A Bayesian approach for determining the optimal semi-metric and bandwidth in scalar-on-function quantile regression with unknown error density and dependent functional data

Extreme values are often associated with tails of a cumulative distribution function, and the study of extreme values and their predictions is an important research topic in climate problems. Through a regression approach, we consider a scalar-on-function nonparametric regression to estimate and predict conditional quantiles, where the regression function can be estimated by the functional Nadaraya-Watson estimator. The accuracy of such an estimator crucially depends on the optimal selections of semi-metric and bandwidth parameters. A Bayesian approach is proposed to simultaneously estimate the bandwidths in the regression function and kernel-form error density. As a by-product of the Bayesian approach, marginal likelihood is used to select the optimal semi-metric. In both independent and dependent functional data, a series of simulation studies shows that the proposed Bayesian approach outperforms the functional cross validation for estimating the regression function, and it performs better than the likelihood cross validation for estimating the error density. The proposed Bayesian approach is utilised in the extreme value analysis for predicting the recurrence interval of maximum temperature at Melbourne Airport, in Australia.