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A note on prediction via estimation of the conditional mode function. (English) Zbl 0614.62045
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).
Reviewer: B.K.Kale

##### MSC:
 62G05 Nonparametric estimation 62G20 Asymptotic properties of nonparametric inference 62M20 Inference from stochastic processes and prediction 60G25 Prediction theory (aspects of stochastic processes)
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