  The problem of error density estimation for a functional single index model
with dependent errors is studied. A Bayesian method is utilized to
simultaneously estimate the bandwidths in the kernel-form error density and
regression function, under an autoregressive error structure. For estimating
both the regression function and error density, empirical studies show that the
functional single index model gives improved estimation and prediction
accuracies than any nonparametric functional regression considered.
Furthermore, estimation of error density facilitates the construction of
prediction interval for the response variable.
