A note on prediction via estimation of the conditional mode function (Q1819855)

Let \(\{(X_ i,Y_ i)\}^{\infty}_{i=1}\) be a stationary \(\Phi\)- mixing process where Y is real valued. The paper considers modal regression defined by the regression function \(\theta\) (x) which equals the mode of the conditional distribution of Y given \(X=x\). For the case where \(X_ i\) is a first order moving average process and \(Y_ i=\beta X_ i+\eta_ i\) where \(\eta_ i\) is a mixture of normals it is pointed out that the classical regression based on \(E(Y| X=x)\) would give biased and inconsistent estimators of \(\beta\). On the other hand the numerical computations show that the modal regression approach gives very accurate estimates of \(\beta\). The proposed estimator \(\theta_ n(x)\) of \(\theta\) (x) is the mode of the estimated conditional density obtained by using suitable kernels. Under suitable regularity conditions on the model as well as the kernels it is shown that \(\theta_ n(x)\) is uniformly and strongly consistent for \(\theta\) (x).