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arxiv: 2605.17052 · v1 · pith:OSEM2IWDnew · submitted 2026-05-16 · 💰 econ.EM

Asymptotic Variance Theory for Trimmed Least Squares and Trimmed Least Absolute Deviations in Censored Panel Models with Fixed Effects

Denis Chetverikov , Jesper R.-V.~S{\o}rensen , Bo Honor\'e This is my paper

Pith reviewed 2026-05-20 15:22 UTC · model grok-4.3

classification 💰 econ.EM
keywords trimmed least squarestrimmed least absolute deviationscensored panel datafixed effectsasymptotic varianceasymptotic normalitybootstrap variance estimator
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The pith

Corrected asymptotic variance formulas for trimmed least squares and trimmed least absolute deviations estimators are derived for censored two-period panel models with fixed effects.

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

This paper re-examines the asymptotic theory for trimmed least squares and trimmed least absolute deviations estimators in censored panel data models that include fixed effects. It demonstrates that the variance formulas appearing in earlier work depend on regularity conditions that were not fully stated or verified in the original derivations. For trimmed least squares the paper identifies an implicit restriction on regressor differences, supplies the proper Hessian, proves asymptotic normality without that restriction, and gives a consistent plug-in variance estimator. For trimmed least absolute deviations it adds a missing conditional-probability term to the variance expression and states the continuity conditions needed for asymptotic normality, while also providing a tuning-parameter-free bootstrap variance estimator.

Core claim

The published asymptotic variance formulas for the trimmed least squares and trimmed least absolute deviations estimators rely on additional regularity conditions not fully stated in the original analysis. For trimmed least squares the published Hessian requires that the regressor-difference index vanish only when the regressor difference itself is zero, a restriction violated for example by a zero parameter vector; the paper derives the correct Hessian, establishes asymptotic normality without imposing this restriction, and obtains a consistent plug-in variance estimator. For trimmed least absolute deviations the published variance formula omits a conditional-probability term, and the paper

What carries the argument

The corrected Hessian matrix for the trimmed least squares estimator together with the adjusted asymptotic variance formula that includes the omitted conditional-probability term for the trimmed least absolute deviations estimator.

If this is right

  • Researchers can apply the corrected Hessian for trimmed least squares to obtain accurate standard errors without the previously implicit restriction on regressor differences.
  • The Hessian estimator originally proposed for trimmed least squares remains consistent for the corrected asymptotic variance.
  • The corrected variance for trimmed least absolute deviations now accounts for the previously omitted conditional-probability term under the stated continuity conditions.
  • A tuning-parameter-free bootstrap provides a practical way to estimate the variance of the trimmed least absolute deviations estimator.
  • Asymptotic normality of both estimators holds in censored two-period panel models once the additional regularity conditions are imposed.

Where Pith is reading between the lines

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

  • Empirical applications that rely on these trimmed estimators for censored panel data may obtain more reliable confidence intervals once the corrected variances are used.
  • The same regularity gaps identified here could be checked in extensions to longer panels or to models with different censoring patterns.
  • Simulation studies comparing the bootstrap variance estimator to the plug-in estimator for trimmed least absolute deviations would help assess finite-sample performance under varying continuity conditions.
  • Related robust estimators in fixed-effects models with censoring or truncation may benefit from parallel re-derivations of their asymptotic variances.

Load-bearing premise

The additional continuity conditions on the underlying distributions that are required for asymptotic normality of the trimmed least absolute deviations estimator and for the validity of the corrected variance formula.

What would settle it

A Monte Carlo experiment that generates data from a censored two-period panel model with a zero parameter vector and checks whether the published Hessian differs from the corrected Hessian or whether asymptotic normality of the trimmed least absolute deviations estimator fails when the continuity conditions are violated.

Figures

Figures reproduced from arXiv: 2605.17052 by Bo Honor\'e, Denis Chetverikov, Jesper R.-V.~S{\o}rensen.

