[Paper Review] Nonparametric estimation of a regression function using the gamma kernel method in ergodic processes

Alessandro Pasquale De Rosa, Maria-Ines Nogueira|arXiv (Cornell University)|May 24, 2016

Statistical Methods and Inference30 references3 citations

TL;DR

This paper establishes strong uniform consistency and asymptotic normality of gamma kernel-based nonparametric regression estimators under general ergodic dependence assumptions, extending prior i.i.d. results to dependent data. The method uses gamma kernels with adaptive bandwidths to handle non-negative support and boundary bias, achieving optimal convergence rates and asymptotic variance under mild mixing conditions.

ABSTRACT

In this paper we consider the nonparametric estimation of density and regression functions with non-negative support using a gamma kernel procedure introduced by Chen (2000). Strong uniform consistency and asymptotic normality of the corresponding estimators are established under a general ergodic assumption on the data generation process. Our results generalize those of Shi and Song (2016), obtained in the classic i.i.d. framework, and the works of Bouezmarni and Rombouts (2008, 2010b) and Gospodinov and Hirukawa (2007) for mixing time series.

Motivation & Objective

• To extend gamma kernel regression estimation from i.i.d. to general ergodic processes, overcoming limitations of symmetric kernels near boundaries.
• To establish strong uniform consistency and asymptotic normality of the gamma kernel regression estimator under weak dependence assumptions.
• To generalize existing results in i.i.d. and mixing time series frameworks to the broader class of ergodic processes.
• To ensure optimal convergence rates and asymptotic variance matching those of i.i.d. settings under mild bandwidth conditions.

Proposed method

• Uses gamma kernel density estimation with shape parameter $\alpha(n,x) = \frac{x}{h_n} + 1$ and scale parameter $\beta(n) = h_n$ to model non-negative support.
• Proposes a ratio estimator $R_n(x) = \frac{\sum_{t=1}^n \Phi(Y_t) K_{\alpha(n,x),\beta(n)}(X_t)}{\sum_{t=1}^n K_{\alpha(n,x),\beta(n)}(X_t)}$ for conditional expectation $R(x) = E[\Phi(Y_1) \mid X_1 = x]$.
• Employs ergodicity as the primary dependence assumption, generalizing beyond mixing processes to include non-mixing ergodic sequences.
• Applies martingale-type decompositions and uniform mixingale approximations to control bias and variance terms.
• Uses moment inequalities and Hölder's inequality to control tail behavior of estimating functions.
• Relies on Lemmas 4.1–4.8 to bound bias, variance, and uniform deviation terms under conditions (H5)–(H8), including moment and bandwidth constraints.

Experimental results

Research questions

• RQ1 Can gamma kernel regression estimators achieve strong uniform consistency under general ergodic dependence?
• RQ2 Does the asymptotic normality of gamma kernel estimators hold in ergodic processes with the same convergence rates as in i.i.d. settings?
• RQ3 How do bandwidth conditions affect the convergence rate and asymptotic variance in dependent data?
• RQ4 Can the gamma kernel method effectively reduce boundary bias in non-negative support regression under weak dependence?
• RQ5 What are the uniform stochastic properties (bias, variance, convergence) of the gamma kernel estimator in non-i.i.d. ergodic processes?

Key findings

• The gamma kernel regression estimator $R_n(x)$ is strongly uniformly consistent over compact subsets of $\mathbb{R}_0^+$ under ergodicity and moment conditions.
• Asymptotic normality holds: $\sqrt{n h_n} (R_n(x) - R(x)) \xrightarrow{d} N(0, \sigma^2(x))$ with $\sigma^2(x)$ matching the i.i.d. asymptotic variance under the same bandwidth choice.
• Convergence rates for bias and variance terms match those in the i.i.d. case when $n h_n \to \infty$ and $n \sqrt{h_n} \to \infty$.
• The bias term is $O(h_n)$ and the variance term is $O(1/(n h_n))$, leading to optimal $MISE$ rate $O(n^{-2/5})$ under $h_n \sim n^{-1/5}$.
• The proof relies on uniform control of $L^2$-moments and tail probabilities via moment inequalities and ergodicity-induced mixingale approximations.
• The results generalize prior findings in i.i.d. (Shi and Song, 2013) and mixing (Bouezmarni and Rombouts, 2010) settings to the broader class of ergodic processes.