Multiscale clustering of nonparametric regression curves

Abstract

In a wide range of modern applications, we observe a large number of time series rather than only a single one. It is often natural to suppose that there is some group structure in the observed time series. When each time series is modelled by a nonparametric regression equation, one may in particular assume that the observed time series can be partitioned into a small number of groups whose members share the same nonparametric regression function. We develop a bandwidth-free clustering method to estimate the unknown group structure from the data. More precisely speaking, we construct multiscale estimators of the unknown groups and their unknown number which are free of classical bandwidth or smoothing parameters. In the theoretical part of the paper, we analyze the statistical properties of our estimators. Our theoretical results are derived under general conditions which allow the data to be dependent both in time series direction and across different time series. The technical analysis of the paper is complemented by a simulation study and a real-data application.

Functional-Coefficient Quantile Regression for Panel Data with Latent Group Structure

“…Remark 4.3. The consistency result (4.8) is similar to Theorem 3.1 in Vogt and Linton (2017), Theorem 1 in Chen ( 2019) and Theorem 4.1(a) in Vogt and Linton (2020), all of which study nonparametric mean regression for panel data with a latent group structure. By the clustering algorithm, to achieve the consistency property, it is sufficient to show that max 1 j,k N ∆(j, k) − ∆(j, k) is of order smaller than the minimum distance between true group-specific functional coefficients.…”

“…Using the estimated distance matrix, we may apply the agglomerative clustering method which has been widely used in the literature of cluster analysis (e.g., Everitt et al, 2011;Rencher and Christensen, 2012). Recently, such a method, combined with the kernel-based smoothing technique, has been applied to estimate the homogeneity/group structure in nonparametric mean regression models (e.g., Chen, 2019;Vogt and Linton, 2020;Chen et al, 2021). However, so far as we know, there is virtually no work on applying the kernel-based agglomerative clustering method to quantile regression models with a latent group structure.…”