Figure 1
Figure 1. Figure 1: Graph of θ 7→ G(θ) in the counterexample −3 −2 −1 1 2 3 −0.3 −0.1 0.1 0.3 θ < 0 θ0 = 0 φ(0) θ > 0 −φ(0) (0, 0) θ G(θ) where the first equality follows from (6), the second from (the here) identical distributions of Y1 and Y2, the third from noting that Y1 is a standard normal random variable truncated from below at zero, and the fourth from integration by parts. The graph of G is shown in [PITH_FULL_IMAGE… view at source ↗
Figure 2
Figure 2. Figure 2: PDFs of Scaled TLAD Estimates (Solid) and Best Normal Approximation (Dashed) [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
read the original abstract

We study inference using trimmed least squares (TLS) and trimmed least absolute deviations (TLAD) estimators of \citet{honore_trimmed_1992} in censored two-period panel-data models with fixed effects. We show that the published asymptotic variance formulas rely on additional regularity conditions that are not fully stated in the original analysis. For TLS, the published Hessian formula requires that the regressor-difference index vanish only when the regressor difference itself is zero, a restriction not explicitly stated in the original paper and violated, for instance, with a zero parameter vector. We derive the correct Hessian, establish asymptotic normality without imposing this restriction, and obtain a consistent plug-in variance estimator. We also show that the Hessian estimator proposed in \citet{honore_trimmed_1992} {\em is} actually consistent for the {\em correct} TLS asymptotic variance. For TLAD, we show that the published variance formula omits a conditional-probability term and that asymptotic normality requires additional continuity conditions. Under these conditions, we derive the corrected asymptotic variance and provide a tuning-parameter-free bootstrap variance estimator.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper studies inference for the trimmed least squares (TLS) and trimmed least absolute deviations (TLAD) estimators of Honore (1992) in censored two-period panel models with fixed effects. It identifies unstated regularity conditions in the original asymptotic variance formulas, derives a corrected Hessian for TLS that permits asymptotic normality without the regressor-difference index restriction, shows that the original Honore Hessian estimator remains consistent for the corrected TLS variance, corrects the TLAD variance formula by adding a conditional-probability term, and supplies a tuning-parameter-free bootstrap variance estimator under additional continuity conditions.

Significance. If the derivations hold, the paper supplies practically useful corrections to the asymptotic theory for two important robust estimators in censored panel data. The demonstration that the published Hessian estimator is consistent for the corrected TLS variance is a non-obvious and immediately applicable result. The bootstrap proposal for TLAD removes the need for tuning parameters, which is a clear implementation advantage. These contributions strengthen the reliability of inference in a setting where censoring and fixed effects are common.

major comments (1)
  1. [Theorem statements for TLAD] The abstract states that asymptotic normality for TLAD requires additional continuity conditions on the underlying distributions; the manuscript should explicitly state these conditions in the theorem statement (likely Theorem 4 or 5) and verify that they are strictly weaker than those implicitly used in Honore (1992).
minor comments (2)
  1. [Notation and definitions] Notation for the trimming indicator and the conditional probability term in the TLAD variance should be introduced once and used consistently across sections.
  2. [Simulation section] The paper should include a brief Monte Carlo illustration showing that the corrected variance estimators achieve nominal coverage where the original formulas do not, to complement the theoretical results.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation of minor revision. We address the single major comment below and will revise the manuscript to incorporate the suggestion.

read point-by-point responses

  1. Referee: [Theorem statements for TLAD] The abstract states that asymptotic normality for TLAD requires additional continuity conditions on the underlying distributions; the manuscript should explicitly state these conditions in the theorem statement (likely Theorem 4 or 5) and verify that they are strictly weaker than those implicitly used in Honore (1992).

    Authors: We agree that the continuity conditions should be stated explicitly. In the revised manuscript we will add the precise continuity requirements directly into the statement of Theorem 5. We will also insert a short remark after the theorem that verifies these conditions are strictly weaker than those implicitly used in Honore (1992): the original paper relies on continuity to ensure the score and Hessian behave as if the conditional-probability term is zero, whereas our conditions allow a non-zero conditional-probability term while still delivering asymptotic normality under weaker smoothness on the conditional distribution of the latent variable. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are independent

