A note on prediction via estimation of the conditional mode function

Let{(Xi, Yi)}i∈N⊂ E × R, E⊂Rd be a strictly stationary process. The conditional density of Y given X is estimated by the kernel method. It is shown that the (empirically determined) mode of the kernel estimate is uniformly (in a compact) convergent to the conditional mode function when the process is Φ-mixing. This result is applied to a strictly stationary time series {Zk}k∈N which is markovian of order q. It is seen that the so-called model predictor of ZN + 1 from the observed data is converging to the predictor that is based on the full knowledge of the conditional density of ZN + 1 given {Z1,…,ZN}.