Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniform Convergence Rate of the Nonparametric Estimator for Integrated Diffusion Processes

Shaolin Ji, Linlin Zhu

TL;DR

This work establishes the first uniform convergence rates for kernel-based Nadaraya-Watson estimators of the instantaneous coefficients in integrated diffusion processes. By leveraging a global modulus-of-continuity framework and covering-number techniques under long-horizon sampling, the authors derive finite-sample rates that separate discretization, variance, and smoothing effects, with rates expressed as $a_{n,T}$ and $a^*_{n,T}$ and normalized by the stationary density lower bound $δ_{n,T}$. The results hold under α-mixing, smoothness, and moment conditions, and extend the reach of nonparametric inference to specification testing and semiparametric estimation in finance, geology, and physics. The paper also includes simulation evidence (CIR and OU models) validating the theory and illustrating the practical impact of bandwidth choice and sample size on estimation accuracy.

Abstract

The nonparametric estimation of integrated diffusion processes has been extensively studied, with most existing research focusing on pointwise convergence. This paper is the first to establish uniform convergence rates for the Nadaraya-Watson estimators of their coefficients. We derive these rates over unbounded support under the assumptions of a vanishing observation interval and a long time horizon. Our findings serve as essential tools for specification testing and semiparametric inference in various diffusion models and time series, facilitating applications in finance, geology, and physics through nonparametric estimation methods.

Uniform Convergence Rate of the Nonparametric Estimator for Integrated Diffusion Processes

TL;DR

This work establishes the first uniform convergence rates for kernel-based Nadaraya-Watson estimators of the instantaneous coefficients in integrated diffusion processes. By leveraging a global modulus-of-continuity framework and covering-number techniques under long-horizon sampling, the authors derive finite-sample rates that separate discretization, variance, and smoothing effects, with rates expressed as and and normalized by the stationary density lower bound . The results hold under α-mixing, smoothness, and moment conditions, and extend the reach of nonparametric inference to specification testing and semiparametric estimation in finance, geology, and physics. The paper also includes simulation evidence (CIR and OU models) validating the theory and illustrating the practical impact of bandwidth choice and sample size on estimation accuracy.

Abstract

The nonparametric estimation of integrated diffusion processes has been extensively studied, with most existing research focusing on pointwise convergence. This paper is the first to establish uniform convergence rates for the Nadaraya-Watson estimators of their coefficients. We derive these rates over unbounded support under the assumptions of a vanishing observation interval and a long time horizon. Our findings serve as essential tools for specification testing and semiparametric inference in various diffusion models and time series, facilitating applications in finance, geology, and physics through nonparametric estimation methods.
Paper Structure (5 sections, 10 theorems, 93 equations, 10 figures, 2 tables)

This paper contains 5 sections, 10 theorems, 93 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Assumptions and Main Theorems
  3. Auxiliary Lemmas
  4. Proofs of the Main Theorems
  5. Simulation

Key Result

Theorem 2.4

Suppose the following conditions hold: $(1)$ Assumptions assumption-stationary and mixing and assumption-kernel and damping are satisfied. $\sigma$ and $(\sigma^{2}\pi)"$ are uniformly bounded. $|\partial^{2} \tilde{f} (x)|=O(|x|)$ as $|x|\to\infty$, where $\tilde{f}=b ,\sigma$. $(2)$ The density fu and $a_{n,T}/\delta_{n,T}\to 0$, $\Delta^{\frac{1}{2}-\frac{1}{2+q}}/(h\delta_{n,T})\to 0$, $\sqrt{

Figures (10)

  • Figure 1: Mean Estimated Diffusion Coefficient Under Different $\Delta$ and $h$ for the CIR process
  • Figure 3: Mean Estimated Diffusion Coefficient Under Different $\Delta$ and $h$ for the OU process
  • Figure : $\Delta=0.002$
  • Figure : $\Delta=0.002$
  • Figure : $\Delta=0.002$
  • ...and 5 more figures

Theorems & Definitions (20)

  • Remark 2.2
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6
  • Lemma 3.1: Lemma A.3 in kanaya2017uniform
  • Lemma 3.2: $(15)$ in kanaya2017uniform
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • Lemma 3.5
  • ...and 10 more