Bounds in Wasserstein Distance for Locally Stationary Functional Time Series (2504.06453v1)

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.

Overview of Nadaraya-Watson Estimation in Locally Stationary Functional Time Series

The paper under review, authored by Jan Nino G. Tinio, Mokhtar Z. Alaya, and Salim Bouzebda, explores the bounds in Wasserstein distance for locally stationary functional time series (LSFTS). This research contributes to the statistical understanding of non-stationary processes where data takes the form of functions or curves over time. The crux of this paper lies in embedding locally stationary phenomena into a framework that employs the Nadaraya-Watson (NW) estimator for conditional distribution estimation, with a focus on the convergence rates of this estimator measured through Wasserstein distance.

Key Contributions and Methodologies

The paper introduces LSFTS as an extension of standard time series methods, emphasizing the capture of time-dependence with time-varying statistical properties. The authors leverage the Nadaraya-Watson estimator, traditionally used for estimating conditional means, to estimate conditional distributions in non-stationary environments. Here, the NW estimator is adapted to handle covariates within a semi-metric space, a challenging yet promising endeavor for functional time series analysis. The research establishes convergence rates for the NW estimator with respect to the Wasserstein distance, which is a pivotal metric from optimal transport theory.

Importantly, several assumptions typical in LSFTS literature are maintained, such as the local stationarity of covariates, the regularity of kernel functions, small ball probability considerations, and mixing conditions. These assumptions facilitate robust theoretical guarantees for the estimator’s performance on function-valued processes, ultimately yielding convergence rates of order $$\mathcal{O}_\mathbb{P}(\frac{1}{T^\frac{1}{2} h\phi(h)} + h)$$, where $h$ is the bandwidth, and $\phi(h)$ denotes the small ball probability.

Numerical Illustrations

Numerical experiments illustrate the theoretical convergence results. Synthetic data are generated using functional autoregressive processes, where LSFTS are modelled with Gaussian noise introduced as small perturbations. These experiments demonstrate decreasing Wasserstein distances with increasing sample sizes, underscoring the efficacy of the NW estimator in approximating conditional distributions more accurately as data volume increases. Real-world datasets, such as sea surface temperatures and stock market indices, further corroborate these findings, showcasing the NW estimator's practical utility in diverse scenarios.

Implications and Future Directions

The implications of this paper are multifaceted, with immediate applications spanning various domains where functional data is prevalent, such as meteorology, finance, and environmental science. The established convergence rates with Wasserstein distance suggest enhanced models for predictive analytics and the potential for improved statistical inference in complex, dynamic environments. The work also paves the way for future research avenues, notably the exploration of integrated kernel approaches, handling missing data in functional time series, and extending the theoretical framework to encompass ergodic functional data. Such expansions undoubtedly enrich the analytical toolkit available to statisticians and data scientists.

In summary, this paper represents a significant advancement in non-parametric estimation for LSFTS, demonstrating a thorough blend of mathematical rigor and practical applicability. The convergence rates obtained not only reinforce but also extend the boundaries of NW estimator applications into the field of functional time series, heralding promising prospects for future analytical innovations.