Resampling for checking linear regression models via non-parametric regression estimation (Q5940725)

Let us consider the fixed regression model \(Y_{t}=m(x_{t})+\varepsilon_{t},\;t=1,\ldots,n,\) and assume that the random errors, \(\varepsilon_{t}\), follow an ARMA-type dependence structure. The purpose of this paper is to study the application of the bootstrap test to check that the unknown regression function, \(m,\) follows a general linear model of the type: NEWLINE\[NEWLINEH_{0}:\;m\in M=m_{\theta}(\cdot)=A^{t}(\cdot)\theta: \theta \in \Theta \subset R^{q},NEWLINE\]NEWLINE with \(A\) being a functional of \(R\) in \(R^{q}\). In a previous paper, ibid. 20, No. 5, 521-541 (1995), the authors proposed a test, \(D=d^{2}(\hat m_{n},m_{\hat\theta_{n}})\), based on the Crámer-von-Mises-type functional distance, where \(\hat m_{n}\) is a Gasser-Müller-type nonparametric estimator of \(m,\) and \(m_{\hat\theta_{n}}\) is a member of the family \(M\) that is closest to \(\hat m_{n}\).NEWLINENEWLINENEWLINEIn this work, two bootstrap algorithms are considered, where the dependence structure of the errors is taken into account. A broad simulation study and an applied example show the good behavior of the bootstrap test.