full rationale

The paper identifies gaps in the regularity conditions of the cited Honoré (1992) analysis and supplies new derivations for the correct TLS Hessian, asymptotic normality under weaker restrictions, and a corrected TLAD variance formula that includes the omitted conditional-probability term. These results are obtained by direct expansion of the estimators' influence functions and Hessian expressions under explicitly stated continuity and support conditions; the proofs do not reduce any target quantity to a previously fitted parameter or to a self-referential definition. Although the author list overlaps with the 1992 reference, the load-bearing steps consist of fresh analytic work rather than an appeal to the original (flawed) formulas as justification. The bootstrap variance estimator for TLAD is likewise constructed without tuning parameters or data-dependent re-use of the same fitted objects. Consequently the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard regularity conditions for asymptotic theory in panel models plus newly stated continuity conditions for TLAD; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Standard regularity conditions for asymptotic normality of trimmed estimators in censored panel data
    Invoked to establish the asymptotic normality results for both TLS and TLAD.
  • domain assumption Additional continuity conditions on the distributions for the TLAD estimator
    Explicitly required in the abstract for the corrected TLAD asymptotic variance and normality.

pith-pipeline@v0.9.0 · 5744 in / 1423 out tokens · 71253 ms · 2026-05-20T15:22:55.705091+00:00 · methodology

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Reference graph

Works this paper leans on

67 extracted references · 67 canonical work pages

  1. [1]

    Bootstrap standard error estimates and inference,

    [58] Hahn, J. and Z. Liao(2021): “Bootstrap standard error estimates and inference,”Econo- metrica, 89, 1963–1977. [24] Honor´e, B. and L. Hu(2017): “Poor (wo)man’s bootstrap,”Econometrica, 1277–1301. [23] Honor´e, B. E.(1992): “Trimmed LAD and least squares estimation of truncated and censored regression models with fixed effects,”Econometrica, 533–565. ...

    work page 2021

  2. [2]

    The function ˙mtls 1 (·,y) defined in (6) is Lipschitz continuous with Lipschitz constant equal to one regardless ofy∈[0,∞)×[0,∞), soM(·,w) inherits these properties via Jensen’s inequality (conditional onW=w)

    work page

  3. [3]

    From the expression (28) forM(·,w), continuity ofF ε|w(·) and the fundamental theo- rem of calculus (and chain rule) imply thatM(·,w) is differentiable at everyt̸= 0 with the 33 derivatives taking the form in (29)

    work page

  4. [4]

    To show the semi-differentiability ofM(·,w) at zero, consider first a sequence{t m}∞ m=1 in (0,∞) converging to zero from above (t m →0 +). Fixϵ >0. Continuity ofF ε|w(·) ensures that there is aδ >0 such that|u−v 1|⩽δimplies|F ε|w(u)−F ε|w(v1)|⩽ϵ. Ast m →0 +, for mlarge enough we have 0< t m ⩽δ, so that 1 tm Z v1+tm v1 Fε|w(u) du−F ε|w(v1) = 1 tm Z v1+tm ...

    work page

  5. [5]

    subgraph,

    If ∆x⊤θ0 ̸= 0, thenM(·,w) is differentiable att= ∆x ⊤θ0 by Item 2, and (31) follows by substitutingt= ∆x ⊤θ0 into (29). If instead ∆x ⊤θ0 = 0, thenv 1(w) =v 2(w) and the left and right derivatives in (30) coincide, soM(·,w) is differentiable att= ∆x ⊤θ0 = 0 with derivative given in (31). A.2 Proof of Theorem 2.2 Proof of Theorem 2.2.As in Honor´ e (1992),...

    work page 1992

  6. [6]

    Appealing to the GLDCT, stacking over the coordinatesj∈[K], we conclude that ∇L(θ0 +τ mϑm)− ∇L(θ 0) τm →E h ¨ℓ11 ∆X ⊤θ0,W ∆X∆X ⊤ i ϑ

    Lemma S1.2.3 shows that ˙ℓ1(·,w) is differentiable att= ∆x⊤θ0, thus yielding the pointwise convergence fm(w)→1{w∈ W} ¨ℓ11 ∆x⊤θ0,w ∆xj∆x⊤ϑ= : f(w). Appealing to the GLDCT, stacking over the coordinatesj∈[K], we conclude that ∇L(θ0 +τ mϑm)− ∇L(θ 0) τm →E h ¨ℓ11 ∆X ⊤θ0,W ∆X∆X ⊤ i ϑ. Since the limit exists for everyϑ∈R K, is linear inϑ, and is independent of ...

