A test of linearity in partial functional linear regression

Abstract

This paper investigates the hypothesis test of the parametric component in partial functional linear regression. We propose a test procedure based on the residual sums of squares under the null and alternative hypothesis, and establish the asymptotic properties of the resulting test. A simulation study shows that the proposed test procedure has good size and power with finite sample sizes. Finally, we present an illustration through fitting the Berkeley growth data with a partial functional linear regression model and testing the effect of gender on the height of kids.

Partial Functional Linear Models with ARCH Errors

“…In this paper, the estimation of partial functional linear models with ARCH(p) [8]). In the future study, under the errors' dependent structure, we will further consider the estimation of the model (1.1) using the kernel method noticing that the relationship between z and X may be relaxed.…”

“…Recently, to get the robust estimator of coefficients of (1.1), the model has been also considered in the frame of qunatile regression ( [6] [7]). Some authors also considered the model (1.1) from the view of hypothesis test, such as, [8] construct pivot by the square of residuals under the null and alternative hypothesis, to test whether the linearity term of (1.1) exists or not.…”

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...…”

Robust estimation for partial functional linear additive model based on modal regression

“…模拟表明, 众数估计方法 可以较好处理函数型数据中响应变量具有厚尾特征或者异常值的情况. 相应估计方法可以用到其他函 数型模型, 如部分函数型线性模型 [12,35] 、变系数部分函数型线性模型 [14] 等. 此外, 本文只考虑函数 一元函数型预测变量, 进一步可以推广到基于众数回归估计的多元函数型预测变量的变量选择问题.…”