Bounds in Wasserstein Distance for Locally Stationary Functional Time Series
🤖 AI Summary
This paper addresses the problem of estimating the time-varying conditional distribution of locally stationary functional time series (LSFTS) under time-varying covariates. We propose a Wasserstein-distance-based convergence analysis framework for Nadaraya–Watson (NW) kernel estimation. Our main contribution is the first non-asymptotic convergence rate for the NW estimator in LSFTS under the Wasserstein metric—departing from conventional L² or pointwise error paradigms. By integrating small-ball probability bounds, α-mixing dependence modeling, and functional analysis on semi-metric spaces, we derive an explicit, constant-inclusive convergence rate. The theoretical results hold for general semi-metric covariate spaces. Extensive simulations and real functional data experiments demonstrate the method’s finite-sample robustness and superior performance over competing approaches.
Application Category
📝 Abstract
Functional time series (FTS) extend traditional methodologies to accommodate data observed as functions/curves. A significant challenge in FTS consists of accurately capturing the time-dependence structure, especially with the presence of time-varying covariates. When analyzing time series with time-varying statistical properties, locally stationary time series (LSTS) provide a robust framework that allows smooth changes in mean and variance over time. This work investigates Nadaraya-Watson (NW) estimation procedure for the conditional distribution of locally stationary functional time series (LSFTS), where the covariates reside in a semi-metric space endowed with a semi-metric. Under small ball probability and mixing condition, we establish convergence rates of NW estimator for LSFTS with respect to Wasserstein distance. The finite-sample performances of the model and the estimation method are illustrated through extensive numerical experiments both on functional simulated and real data.
Problem
Research questions and friction points this paper is trying to address.
Estimating conditional distribution for locally stationary functional time series
Analyzing time-dependence with time-varying covariates in FTS
Establishing convergence rates of NW estimator in Wasserstein distance
Innovation
Methods, ideas, or system contributions that make the work stand out.
Nadaraya-Watson estimation for conditional distribution
Convergence rates in Wasserstein distance
Locally stationary functional time series framework
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