    work page 1973

  7. [7]

    dy∗ 1 dy∗ 2 = 2τ −1 m Z R 1{y∗ 2 >0} Z R 1{y∗ 1 ⩾0}1{y ∗ 1 −y ∗ 2 ∈[t, t+τ m]}fY ∗|w(y∗ 1, y∗

    work page

  8. [8]

    dy∗ 1 dy∗ 2 = 2τ −1 m Z R 1{y∗ 2 >0} Z R 1{u+y ∗ 2 ⩾0}1{u∈[t, t+τ m]}fY ∗|w(u+y ∗ 2, y∗

    work page

  9. [9]

    du dy∗ 2 = Z R 21{y∗ 2 >0}τ −1 m "Z [t,t+τm] 1{u⩾−y ∗ 2}fY ∗|w(u+y ∗ 2, y∗

    work page

  10. [10]

    du # dy∗ 2,(41) where we have used non-negativity to invoke Tonelli’s theorem, absolute continuity to modify the inner integral on a Lebesgue null set inR(which changes withy ∗ 2), and the change of variablesu :=y ∗ 1 −y ∗

    work page

  11. [11]

    Consider the measure space (R,B, λ) and define the (outer integrand) functionf m by fm(y∗

    work page

  12. [12]

    := 21{y∗ 2 >0}τ −1 m Z [t,t+τm] 1{u⩾−y ∗ 2}fY ∗|w(u+y ∗ 2, y∗

    work page

  13. [13]

    Thenf m is non-negative and bounded from above byg m defined by gm(y∗

    du. Thenf m is non-negative and bounded from above byg m defined by gm(y∗

    work page

  14. [14]

    := 2τ −1 m Z [t,t+τm] fY ∗|w(u+y ∗ 2, y∗

    work page

  15. [15]

    Assumption 3.5 implies thatf ∆Y ∗|w(·) =f ∆ε|w(· −∆x ⊤θ0) is bounded by a constantC, so Tonelli’s theorem yields Z R gm(y∗

    du. Assumption 3.5 implies thatf ∆Y ∗|w(·) =f ∆ε|w(· −∆x ⊤θ0) is bounded by a constantC, so Tonelli’s theorem yields Z R gm(y∗

    work page

  16. [16]

    dy∗ 2 = 2τ −1 m Z [t,t+τm] Z R fY ∗|w(u+y ∗ 2, y∗

    work page

  17. [17]

    Sincef Y ∗|w(·,·) =f ε|w(·−a−x ⊤ 1 θ0,·−a−x ⊤ 2 θ0) is continuous (Assumption 3.7) and1{·⩾−y ∗ 2}isrightcontinuous, both inner integrands u7→f Y ∗|w(u+y ∗ 2, y∗

    dy∗ 2 du = 2τ −1 m Z [t,t+τm] f∆Y ∗|w(u) du⩽C, showing thatg m (and thusf m) is integrable. Sincef Y ∗|w(·,·) =f ε|w(·−a−x ⊤ 1 θ0,·−a−x ⊤ 2 θ0) is continuous (Assumption 3.7) and1{·⩾−y ∗ 2}isrightcontinuous, both inner integrands u7→f Y ∗|w(u+y ∗ 2, y∗

    work page

  18. [18]

    andu7→1{u⩾−y ∗ 2}fY ∗|w(u+y ∗ 2, y∗

    work page

  19. [19]

    Asτ m →0 +, it follows from right continuity that both gm(y∗ 2)→2f Y ∗|w(t+y ∗ 2, y∗

    are right continuous for each 50 y∗ 2 ∈R. Asτ m →0 +, it follows from right continuity that both gm(y∗ 2)→2f Y ∗|w(t+y ∗ 2, y∗

    work page

  20. [20]

    =: g(y ∗ 2), pointwise iny ∗ 2 ∈Rand fm(y∗ 2)→21{y ∗ 2 >0}1{t⩾−y ∗ 2}fY ∗|w(t+y ∗ 2, y∗

    work page

  21. [21]

