  In this paper, a functional partial quantile regression approach, a quantile
regression analog of the functional partial least squares regression, is
proposed to estimate the function-on-function linear quantile regression model.
A partial quantile covariance function is first used to extract the functional
partial quantile regression basis functions. The extracted basis functions are
then used to obtain the functional partial quantile regression components and
estimate the final model. In our proposal, the functional forms of the
discretely observed random variables are first constructed via a
finite-dimensional basis function expansion method. The functional partial
quantile regression constructed using the functional random variables is
approximated via the partial quantile regression constructed using the basis
expansion coefficients. The proposed method uses an iterative procedure to
extract the partial quantile regression components. A Bayesian information
criterion is used to determine the optimum number of retained components. The
proposed functional partial quantile regression model allows for more than one
functional predictor in the model. However, the true form of the proposed model
is unspecified, as the relevant predictors for the model are unknown in
practice. Thus, a forward variable selection procedure is used to determine the
significant predictors for the proposed model. Moreover, a case-sampling-based
bootstrap procedure is used to construct pointwise prediction intervals for the
functional response. The predictive performance of the proposed method is
evaluated using several Monte Carlo experiments under different data generation
processes and error distributions. Through an empirical data example, air
quality data are analyzed to demonstrate the effectiveness of the proposed
method.
