Summary Periodicity and trend are features describing an observed sequence, and extracting these features is an important issue in many scientific fields. However, it is not an easy task for existing methods to analyse simultaneously the trend and dynamics of the periodicity such as time varying frequency and amplitude, and the adaptivity of the analysis to such dynamics and robustness to heteroscedastic dependent errors are not guaranteed. These tasks become even more challenging when there are multiple periodic components. We propose a non‐parametric model to describe the dynamics of multicomponent periodicity and investigate the recently developed synchro‐squeezing transform in extracting these features in the presence of a trend and heteroscedastic dependent errors. The identifiability problem of the non‐parametric periodicity model is studied, and the adaptivity and robustness properties of the synchro‐squeezing transform are theoretically justified in both discrete and continuous time settings. Consequently we have a new technique for decoupling the trend, periodicity and heteroscedastic, dependent error process in a general non‐parametric set‐up. Results of a series of simulations are provided, and the incidence time series of varicella and herpes zoster in Taiwan and respiratory signals observed from a sleep study are analysed.

Many popular curve estimators based on smoothing have difficulties caused by boundary effects. These effects are visually disturbing in practice and can play a dominant role in theoretical analysis. Local polynomial regression smoothers are known to correct boundary effects automatically. Some analogs are implemented for density estimation and the resulting estimators also achieve automatic boundary corrections. In both settings of density and regression estimation, we investigate best weight functions for local polynomial fitting at the endpoints and find a simple solution. The solution is universal for general degree of local polynomial fitting and general order of estimated derivative. Furthermore, such local polynomial estimators are best among all linear estimators in a weak minimax sense, and they are highly efficient even in the usual linear minimax sense.

Nonparametric tests of modality are a distribution-free way of assessing evidence about inhomogeneity in a population, provided that the potential subpopulations are suf®ciently well separated. They include the excess mass and dip tests, which are equivalent in univariate settings and are alternatives to the bandwidth test. Only very conservative forms of the excess mass and dip tests are available at present, however, and for that reason they are generally not competitive with the bandwidth test. In the present paper we develop a practical approach to calibrating the excess mass and dip tests to improve their level accuracy and power substantially. Our method exploits the fact that the limiting distribution of the excess mass statistic under the null hypothesis depends on unknowns only through a constant, which may be estimated. Our calibrated test exploits this fact and is shown to have greater power and level accuracy than the bandwidth test has. The latter tends to be quite conservative, even in an asymptotic sense. Moreover, the calibrated test avoids dif®culties that the bandwidth test has with spurious modes in the tails, which often must be discounted through subjective intervention of the experimenter.

High-dimensional data analysis has been an active area, and the main focuses have been variable selection and dimension reduction. In practice, it occurs often that the variables are located on an unknown, lower-dimensional nonlinear manifold. Under this manifold assumption, one purpose of this paper is regression and gradient estimation on the manifold, and another is developing a new tool for manifold learning. To the first aim, we suggest directly reducing the dimensionality to the intrinsic dimension d of the manifold, and performing the popular local linear regression (LLR) on a tangent plane estimate. An immediate consequence is a dramatic reduction in the computation time when the ambient space dimension p d. We provide rigorous theoretical justification of the convergence of the proposed regression and gradient estimators by carefully analyzing the curvature, boundary, and non-uniform sampling effects.A bandwidth selector that can handle heteroscedastic errors is proposed. Tothe second aim, we analyze carefully the behavior of our regression estimator both in the interior and near the boundary of the manifold, and make explicit its relationship with manifold learning, in particular estimating the LaplaceBeltrami operator of the manifold. In this context, we also make clear that it is important to use a smaller bandwidth in the tangent plane estimation than in the LLR. Simulation studies and the Isomap face data example are used to illustrate the computational speed and estimation accuracy of our methods.

Ultra-high dimensional longitudinal data are increasingly common and the analysis is challenging both theoretically and methodologically. We offer a new automatic procedure for finding a sparse semivarying coefficient model, which is widely accepted for longitudinal data analysis. Our proposed method first reduces the number of covariates to a moderate order by employing a screening procedure, and then identifies both the varying and constant coefficients using a group SCAD estimator, which is subsequently refined by accounting for the within-subject correlation. The screening procedure is based on working independence and B-spline marginal models. Under weaker conditions than those in the literature, we show that with high probability only irrelevant variables will be screened out, and the number of selected variables can be bounded by a moderate order. This allows the desirable sparsity and oracle properties of the subsequent structure identification step. Note that existing methods require some kind of iterative screening in order to achieve this, thus they demand heavy computational effort and consistency is not guaranteed. The refined semivarying coefficient model employs profile