    Also, sincef ∆Y ∗|w(·) =f ∆ε|w(· −∆x ⊤θ0) is continuous (Assumption 3.7), Z R gm(y∗

    =: f(y ∗ 2) pointwise iny ∗ 2 ∈R. Also, sincef ∆Y ∗|w(·) =f ∆ε|w(· −∆x ⊤θ0) is continuous (Assumption 3.7), Z R gm(y∗

    work page

  22. [22]

    dy∗ 2 = 2τ −1 m Z [t,t+τm] f∆Y ∗|w(u) du →2f ∆Y ∗|w(t) = 2 Z R fY ∗|w(t+y ∗ 2, y∗

    work page

  23. [23]

    It thus follows from the Generalized Lebesgue Dominated Convergence Theorem (GLDCT) in Theorem S2.1 thatfis integrable and R fm dλ→ R fdλ

    dy∗ 2 <∞. It thus follows from the Generalized Lebesgue Dominated Convergence Theorem (GLDCT) in Theorem S2.1 thatfis integrable and R fm dλ→ R fdλ. The latter convergence translates to Ma(t+τ m,w)−M a(t,w) τm →2 Z R 1{y∗ 2 >0}1{t⩾−y ∗ 2}fY ∗|w(t+y ∗ 2, y∗

    work page

  24. [24]

    dy∗ 2, showing thatM a(·,w) isright differentiableattwithright derivative ˙Ma,1+(t,w) = 2 Z +∞ max{0,−t} fY ∗|w(t+y ∗ 2, y∗

    work page

  25. [25]

    dy∗ 2. Part a,leftdifferentiability:Express the difference quotient as Ma(t−τ m,w)−M a(t,w) (−τm) =−2τ −1 m Ew 1{Y ∗ 1 >0}1{Y ∗ 2 >0} 1{∆Y ∗ ⩽t−τ m} −1{∆Y ∗ ⩽t} = Z R 21{y∗ 2 >0}τ −1 m "Z [t−τm,t] 1{u⩾−y ∗ 2}fY ∗|w(u+y ∗ 2, y∗

    work page

  26. [26]

    du # dy∗ 2,(42) = Z R 21{y∗ 2 >0}τ −1 m "Z [t−τm,t] 1{u >−y ∗ 2}fY ∗|w(u+y ∗ 2, y∗

    work page

  27. [27]

    du # dy∗ 2, where (42) follows by the same argument as that leading to (41), with (t, t+τ m) replaced by (t−τ m, t). We then proceed as with the proof of right differentiability, where we now use 51 left continuity of1{·>−y ∗ 2}instead of right continuity of1{·⩾−y ∗ 2}to conclude that Ma(t−τ m,w)−M a(t,w) (−τm) →2 Z R 1{y∗ 2 >0}1{t >−y ∗ 2}fY ∗|w(t+y ∗ 2, y∗

    work page

  28. [28]

    Hence,M a(·,w) isleft differentiableattwithleft derivative ˙Ma,1−(t,w) = 2 Z +∞ max{0,−t} fY ∗|w(t+y ∗ 2, y∗

    dy∗ 2. Hence,M a(·,w) isleft differentiableattwithleft derivative ˙Ma,1−(t,w) = 2 Z +∞ max{0,−t} fY ∗|w(t+y ∗ 2, y∗

    work page

  29. [29]

    dy∗ 2. Part a,two-sideddifferentiability:The left and right derivatives exist and agree for all t∈R, soM a(·,w) is differentiable with derivative given by ˙Ma,1(t,w) = 2 Z +∞ max{0,−t} fY ∗|w(t+y ∗ 2, y∗

    work page

  30. [30]

    In case thatt <0, using the change of variablesz :=t+y ∗ 2, we havey ∗ 2 =z−t, and the range of integration becomes [0,+∞)

    dy∗ 2. In case thatt <0, using the change of variablesz :=t+y ∗ 2, we havey ∗ 2 =z−t, and the range of integration becomes [0,+∞). We can therefore express this derivative in the (more symmetric looking) form ˙Ma,1(t,w) = 2 Z +∞ 0 fY ∗|w z+ max{0, t}, z−min{0, t} dz. Part b,rightdifferentiability:Express the function as Mb(t,w) = E w 1{Y ∗ 1 >0}1{Y ∗ 2 ⩽0...

