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arxiv: 1907.04956 · v1 · pith:V7TSIQ4Ynew · submitted 2019-07-10 · 🧮 math.ST · stat.TH

Nonparametric estimation of the conditional density function with right-censored and dependent data

Xianzhu Xiong , Meijuan Ou This is my paper

Pith reviewed 2026-05-24 23:05 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords conditional density estimationright-censored dataalpha-mixinglocal linear estimatorasymptotic normalitynonparametric estimationdependent data
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The pith

Asymptotic normality of local constant and local linear estimators for the conditional density is established under right censoring and α-mixing dependence.

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

This paper establishes the asymptotic normality of local constant and local linear estimators for the conditional density function when observations are right-censored and form a stationary α-mixing sequence. The result, paired with the condition that the product of sample size and bandwidths diverges, implies consistency of the estimators. It also permits the construction of confidence intervals. The findings extend prior work by relaxing independence assumptions to allow for a broader class of dependent data. Simulations are used to check the finite-sample performance of these estimators.

Core claim

Under the assumption that the observations form a stationary α-mixing sequence, the local constant and the local linear estimators of the conditional density function with right-censored data are asymptotically normal. This normality, together with lim nh_n b_n = ∞, implies consistency and enables construction of confidence intervals. The local linear estimator result is relaxed from i.i.d. to α-mixing, and from ρ-mixing to α-mixing.

What carries the argument

The local constant and local linear kernel estimators of the conditional density function, whose asymptotic normality is derived under α-mixing and right-censoring.

Load-bearing premise

The data observations must form a stationary sequence with α-mixing coefficients that decay sufficiently fast for the central limit theorem to apply.

What would settle it

Generating data from a process whose dependence violates the α-mixing rate requirement and checking whether the estimators deviate from the predicted normal limiting distribution.

read the original abstract

In this paper, we study the local constant and the local linear estimators of the conditional density function with right-censored data which exhibit some type of dependence. It is assumed that the observations form a stationary $\alpha-$mixing sequence. The asymptotic normality of the two estimators is established, which combined with the condition that $\lim\limits_{n\rightarrow\infty}nh_nb_n=\infty$ implies the consistency of the two estimators and can be employed to construct confidence intervals for the conditional density function. The result on the local linear estimator of the conditional density function in Kim et al. (2010) is relaxed from the i.i.d. assumption to the $\alpha-$mixing setting, and the result on the local linear estimator of the conditional density function in Spierdijk (2008) is relaxed from the $\rho$-mixing assumption to the $\alpha-$mixing setting. Finite sample behavior of the estimators is investigated by simulations.

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 develops local constant and local linear kernel estimators for the conditional density function under right censoring when observations form a stationary α-mixing sequence. It establishes asymptotic normality of both estimators; combined with the bandwidth condition lim n→∞ n h_n b_n = ∞ this yields consistency and permits construction of confidence intervals. The results relax the i.i.d. assumption in Kim et al. (2010) and the ρ-mixing assumption in Spierdijk (2008) to the weaker α-mixing setting. Finite-sample behavior is examined via simulation.

Significance. If the asymptotic normality derivations hold, the work meaningfully weakens the dependence assumptions under which nonparametric conditional density estimation is justified for censored data. This extension is useful for applications involving dependent survival data and supplies the theoretical basis for interval estimation that was previously unavailable under α-mixing.

major comments (1)
  1. [Abstract / main theorems] The central claim rests on the stationary α-mixing assumption (abstract). The manuscript must explicitly state the decay rate required of the mixing coefficients α(n) so that the covariance terms arising from the product-limit estimator for the censoring distribution remain negligible in the CLT; without this rate the normality result cannot be verified.
minor comments (2)
  1. [Abstract] Clarify the precise definitions of the bandwidth sequences h_n (for the covariate) and b_n (for the response) and their roles in the local constant versus local linear estimators.
  2. [Simulation study] The simulation section should report the specific mixing coefficients or dependence strength used to generate the dependent samples so that readers can assess how close the finite-sample results are to the theoretical regime.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comment. We address the point below.

read point-by-point responses

  1. Referee: [Abstract / main theorems] The central claim rests on the stationary α-mixing assumption (abstract). The manuscript must explicitly state the decay rate required of the mixing coefficients α(n) so that the covariance terms arising from the product-limit estimator for the censoring distribution remain negligible in the CLT; without this rate the normality result cannot be verified.

    Authors: We agree that an explicit decay rate on the mixing coefficients is required for the covariance terms involving the product-limit estimator to be negligible in the central limit theorem. The current version states the α-mixing assumption in the abstract and assumptions but does not specify the rate. In the revised manuscript we will add a concrete condition (for example, α(n) = O(n^{-r}) for r > 1 + δ with δ > 0 chosen to dominate the bandwidth terms) to the list of assumptions and restate it in the theorems so that the CLT holds as claimed. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives asymptotic normality for local constant and local linear conditional density estimators under stationary α-mixing, right censoring, and the bandwidth condition lim nh_n b_n = ∞. These derivations follow from standard kernel bias-variance decompositions combined with established α-mixing covariance bounds and product-limit censoring adjustments; they do not reduce to any quantity defined in terms of the target result itself. No self-citations appear in the load-bearing steps, no fitted parameters are relabeled as predictions, and the relaxations of Kim et al. (2010) and Spierdijk (2008) are external extensions rather than internal loops. The argument is therefore self-contained against external probabilistic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the α-mixing condition and standard kernel estimation assumptions for censored data; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Observations form a stationary α-mixing sequence
    Stated explicitly in the abstract as the dependence structure required for the asymptotic normality proof.
  • domain assumption Right-censoring mechanism is independent of the variables of interest
    Implicit in all right-censored density estimation; required for the estimators to target the correct conditional density.

pith-pipeline@v0.9.0 · 5692 in / 1340 out tokens · 18386 ms · 2026-05-24T23:05:54.330058+00:00 · methodology

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Reference graph

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