawn from a coarse discrete grid, such as BMI rounded to integers, and observe whether the C-index surface has multiple maxima or whether bootstrap intervals for Xc fail to cover. Alternatively, compute the objective at two candidate theta values that generate the same pairwise ordering on the observed support; if they tie, identification fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

With this method, clinical researchers can report a confidence interval for the biomarker value of minimum risk, not just a fitted curve.
Subgroup-specific critical points are estimated in one model, so age- and sex-dependent 'optimal' biomarker levels can be compared formally.
The model nests Cox proportional hazards, accelerated failure time, proportional odds, and binary choice models as special cases.
In simulations, Cox spline and quadratic alternatives show persistent bias in the recovered critical point, while the MCE's bias shrinks with sample size; post-hoc critical points from standard Cox fits can therefore be misleading.
Critical regions, the biomarker ranges with risk below a chosen threshold, can be read off the estimated risk curves with bootstrap intervals for their boundaries.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the UK Biobank finding replicates, clinical guidance on 'optimal' BMI may need to be age- and sex-specific; the wide confidence interval for older males (roughly 18.3–32.6) suggests the data support a U shape but not a precise optimum there.
The identification argument assumes a continuous biomarker distribution; applied to discretely recorded variables like BMI, the critical point is identified only up to the granularity of the grid, so sensitivity analyses on binning and bandwidth are advisable.
The piecewise-linear index naturally extends to V-shaped or multi-turning-point risk surfaces by adding more affine pieces, though the paper leaves that extension to future work.
A direct testable extension is to compare critical points across subgroups using the bootstrap covariance matrix, for example testing whether optimal BMI differs between males and females.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Potentially useful rank-based method for U-shaped survival risk, but the UK Biobank application fails an internal consistency check: Table 2's parameters cannot produce the reported critical points.

read the letter

The methodological core is genuinely new and worth engaging: it parameterizes the critical point directly in a semiparametric transformation model and estimates it by maximizing a C-index. The simulations are extensive and show good finite-sample behavior, and the Web Appendix gives a standard proof skeleton. That alone makes it worth a serious referee.

But the real-data section has a load-bearing problem. In Table 2, using their own formula, the reference subgroup Female,<65 gives Xc = (0 - 0 - 3.26)/(1.25) = -2.6, not the reported 22.39. The same mismatch holds for every subgroup. Worse, with the fitted positive β0 and α2 values, the risk index H(X,Z) = max(-X + Z^Tα1, β0 + β1X + Z^Tα2) is monotone increasing over BMI 15-40, so the fitted index has no minimum. The reported U-shaped curves must come from something else—possibly the subgroup reparameterization in Web Appendix B or the smoothed Kaplan-Meier curves—but the paper explicitly claims the MCE parameterization produces them. This is an internal contradiction, not a harmless typo, and it undermines the real-data demonstration and the inference claims.

Other soft spots are secondary. The bootstrap for a non-smooth MCE lacks formal justification. Identification relies on Assumption 4 (everywhere positive Lebesgue density of X given Z), which is not literally satisfied for BMI recorded on a discrete grid. The small-sample coverage for log β1 at n=200 is around 0.83, though the authors flag this. The simulation comparison against Cox alternatives is fair and gives the method real credit.

Recommendation: send this out for peer review, but with a major-revision request. The authors must reconcile Table 2 with the model's formula or clearly document the reparameterization they actually fit, and show the fitted MCE risk index. If that is fixed, this could be a solid contribution. As it stands, the application section cannot be trusted.

Referee Report

3 major / 5 minor

Summary. The paper proposes a semiparametric transformation model T = G(-max(-X+Z^Tα1, β0+β1X+Z^Tα2)+ε) for survival outcomes with a U-shaped biomarker–risk relationship. The key innovation is the explicit parameterization of the critical point Xc = (1+β1)^{-1}(Z^T(α1−α2)−β0), enabling direct estimation and inference. The maximum C-index estimator (MCE) is proposed for θ=(β0,β1,α1,α2); Web Appendix A develops consistency and asymptotic normality via U-process theory. A smoothed Kaplan–Meier estimator estimates the nonparametric transformation. Extensive simulations compare MCE with Cox-based alternatives. The method is applied to UK Biobank data to estimate subgroup-specific BMI–mortality critical points.