    work page

  31. [31]

    dy∗ 1 dy∗ 2 = Z R 1{y∗ 2 ⩽0} τ −1 m Z R 1{y∗ 1 ⩾0}1{y ∗ 1 ∈[t, t+τ m]}fY ∗|w(y∗ 1, y∗

    work page

  32. [32]

    dy∗ 1 dy∗ 2 = Z R 1{y∗ 2 ⩽0} " τ −1 m Z [t,t+τm] 1{y∗ 1 ⩾0}f Y ∗|w(y∗ 1, y∗

    work page

  33. [33]

    dy∗ 1 # dy∗ 2.(43) 52 Consider the measure space (R,B, λ) and define the (outer integrand) functionf m by fm(y∗

    work page

  34. [34]

    :=1{y ∗ 2 ⩽0}τ −1 m Z [t,t+τm] 1{y∗ 1 ⩾0}f Y ∗|w(y∗ 1, y∗

    work page

  35. [35]

    Thenf m is non-negative and bounded from above byg m defined by gm(y∗

    dy∗ 1. Thenf m is non-negative and bounded from above byg m defined by gm(y∗

    work page

  36. [36]

    :=τ −1 m Z [t,t+τm] fY ∗|w(y∗ 1, y∗

    work page

  37. [37]

    Asf Y ∗ 1 |w(·) =f ε1|w(· −a−x ⊤ 1 θ0) is assumed bounded by a constantC(Assumption 3.5), Tonelli’s theorem yields Z R gm(y∗

    dy∗ 1. Asf Y ∗ 1 |w(·) =f ε1|w(· −a−x ⊤ 1 θ0) is assumed bounded by a constantC(Assumption 3.5), Tonelli’s theorem yields Z R gm(y∗

    work page

  38. [38]

    dy∗ 2 =τ −1 m Z [t,t+τm] Z R fY ∗|w(y∗ 1, y∗

    work page

  39. [39]

    dy∗ 2 dy∗ 1 =τ −1 m Z [t,t+τm] fY ∗ 1 |w(y∗

    work page

  40. [40]

    Sincef Y ∗|w(·,·) =f ε|w(·−a−x ⊤ 1 θ0,·−a−x ⊤ 2 θ0) is continuous (Assumption 3.7) and1{·⩾0}isrightcontinuous, both inner integrands y∗ 1 7→f Y ∗|w(y∗ 1, y∗

    dy∗ 1 ⩽C, showing thatg m (and thusf m) is integrable. Sincef Y ∗|w(·,·) =f ε|w(·−a−x ⊤ 1 θ0,·−a−x ⊤ 2 θ0) is continuous (Assumption 3.7) and1{·⩾0}isrightcontinuous, both inner integrands y∗ 1 7→f Y ∗|w(y∗ 1, y∗

    work page

  41. [41]

    andy ∗ 1 7→1{y ∗ 1 ⩾0}f Y ∗|w(y∗ 1, y∗

    work page

  42. [42]

    Asτ m →0 +, it follows from right continuity that both gm(y∗ 2)→f Y ∗|w(t, y∗

    are right continuous for eachy ∗ 2 ∈R. Asτ m →0 +, it follows from right continuity that both gm(y∗ 2)→f Y ∗|w(t, y∗

    work page

  43. [43]

    =: g(y ∗ 2), pointwise iny ∗ 2 ∈Rand fm(y∗ 2)→1{y ∗ 2 ⩽0}1{t⩾0}f Y ∗|w(t, y∗

    work page

  44. [44]

    Also, sincef Y ∗ 1 |w(·) =f ε1|w(· −a−x ⊤ 1 θ0) is continuous (Assumption 3.7), Z R gm(y∗

    =: f(y ∗ 2) pointwise iny ∗ 2 ∈R. Also, sincef Y ∗ 1 |w(·) =f ε1|w(· −a−x ⊤ 1 θ0) is continuous (Assumption 3.7), Z R gm(y∗

    work page

  45. [45]

    dy∗ 2 =τ −1 m Z [t,t+τm] fY ∗ 1 |w(y∗

    work page

  46. [46]

    dy∗ 1 →f Y ∗ 1 |w(t) = Z R fY ∗|w(t, y∗

    work page

  47. [47]

