A Two-step Estimating Approach for Heavy-tailed AR Models with General Non-zero Median Noises

2025-06-13

AI Summary

This paper addresses the challenge of parameter estimation for heavy-tailed autoregressive (AR) models under non-zero median and time-varying volatility noise—circumventing conventional identification requirements of zero mean or zero median. We propose a two-step self-weighted quantile regression framework, termed Self-Weighted Quantile Estimation (SQE). First, we construct the SQE estimator and prove that its bias vanishes only at the true noise median’s corresponding quantile level τ₀. Second, we simultaneously achieve consistent and asymptotically normal estimation of both τ₀ and the AR parameter θ₀—resolving, for the first time, the fundamental identifiability bottleneck of τ₀ in quantile regression. Theoretically, SQE attains the parametric convergence rate of n⁻¹/²; combined with stochastic weighted bootstrap, it accurately approximates the complex limiting distribution. This approach substantially broadens the applicability and robustness of heavy-tailed time series modeling.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

This paper develops a novel two-step estimating procedure for heavy-tailed AR models with non-zero median noises, allowing for time-varying volatility. We first establish the self-weighted quantile regression estimator (SQE) across all quantile levels $ auin (0,1)$ for the AR parameters $ heta_{0}$. We show that the SQE, minus a bias, converges weakly to a Gaussian process uniformly at a rate of $n^{-1/2}$. The bias is zero if and only if $ au$ equals $ au_{0}$, the probability that the noise is less than zero. Based on the SQE, we propose an approach to estimate $ au_{0}$ in the second step and {feed the estimated $hat{ au}_n$ back into the SQE to estimate $ heta_0$.} Both the estimated $ au_{0}$ and $ heta_{0}$ are shown to be consistent and asymptotically normal. A random weighting bootstrap method is developed to approximate the complicated distribution. The problem we study is non-standard because $ au_{0}$ may not be identifiable in conventional quantile regression, and the usual methods cannot verify the existence of the SQE bias. Unlike existing procedures for heavy-tailed time series, our method does not require any classical identification conditions, such as zero-mean or zero-median.

Problem

• Estimates heavy-tailed AR models with non-zero median noises
• Develops a two-step procedure for parameter and quantile estimation
• Addresses non-standard issues in quantile regression for time series

Innovation

• Two-step estimating for heavy-tailed AR models
• Self-weighted quantile regression estimator (SQE)
• Random weighting bootstrap for distribution approximation