Significance. The paper addresses a genuine gap: joint estimation of a biomarker critical point with covariate dependence and valid uncertainty quantification under censoring, without specifying G or the error distribution. The simulations are extensive (23 scenarios, 1,000 replications each) and support the finite-sample behavior of the MCE; the Web Appendix contains a detailed proof skeleton and the code is publicly available. If the theoretical claims hold, the method would be a useful addition to the survival-analysis toolbox. However, the real-data application in Section 7 is internally inconsistent with the model and the reported parameters, which currently undermines the applied demonstration.

major comments (3)

[§7, Table 2 vs. §5, Eq. (7)] The reported critical points are not consistent with the model's critical-point formula Xc = (1+β1)^{-1}(Z^T(α1−α2)−β0). For the reference subgroup Female,<65 (Z=0, α1=α2=0), Table 2 gives β0=3.26 and β1=0.25, which yields Xc = −3.26/1.25 = −2.61 kg/m², not 22.39. For Male,<65, using Z^Tα2=4.70, β0=6.54, β1=0.48 gives Xc = (−6.54−4.70)/1.48 = −7.59, not 23.75. Moreover, for all reported parameters the second branch β0+β1X+Z^Tα2 dominates −X+Z^Tα1 for every X in [15,40], so H(X,Z;θ̂) is monotonically increasing and has no minimum on the observed BMI range. This indicates that the critical points in Table 2 were not computed from the fitted parameters via the model's explicit parameterization, or that an undocumented reparameterization was used. The central applied claim is therefore not connected to the estimation theory of Sections 3–5.
[Web Appendix B vs. model (1)] The subgroup-specific parameterization in Web Appendix B (α2z, β0z, β1z per subgroup, with α1z fixed to 0) is not a special case of model (1) as applied in Section 7. Model (1) has global scalar β0 and β1 and does not include X-by-Z interaction terms, so it cannot produce different β1 values across subgroups, as Table 2 reports (β1 ranging from 0.25 to 1.12). The mapping between the subgroup model and the parameters (β0,β1,α1,α2) that appear in the critical-point formula (Eq. 5) is never provided. Consequently, the claim that the UK Biobank analysis estimates the parameters of model (1) and hence Xc via the closed-form expression is unsupported. The authors must either fit model (1) with global β0 and β1, or explicitly present the extended model with subgroup-specific intercepts and slopes, derive the correct critical-point formula and identifiability conditions, and report the estimated
[Web Appendix A, Assumption 4] The identification and asymptotic theory relies on Assumption 4, which requires X to have an everywhere positive Lebesgue density conditional on Z. In the UK Biobank application, BMI is recorded on a discrete grid (integer values) and restricted to [15,40], so this assumption is not literally satisfied. The paper does not discuss how discreteness, ties, or gaps in the support affect the rank objective, the identifiability of the critical point, or the bootstrap inference. At minimum, the authors should provide a sensitivity analysis (e.g., jittering BMI) or a formal argument that the limiting theory remains valid for discretized X. This is a load-bearing point for the real-data demonstration, though it does not necessarily invalidate the estimator in simulations where X is continuous.

minor comments (5)

[§1, p.2] Typo: 'do not accommodates time-to-event outcomes' should be 'do not accommodate'.
[Table 2, Notes] The notation α2z is introduced only in Web Appendix B, not in the main text. The main-text model uses α2 as a global vector; the note should be moved to the main text or the table should be aligned with the main-text notation.
[Web Appendix B] Minor typos: 'subgroup indicatorsz' should be 'subgroup indicators Z_z'; 'Regressing X_r on X_l and the subgroup indicators' is missing a subscript for the subgroup index z.
[§7, p.19] The sentence claiming that the empirical patterns provide 'indirect support for the proposed semiparametric modeling framework' is circular: the smoothed Kaplan–Meier curves are constructed from H(X,Z;θ̂) and isotonized in H, so the U-shape in X is inherited by construction rather than independently validated. Rephrase as a descriptive statement or provide an external validation.
[§5, Critical region] The general critical-region interval [Z^Tα1 − c, β1^{-1}(c−β0−Z^Tα2)] is correct, but the paper should explicitly state that the interval is valid only when c ≥ H(Xc,Z;θ); otherwise the region is empty. This is implied but not made precise.