    The GLDCT (Theorem S2.1) therefore shows thatfis integrable and R fm dλ→ R fdλ, the latter convergence meaning that Mb(t+τ m,w)−M b(t,w) τm → Z R 1{y∗ 2 ⩽0}1{t⩾0}f Y ∗|w(t, y∗

    dy∗ 2 <∞. The GLDCT (Theorem S2.1) therefore shows thatfis integrable and R fm dλ→ R fdλ, the latter convergence meaning that Mb(t+τ m,w)−M b(t,w) τm → Z R 1{y∗ 2 ⩽0}1{t⩾0}f Y ∗|w(t, y∗

    work page

  48. [48]

    53 Hence,M b(·,w) isright differentiableattwithright derivative ˙Mb,1+(t,w) =1{t⩾0} Z 0 −∞ fY ∗|w(t, z) dz

    dy∗ 2. 53 Hence,M b(·,w) isright differentiableattwithright derivative ˙Mb,1+(t,w) =1{t⩾0} Z 0 −∞ fY ∗|w(t, z) dz. Part b,leftdifferentiability:Express the difference quotient as Mb(t−τ m,w)−M b(t,w) (−τm) = (−τm)−1Ew 1{Y ∗ 1 >0}1{Y ∗ 2 ⩽0} 1{Y ∗ 1 ⩽t−τ m} −1{Y ∗ 1 ⩽t} = Z R 1{y∗ 2 ⩽0} " τ −1 m Z [t−τm,t] 1{y∗ 1 ⩾0}f Y ∗|w(y∗ 1, y∗

    work page

  49. [49]

    dy∗ 1 # dy∗ 2 (44) = Z R 1{y∗ 2 ⩽0} " τ −1 m Z [t−τm,t] 1{y∗ 1 >0}f Y ∗|w(y∗ 1, y∗

    work page

  50. [50]

    dy∗ 1 # dy∗ 2, where (44) follows by the same argument as that leading to (43), with (t, t+τ m) replaced by (t−τ m, t). We then proceed as with the proof of right differentiability, where we now use left continuity of1{·>0}instead of right continuity of1{·⩾0}to conclude that Mb(t−τ m,w)−M b(t,w) (−τm) → Z R 1{y∗ 2 ⩽0}1{t >0}f Y ∗|w(t, y∗

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  51. [51]

    Hence,M b(·,w) isleft differentiableattwithleft derivative ˙Mb,1−(t,w) =1{t >0} Z 0 −∞ fY ∗|w(t, z) dz

    dy∗ 2. Hence,M b(·,w) isleft differentiableattwithleft derivative ˙Mb,1−(t,w) =1{t >0} Z 0 −∞ fY ∗|w(t, z) dz. Part c,rightdifferentiability:Express the function as Mc(t,w) = E w 1{Y ∗ 1 ⩽0}1{Y ∗ 2 >0} 1−1{Y ∗ 2 ⩽−t} . 54 Then using absolute continuity, non-negativity and Tonelli’s theorem, we get Mc(t+τ m,w)−M c(t,w) τm =τ −1 m Ew 1{Y ∗ 1 ⩽0}1{Y ∗ 2 >0} ...

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  52. [52]

    dy∗ 2 dy∗ 1 = Z R 1{y∗ 1 ⩽0} " τ −1 m Z [−t−τm,−t] 1{y∗ 2 >0}f Y ∗|w(y∗ 1, y∗

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  53. [53]

    dy∗ 2 # dy∗ 1.(45) Consider the measure space (R,B, λ) and define the (outer integrand) functionf m by fm(y∗

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  54. [54]

    :=1{y ∗ 1 ⩽0}τ −1 m Z [−t−τm,−t] 1{y∗ 2 >0}f Y ∗|w(y∗ 1, y∗

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  55. [55]

    Thenf m is non-negative and bounded from above byg m defined by gm(y∗

    dy∗ 2. Thenf m is non-negative and bounded from above byg m defined by gm(y∗

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  56. [56]

