The multivariate adaptive regression spline (MARS) is one of the popular estimation methods for nonparametric multivariate regressions. However, as MARS is based on marginal splines, to incorporate interactions of covariates, products of the marginal splines must be used, which leads to an unmanageable number of basis functions when the order of interaction is high and results in low estimation efficiency. In this paper, we improve the performance of MARS by using linear combinations of the covariates which achieve sufficient dimension reduction. The special basis functions of MARS facilitate calculation of gradients of the regression function, and estimation of the linear combinations is obtained via eigen-analysis of the outer-product of the gradients. Under some technical conditions, the asymptotic theory is established for the proposed estimation method. Numerical studies including both simulation and empirical applications show its effectiveness in dimension reduction and improvement over MARS and other commonly-used nonparametric methods in regression estimation and prediction.

Modelling functional time series, time series of random functions defined within a finite interval, has became one of the main frontiers of developments in time series models. Various functional linear and nonlinear time series models have been proposed and extensively studied in the past two decades (e.g., Bosq, 2000; Hörmann and Kokoszka, 2010; Horváth and Kokoszka, 2012; Hörmann, Horváth and Reeder, 2013; Li, Robinson and Shang, 2020). These models together with relevant methodologies have been applied to various fields such as biology, demography, economics, environmental science and finance. However, the model frameworks and methodologies developed in the aforementioned literature heavily rely on the stationarity assumption, which is often rejected when testing the functional time series data in practice. For example, Horváth, Kokoszka and Rice (2014) find evidence of nonstationarity for intraday price curves of some stocks collected in the US market; Aue, Rice and Sönmez (2018) reject the null hypothesis of stationarity for the temperature curves collected in Australia; and Li, Robinson and Shang (2023) reveal evidence of nonstationary feature for the functional time series constructed from the age-and sex-specific life-table death counts. It thus becomes imperative to test whether the collected functional time series are stationary. The primary interest of this paper is to test whether there exist structural breaks in the mean function over time and subsequently estimate locations of breaks if they do exist. There have been increasing interests on detecting and estimating structural breaks in functional time series. Broadly speaking, there are two types of detection techniques.