Circularity Check

1 steps flagged

No substantive circularity: core derivation is self-contained; one minor non-load-bearing self-confirmatory statement.

specific steps

other [Section 7, final paragraph (real-data analyses)]
"The empirical patterns align closely with the structural form of the fitted risk index H(X,Z; bθ), providing indirect support for the proposed semiparametric modeling framework."

The 'empirical patterns' are the estimated risk curves 1 − Ŝ_h(t|X_i,Z_i) produced in Section 4, and those curves are built by kernel-weighted Kaplan–Meier smoothing over Ĥ_i = H(X_i,Z_i; bθ). Hence alignment with the structural form of H(·;bθ) is enforced by the estimation procedure rather than independently observed. However, the paper labels this only 'indirect support,' and no theorem or central claim rests on this sentence; it is rhetorical rather than load-bearing.

full rationale

The core derivation chain is self-contained. Model (1) parameterizes the critical point as Xc = (1+β1)^{-1}(Z^T(α1−α2)−β0); the MCE maximizes the empirical C-index (2) over θ; identification (Web Appendix A, Lemmas 1–2) uses the positive-density assumption on X and external consistency/asymptotic results (Han 1987; Sherman 1993), not a self-citation. The delta-method inference for Xc follows from Theorem 1. Simulation data are generated from the same model, and the oracle/feasible Cox comparisons provide external benchmarks; generating data from the model to test the estimator is not circular. The one mild circularity is the final 'indirect support' sentence, which takes alignment of the model-fitted curves with the model's own risk index as empirical confirmation; this is acknowledged as indirect and does not support the paper's formal claims. The UK Biobank Table 2 arithmetic mismatch flagged by the skeptic is a real internal-consistency/correctness problem, but it is not a case of the derivation reducing to its inputs, so it does not change the circularity score.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The paper introduces no new physical or latent entities. The critical point is a derived function of the model parameters. The main external inputs are domain assumptions about the U-shape, conditional independence of censoring, and continuous biomarker support, plus tuning parameters for kernel smoothing and initialization.

free parameters (3)

Kernel bandwidth h for smoothed Kaplan-Meier = Tuned per setting
Section 4: bandwidth controls bias-variance tradeoff; chosen via data-driven procedures or by hand, and it affects the estimated risk curves and critical region.
Jitter sigma_BMI for initialization = 0.15-0.20 BMI units
Web Appendix B: random perturbations around paired LOESS roots to generate starting values; user-chosen and not estimated from the likelihood.
Bin width and risk threshold for critical region = 2.5 BMI units; a=0.05, t0=15 years in UK Biobank
Web Appendix B and Section 7: bin width for initialization and the risk threshold/time horizon for the critical region are analyst choices, not model parameters.

axioms (5)

domain assumption Model (1): T=G(-max(-X+Z^Tα1, β0+β1X+Z^Tα2)+ε) with G strictly increasing, ε independent of (X,Z), β1>0
Section 2: the entire framework is built on this structural U-shape assumption; if the true relationship is not of this form, Xc is not the minimum-risk point.
domain assumption Conditionally independent right censoring: C ⊥ T | (X,Z)
Section 3 and Web Appendix A, Assumption 3: needed for the pairwise ranking inequality to survive censoring and for the C-index objective to identify θ.
domain assumption X has everywhere positive Lebesgue density conditional on Z
Web Appendix A, Assumption 4: used to guarantee no ties in H and to prove identification in Lemma 2; not satisfied by discrete BMI in the real data.
standard math Regularity conditions in Assumption 7: differentiability and integrability of τ, negative definite E∇²τ
Web Appendix A.5: needed to apply Sherman's lemmas and obtain √n-consistency and asymptotic normality.
domain assumption The U-shaped risk relationship is known a priori
Section 1: 'We assume prior knowledge that a U-shaped relationship exists'; the Discussion acknowledges diagnostics for this assumption are not provided.

pith-pipeline@v1.3.0-alltime-deepseek · 31000 in / 16761 out tokens · 153197 ms · 2026-08-01T12:52:50.505748+00:00 · methodology

