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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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[1] Billingsley, P., Convergence of probability measures, (1968), Wiley New York · Zbl 0172.21201
[2] Collomb, G., Estimation non-parametrique de la regression: revue bibliographique, Internat. statist. rev., 49, 75-93, (1981) · Zbl 0471.62039
[3] Collomb, G., Proprietés de convergence presque complète du predicteur a noyan, Z. wahrsch. verw. geb., 66, 441-460, (1984) · Zbl 0525.62046
[4] Collomb, G.; Härdle, W., Strong uniform convergence rates in robust nonparametric time series analysis and prediction: kernel regression estimation from dependent observations, Stochastic process. appl., 23, 77-89, (1986) · Zbl 0612.62127
[5] Doob, J.L., Stochastic processes, (1953), Wiley New York · Zbl 0053.26802
[6] Martin, R.D., Robust methods for time series, () · Zbl 0482.62079
[7] Robinson, P.M., Robust nonparametric autoregression, () · Zbl 0573.62037
[8] Silverman, B.W., Kernel density estimation using the fast Fourier transform, Appl. statist., 31, 93-97, (1982) · Zbl 0483.62032
[9] Watson, G.S., Smooth regression analysis, Sankhya A, 26, 359-372, (1964) · Zbl 0137.13002
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