# Estimating and visualizing conditional densities

We consider the kernel estimator of conditional density and derive its asymptotic bias, variance and mean-square error. Optimal bandwidths (with respect to integrated mean-square error) are found and it is shown that the convergence rate of the density estimator is order n-2/3. We also note that the conditional mean function obtained from the estimator is equivalent to a kernel smoother. Given the undesirable bias properties of kernel smoothers, we seek a modified conditional density estimator which has mean equivalent to some other nonparametric regression smoother with better bias properties. It is also shown that our modified estimator has smaller mean square error than the standard estimator in some commonly occurring situations. Finally, three graphical methods for visualizing conditional density estimators are discussed and applied to a data set consisting of maximum daily temperatures in Melbourne, Australia.

**Keywords:** automatic forecasting, exponential smoothing, prediction intervals, state space models.