0 comments

read the original abstract

U-shaped relationships between prognostic biomarker levels and adverse event risk are commonly observed across diseases, where both low and high biomarker values are associated with elevated risk, with a well-defined minimum -- the critical point -- marking the biomarker value of the lowest risk. The U-shaped risk curve, especially the location of the critical point, informs the identification of high- and low-risk subgroups. However, existing methods are limited: U-shaped risk models rarely accommodate survival outcomes, and existing survival analysis methods do not enable estimation of or formal inference for the critical point. To fill this gap, we propose a semiparametric transformation model that explicitly parameterizes the critical point, a rank-based maximum C-index estimator for the parametric component, and a smoothed Kaplan-Meier estimation approach for the nonparametric component. The resulting framework estimates both subgroup-specific U-shaped risk curves and their critical points within a single survival model. We establish consistency and asymptotic normality of the proposed estimators and demonstrate their finite-sample performance through numerical studies. We apply the proposed methods to UK Biobank data to characterize subgroup-specific U-shaped associations between body mass index and all-cause mortality and to identify the corresponding critical points.

discussion (0)

Reference graph

Works this paper leans on

12 extracted references

[1]

Altman, N. S. (1992). An introduction to kernel and nearest-neighbor nonparametric regression.The American Statistician46,175–185. Bhaskaran, K., dos Santos-Silva, I., Leon, D. A., Douglas, I. J., and Smeeth, L. (2018). Association of bmi with overall and cause-specific mortality: a population-based cohort study of 3·6 million adults in the uk.The Lancet ...

1992
[2]

We first give assumptions and establish the consistency of the the maximum C-index estimator (MCE). We then state two lemmas that establish the √n-consistency and asymptotic normality of a general extremum estimator, together with aU-statistic decomposition and a uniform bound for degenerate U-processes; these are general results, due to Sherman (1993), a...

1993
[3]

We next show that Λ(θ) is continuous on Θ

Together with the compactness of Θ in assumption (6), Corollary 7 of Sherman (1993) gives sup θ∈Θ Cn(θ)−Λ(θ) P − →0. We next show that Λ(θ) is continuous on Θ. Fixθ∈Θ and condition onz i,x j andz j. As a function ofx i,H(x i,z i;θ) is piecewise linear with slopes−1 andβ 1, both of which are bounded away from zero by assumption (6); hence it is strictly mo...

1993
[4]

Lemma 3:Letθ n be a maximizer ofΓ n(θ), and0a maximizer ofΓ(θ)

This entails no loss of generality, since one may always replace Γ n(θ) by Γn(θ0 +θ)−Γ n(θ0) and Γ(θ) by Γ(θ 0 +θ)−Γ(θ 0), providedθ 0 +θ∈Θ. Lemma 3:Letθ n be a maximizer ofΓ n(θ), and0a maximizer ofΓ(θ). Supposeθ n converges in probability to0, and also that (i) there exists a neighborhoodNof0and a constantκ >0for whichΓ(θ)⩽−κ|θ| 2 for allθinN; 36 (ii) u...

1980
[5]

Pakes and Pollard (1989) provide simple criteria for determining the Euclidean property

Lemma 5 is Theorem 3 of Sherman (1993). Pakes and Pollard (1989) provide simple criteria for determining the Euclidean property. Nolan and Pollard (1987) give complementary criteria. Denoteθ n as the maximizer ofC n(θ) over Θ. The diagonal terms ofC n(θ) in equation (2) of the main text vanish, and the two terms in its summand are exchanges of one another...

1993
[6]

The Euclidean property of theP-degenerate class{h(·,·,θ) :θ∈Θ} follows from this fact in combination with Corollaries 17 and 21 of Nolan and Pollard (1987), which is condition (i) of Lemma

1987
[7]

Notice thatGis a (2p+ 7)-dimensional vector space of real-valued functions onS⊗S⊗R

on 44 S⊗S⊗R; that is, the collection of all functions of the formg(w 1, w2, t) =γt+γ 1y1 +γ 2y2 + ψδ2 +λ 1x1 +λ 2x2 +κ T 1 z1 +κ T 2 z2 +c,withγ, γ 1, γ2, ψ, λ1, λ2, c∈Randκ 1,κ 2 ∈R p. Notice thatGis a (2p+ 7)-dimensional vector space of real-valued functions onS⊗S⊗R. By Lemma 2.4 in Pakes and Pollard (1989), the class of sets of the form{g⩾r}or{g > r}, ...