    :=τ −1 m Z [−t−τm,−t] fY ∗|w(y∗ 1, y∗

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  57. [57]

    Asf Y ∗ 2 |w(·) =f ε2|w(· −a−x ⊤ 2 θ0) is bounded byC(Assumption 3.5), Tonelli’s theorem yields Z R gm(y∗

    dy∗ 2. Asf Y ∗ 2 |w(·) =f ε2|w(· −a−x ⊤ 2 θ0) is bounded byC(Assumption 3.5), Tonelli’s theorem yields Z R gm(y∗

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  58. [58]

    dy∗ 1 =τ −1 m Z [−t−τm,−t] Z R fY ∗|w(y∗ 1, y∗

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  59. [59]

    dy∗ 1 dy∗ 2 =τ −1 m Z [−t−τm,−t] fY ∗ 2 |w(y∗

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  60. [60]

    Sincef Y ∗|w(·,·) =f ε|w(·−a−x ⊤ 1 θ0,·−a−x ⊤ 2 θ0) is continuous (Assumption 3.7) and1{·>0}isleftcontinuous, both inner integrandsy ∗ 2 7→ fY ∗|w(y∗ 1, y∗

    dy∗ 2 ⩽C, showing thatg m (and thusf m) is integrable. Sincef Y ∗|w(·,·) =f ε|w(·−a−x ⊤ 1 θ0,·−a−x ⊤ 2 θ0) is continuous (Assumption 3.7) and1{·>0}isleftcontinuous, both inner integrandsy ∗ 2 7→ fY ∗|w(y∗ 1, y∗

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  61. [61]

    andy ∗ 2 7→1{y ∗ 2 >0}f Y ∗|w(y∗ 1, y∗

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  62. [62]

    are left continuous for eachy ∗ 1 ∈R. As τm →0 +, it follows from left continuity that both gm(y∗ 1)→f Y ∗|w(y∗ 1,−t) = : g(y ∗ 1) pointwise iny ∗ 1 ∈Rand fm(y∗ 1)→1{y ∗ 1 ⩽0}1{−t >0}f Y ∗|w(y∗ 1,−t) = : f(y ∗ 1) pointwise iny ∗ 1 ∈R. Also, sincef Y ∗ 2 |w(·) =f ε2|w(· −a−x ⊤ 2 θ0) is continuous (Assumption 55 3.7), Z R gm(y∗

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  63. [63]

    dy∗ 1 =τ −1 m Z [−t−τm,−t] fY ∗ 2 |w(y∗

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  64. [64]

    dy∗ 2 →f Y ∗ 2 |w(−t) = Z R fY ∗|w(y∗ 1,−t) dy ∗ 1 = Z R g(y ∗

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  65. [65]

    dy∗ 1 <∞. The GLDCT (Theorem S2.1) now shows thatfintegrable and R fm dλ→ R fdλ, the latter convergence meaning that Mc(t+τ m,w)−M c(t,w) τm → Z R 1{y∗ 1 ⩽0}1{−t >0}f Y ∗|w(y∗ 1,−t) dy ∗ 1, showing thatM c(·,w) isright differentiableattwithright derivative ˙Mc,1+(t,w) =1{t <0} Z 0 −∞ fY ∗|w(z,−t) dz. Part c,leftdifferentiability:Express the difference quo...

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  66. [66]

    dy∗ 2 # dy∗ 1 (46) = Z R 1{y∗ 1 ⩽0} " τ −1 m Z [−t,−t+τm] 1{y∗ 2 ⩾0}f Y ∗|w(y∗ 1, y∗

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  67. [67]

    where (46) follows by the same argument as that leading to (45), with (−t−τ m,−t) replaced by (−t,−t+τ m)

    dy∗ 2 # dy∗ 1. where (46) follows by the same argument as that leading to (45), with (−t−τ m,−t) replaced by (−t,−t+τ m). We then proceed as with the proof of right differentiability, where we now use right continuity of1{·⩾0}instead of left continuity of1{·>0}to conclude that Mc(t−τ m,w)−M c(t,w) (−τm) → Z R 1{y∗ 1 ⩽0}1{−t⩾0}f Y ∗|w(y∗ 1,−t) dy ∗ 1. Henc...

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