Quadratic inference with dense functional responses

Abstract

We address the challenge of estimation in the context of constant linear effect models with dense functional responses. In this framework, the conditional expectation of the response curve is represented by a linear combination of functional covariates with constant regression parameters. In this paper, we present an alternative solution by employing the quadratic inference approach, a well-established method for analyzing correlated data, to estimate the regression coefficients. Our approach leverages non-parametrically estimated basis functions, eliminating the need for choosing working correlation structures. Furthermore, we demonstrate that our method achieves a parametric $\sqrt{n}$ -convergence rate, contingent on an appropriate choice of bandwidth. This convergence is observed when the number of repeated measurements per trajectory exceeds a certain threshold, specifically, when it surpasses $n^{a_0}$ , with $n$ representing the number of trajectories. Additionally, we establish the asymptotic normality of the resulting estimator. The performance of the proposed method is compared with that of existing methods through extensive simulation studies, where our proposed method outperforms. Real data analysis is also conducted to demonstrate the proposed method.

Keywords: Constant Linear-Effect Models; Functional Principal Component Analysis; Quadratic Inference; Semi-parametric Functional Regression.

Fig. 1:

Beijing2017-data: (Left) Reading of hourly PM_2.5 measures for twelve different locations over 608 hourly time-points during January 2017. (Right) Scree plots of a fraction of variance explained (FVE) to determine the optimal number of components to retain in functional principal component analysis. The red line indicates the cumulative fraction of variance across the number of components.