Testing independence and goodness-of-fit jointly for functional linear models

A Literature Review of Semi-functional Partial Linear Regression Models

“… Robust: Existing outliers in the data or violations from distributional assumptions yield to the robust methods such as the sieve M-estimator for semi-functional linear model [43], with polynomial splines to approximate the slope parameter and resistance to heavy-tailed errors or outliers in the response [44], different estimators such as M-estimators with bi-square function, GM-estimator with Huber function, LMS-estimator and LTS-estimators [45], estimation based on exponential squared loss and FPCA [46], estimation based on the class of scale mixtures of normal (SMN) distributions for measurement errors and Bayesian framework with MCMC algorithm [47], Robust MM-estimators with B-Spline approximation [48], with modal regression [49] and a modified Huber's function with tail function with a data-driven procedure for selecting the tuning parameters [50].  Testing: Different hypothesis and testing statistics are developed ,such as: testing the linear component [51,52] with B-spline [53], functional covariates [54], densely and sparsely observed single and multiple functional covariates with four tests such as Wald, Score, likelihood ratio and F [55], Goodness-of-fit tests with wild bootstrap resampling, false discovery rate and independence test with generalized distance covariance or new metric, functional martingale difference divergence (FMDD), [56][57][58], series correlation test [59].  Quantile regression: Some extensions consider quantile regression property such as: proposed functional partially linear quantile regression model (FPLQRM) that has the linear variables which may be categorical [60], estimating the slope function between a dependent variable and both vector and functional random variable with FPCA [61], and piecewise polynomial [62] and kNN quantile method [63], functional composite quantile regression (CQR) with simple partial quantile regression (SIMPQR) algorithm and partial quantile regression (PQR) basis [64], composite quantile estimation with strictly stationary process errors [65] and with polynomial splines [66], Hill estimator for extreme quantile estimation with heavy-tailed distributions [67], developed quantile rank score test for a parametric component of the model [68], varying-coefficient p...…”

A review of goodness-of-fit tests for models involving functional data

“…A recent contribution by Lai et al (2020) is devoted to the testing a modified null hypothesis: H 0 : "X is independent of ε and m ∈ L", using the recent results related with the distance covariance (see Székely et al (2007), Lyons (2013), and Sejdinovic et al (2013)). Consider (X , ρ X ) and (Y, ρ Y ) two semimetric spaces of negative type, where ρ X and ρ Y are the corresponding semimetrics.…”

Representation Theorem and Functional CLT for RKHS-Based Function-on-Function Regressions

“…Existing methods, such as projection, dimension-reduction, and functional linear regression analysis, are not adapted for such data. Overviews can be found in the book by Horváth and Kokoszka [2] and some recently published papers such as Yuan et al [3] and Lai et al [4].…”