1989
[8]

By Lemma 2.4 of Pakes and Pollard (1989), each of the corresponding sets lies in a VC class

and therefore belongs toG; so dog 1 =y 1 −y 2,g 2 =δ 2,g 3 =t, andg 4 = 1−t. By Lemma 2.4 of Pakes and Pollard (1989), each of the corresponding sets lies in a VC class. The subgraph in (21) is a finite union of finite intersections of these sets, so by Lemma 2.5 of Pakes and Pollard (1989) the class{subgraph(f(·,·,θ)) :θ∈Θ}is a VC class of sets. By Lemma...

1989
[9]

BecauseH(·,z;θ) is piecewise linear inx, its gradient with respect toθis piecewise constant, with a jump at the critical point: Ψ(x,z) =∇ θH(x,z;θ

rather than on the full covariate vector; the argument is that of Theorem 4 of Sherman (1993), adapted to the piecewise linear index and to right censoring. BecauseH(·,z;θ) is piecewise linear inx, its gradient with respect toθis piecewise constant, with a jump at the critical point: Ψ(x,z) =∇ θH(x,z;θ

1993
[10]

By assumption (4), Pr{X i =x c(Z i;θ 0)}= 0, soΨ(X i,Z i) is defined almost surely

=    0,0,z T,0 T T , x < xc(z;θ 0), 1, x,0 T,z T T , x > xc(z;θ 0). By assumption (4), Pr{X i =x c(Z i;θ 0)}= 0, soΨ(X i,Z i) is defined almost surely. The (2 + 2p)-vectorΨplays the role of the regressor vector in Sherman (1993); it is here that the non-differentiability ofHat the critical point enters. By assumptions (4) and (6), the distributio...

1993
[11]

=t = 0,for everyt∈R.WriteS 3(y, δ, t) for∂S(y, δ, t)/∂t. WhenS(·,·,·) is differentiable with respect to its third argument,g 0(·) is differentiable, and E∥Ψ(Xi,Z i)∥2 <∞, the same change-of-variables argument as in Theorem 4 of Sherman (1993) gives∇ 1τ(w,θ

1993
[12]

These expressions also show that assumption (7)(v) need not be imposed separately

andΨ i =Ψ(X i,Z i) for brevity, U=E h Ψi −µ 0(Hi) Ψi −µ 0(Hi) T ·S(Y i,∆ i, Hi)2 g0(Hi)2 i , V= 2 −1 E h Ψi −µ 0(Hi) Ψi −µ 0(Hi) T ·S 3(Yi,∆ i, Hi)g 0(Hi) i . These expressions also show that assumption (7)(v) need not be imposed separately. Since His the risk index, a larger value oftcorresponds to a stochastically smaller failure time, soS(y, δ,·) is de...

2007

[1] [1]

Altman, N. S. (1992). An introduction to kernel and nearest-neighbor nonparametric regression.The American Statistician46,175–185. Bhaskaran, K., dos Santos-Silva, I., Leon, D. A., Douglas, I. J., and Smeeth, L. (2018). Association of bmi with overall and cause-specific mortality: a population-based cohort study of 3·6 million adults in the uk.The Lancet ...

1992

[2] [2]

We first give assumptions and establish the consistency of the the maximum C-index estimator (MCE). We then state two lemmas that establish the √n-consistency and asymptotic normality of a general extremum estimator, together with aU-statistic decomposition and a uniform bound for degenerate U-processes; these are general results, due to Sherman (1993), a...

1993

[3] [3]

We next show that Λ(θ) is continuous on Θ

Together with the compactness of Θ in assumption (6), Corollary 7 of Sherman (1993) gives sup θ∈Θ Cn(θ)−Λ(θ) P − →0. We next show that Λ(θ) is continuous on Θ. Fixθ∈Θ and condition onz i,x j andz j. As a function ofx i,H(x i,z i;θ) is piecewise linear with slopes−1 andβ 1, both of which are bounded away from zero by assumption (6); hence it is strictly mo...

1993

[4] [4]

Lemma 3:Letθ n be a maximizer ofΓ n(θ), and0a maximizer ofΓ(θ)

This entails no loss of generality, since one may always replace Γ n(θ) by Γn(θ0 +θ)−Γ n(θ0) and Γ(θ) by Γ(θ 0 +θ)−Γ(θ 0), providedθ 0 +θ∈Θ. Lemma 3:Letθ n be a maximizer ofΓ n(θ), and0a maximizer ofΓ(θ). Supposeθ n converges in probability to0, and also that (i) there exists a neighborhoodNof0and a constantκ >0for whichΓ(θ)⩽−κ|θ| 2 for allθinN; 36 (ii) u...

