Statistics and Its Interface

Volume 16 (2023)

Number 2

Special issue on recent developments in complex time series analysis – Part II

Guest editors: Robert T. Krafty (Emory Univ.), Guodong Li (Univ. of Hong Kong), Anatoly Zhigljavsky (Cardiff Univ.)

Uniform consistency for local fitting of time series non-parametric regression allowing for discrete-valued response

Pages: 305 – 318

DOI: https://dx.doi.org/10.4310/22-SII745

Authors

Rong Peng (Business School, University of Edinburgh, United Kingdom; and School of Mathematical Sciences, University of Southampton, United Kingdom)

Zudi Lu (S3RI and School of Mathematical Sciences, University of Southampton, United Kingdom)

Abstract

Local linear kernel fitting is a popular nonparametric technique for modelling nonlinear time series data. Investigations into it, although extensively made for continuousvalued case, are still rare for the time series that are discrete-valued. In this paper, we propose and develop the uniform consistency of local linear maximum likelihood (LLML) fitting for time series regression allowing response to be discrete-valued under $\beta$-mixing dependence condition. Specifically, the uniform consistency of LLML estimators is established under time series conditional exponential family distributions with aid of a beta-mixing empirical process through local estimating equations. The rate of convergence is also provided under mild conditions. Performances of the proposed method are demonstrated by a Monte-Carlo simulation study and an application to COVID-19 data. There is a huge potential for the developed theory contributing to further development of discrete-valued response semiparametric time series models.

Keywords

uniform consistency, discrete-valued time series, exponential family, local linear fitting, $\beta$-mixing, non-parametric

2010 Mathematics Subject Classification

Primary 62M10, 62M20. Secondary 62G05.

Received 31 May 2021

Accepted 13 June 2022

Published 13 April 2023