[Paper Review] A strong uniform convergence rate of a kernel conditional quantile estimator under random left-truncation and dependent data

Elias Ould-Saı̈d, Djabrane Yahia+1 more|ArXiv.org|Oct 7, 2008

Statistical Methods and Inference16 references3 citations

TL;DR

This paper establishes a strong uniform convergence rate for a kernel-based conditional quantile estimator under random left-truncation and α-mixing dependence. By extending prior i.i.d. results, it proves almost sure uniform convergence of the estimator at a rate combining logarithmic and bandwidth terms, under weak mixing conditions, enabling robust nonparametric quantile inference in dependent, truncated data settings such as survival analysis or econometrics.

ABSTRACT

In this paper we study some asymptotic properties of the kernel conditional quantile estimator with randomly left-truncated data which exhibit some kind of dependence. We extend the result obtained by Lemdani, Ould-Saïd and Poulin [16] in the iid case. The uniform strong convergence rate of the estimator under strong mixing hypothesis is obtained.

Motivation & Objective

• To extend the strong uniform consistency of kernel conditional quantile estimators to settings with random left-truncation and dependent data.
• To establish a non-asymptotic convergence rate for the estimator under α-mixing dependence, a weak form of dependence common in time series.
• To provide theoretical guarantees for nonparametric quantile regression in truncated, dependent data, relevant for survival analysis and econometrics.
• To generalize prior results in the i.i.d. case (Lemdani et al., 2008) to a more general, dependent data model.
• To support robust inference in the presence of heavy-tailed errors and outliers via conditional quantile estimation under truncation.

Proposed method

• Proposes a kernel conditional quantile estimator based on a weighted empirical distribution function, adjusted for truncation via inverse probability weighting.
• Uses a bivariate kernel estimator for the joint density of (X, Y), with bandwidth h, and conditions on the kernel function (e.g., bounded, compact support).
• Applies a strong mixing condition (α-mixing) to model weak dependence in the data, with mixing coefficients decaying at a polynomial rate.
• Derives the bias and variance of the estimator using Taylor expansions and moment bounds, accounting for truncation and dependence.
• Employs Borel-Cantelli lemma and maximal inequalities to control the uniform deviation of the empirical process.
• Combines bias and variance bounds to derive a uniform convergence rate of the form O_p( max{ sqrt(log n / (n h)) , h^2 } ) under α-mixing.

Experimental results

Research questions

• RQ1 What is the rate of strong uniform convergence of the kernel conditional quantile estimator under random left-truncation and α-mixing dependence?
• RQ2 How does the dependence structure (α-mixing) affect the convergence rate compared to the i.i.d. case?
• RQ3 Can the uniform consistency and convergence rate be established under weaker moment and mixing conditions than those in prior work?
• RQ4 What is the impact of truncation on the bias and variance of the kernel estimator in a dependent setting?
• RQ5 How does the estimator perform in terms of uniform deviation over the design space under non-i.i.d. data?

Key findings

• The paper establishes a strong uniform convergence rate of O_p( max{ sqrt(log n / (n h)) , h^2 } ) for the kernel conditional quantile estimator under α-mixing and random left-truncation.
• The convergence rate is derived under the assumption that the α-mixing coefficients decay polynomially, i.e., α(n) = O(n^{-δ}) for some δ > 0.
• The bias term is shown to be O(h^2) under regularity conditions on the density and kernel function, independent of dependence.
• The variance term is controlled via maximal inequalities and moment bounds, with the uniform deviation bounded by O_p( sqrt(log n / (n h)) ) under α-mixing.
• The result extends the i.i.d. convergence rate of Lemdani et al. (2008) to the dependent case, preserving the same rate structure.
• The proof relies on decomposing the estimator into bias, variance, and empirical process components, with the latter controlled via Borel-Cantelli and moment conditions.