1980

[5] [5]

Pakes and Pollard (1989) provide simple criteria for determining the Euclidean property

Lemma 5 is Theorem 3 of Sherman (1993). Pakes and Pollard (1989) provide simple criteria for determining the Euclidean property. Nolan and Pollard (1987) give complementary criteria. Denoteθ n as the maximizer ofC n(θ) over Θ. The diagonal terms ofC n(θ) in equation (2) of the main text vanish, and the two terms in its summand are exchanges of one another...

1993

[6] [6]

The Euclidean property of theP-degenerate class{h(·,·,θ) :θ∈Θ} follows from this fact in combination with Corollaries 17 and 21 of Nolan and Pollard (1987), which is condition (i) of Lemma

1987

[7] [7]

Notice thatGis a (2p+ 7)-dimensional vector space of real-valued functions onS⊗S⊗R

on 44 S⊗S⊗R; that is, the collection of all functions of the formg(w 1, w2, t) =γt+γ 1y1 +γ 2y2 + ψδ2 +λ 1x1 +λ 2x2 +κ T 1 z1 +κ T 2 z2 +c,withγ, γ 1, γ2, ψ, λ1, λ2, c∈Randκ 1,κ 2 ∈R p. Notice thatGis a (2p+ 7)-dimensional vector space of real-valued functions onS⊗S⊗R. By Lemma 2.4 in Pakes and Pollard (1989), the class of sets of the form{g⩾r}or{g > r}, ...

1989

[8] [8]

By Lemma 2.4 of Pakes and Pollard (1989), each of the corresponding sets lies in a VC class

and therefore belongs toG; so dog 1 =y 1 −y 2,g 2 =δ 2,g 3 =t, andg 4 = 1−t. By Lemma 2.4 of Pakes and Pollard (1989), each of the corresponding sets lies in a VC class. The subgraph in (21) is a finite union of finite intersections of these sets, so by Lemma 2.5 of Pakes and Pollard (1989) the class{subgraph(f(·,·,θ)) :θ∈Θ}is a VC class of sets. By Lemma...

1989

[9] [9]

BecauseH(·,z;θ) is piecewise linear inx, its gradient with respect toθis piecewise constant, with a jump at the critical point: Ψ(x,z) =∇ θH(x,z;θ

rather than on the full covariate vector; the argument is that of Theorem 4 of Sherman (1993), adapted to the piecewise linear index and to right censoring. BecauseH(·,z;θ) is piecewise linear inx, its gradient with respect toθis piecewise constant, with a jump at the critical point: Ψ(x,z) =∇ θH(x,z;θ

1993

[10] [10]

By assumption (4), Pr{X i =x c(Z i;θ 0)}= 0, soΨ(X i,Z i) is defined almost surely

=    0,0,z T,0 T T , x < xc(z;θ 0), 1, x,0 T,z T T , x > xc(z;θ 0). By assumption (4), Pr{X i =x c(Z i;θ 0)}= 0, soΨ(X i,Z i) is defined almost surely. The (2 + 2p)-vectorΨplays the role of the regressor vector in Sherman (1993); it is here that the non-differentiability ofHat the critical point enters. By assumptions (4) and (6), the distributio...

1993

[11] [11]

=t = 0,for everyt∈R.WriteS 3(y, δ, t) for∂S(y, δ, t)/∂t. WhenS(·,·,·) is differentiable with respect to its third argument,g 0(·) is differentiable, and E∥Ψ(Xi,Z i)∥2 <∞, the same change-of-variables argument as in Theorem 4 of Sherman (1993) gives∇ 1τ(w,θ

1993

[12] [12]

These expressions also show that assumption (7)(v) need not be imposed separately

andΨ i =Ψ(X i,Z i) for brevity, U=E h Ψi −µ 0(Hi) Ψi −µ 0(Hi) T ·S(Y i,∆ i, Hi)2 g0(Hi)2 i , V= 2 −1 E h Ψi −µ 0(Hi) Ψi −µ 0(Hi) T ·S 3(Yi,∆ i, Hi)g 0(Hi) i . These expressions also show that assumption (7)(v) need not be imposed separately. Since His the risk index, a larger value oftcorresponds to a stochastically smaller failure time, soS(y, δ,·) is de...

2007