Generalised kernel smoothing for non-negative stationary ergodic processes

Abstract

In this paper, we consider a generalised kernel smoothing estimator of the regression function with non-negative support, using gamma probability densities as kernels, which are non-negative and have naturally varying shapes. It is based on a generalisation of Hille's lemma and a perturbation idea that allows us to deal with the problem at the boundary. Its uniform consistency and asymptotic normality are obtained at interior and boundary points, under a stationary ergodic process assumption, without using traditional mixing conditions. The asymptotic mean squared error of the estimator is derived and the optimal value of smoothing parameter is also discussed. Graphical illustrations of the proposed estimator are provided for simulated as well as for real data. A simulation study is also carried out to compare our method with the competing local linear method.

Keywords:

• ergodic processes
• Hille's lemma
• gamma density
• martingale difference
• mixing
• normality
• prediction
• regression function

AMS 1991 Subject Classification :

• Primary: 62G05
• 62M20
• Secondary: 60J15

Acknowledgements

The authors gratefully acknowledge financial support from the Statistical Laboratory of CRM, Montreal (for N. Laïb's trip to Montreal) and the NSERC Discovery grants (Y. Chaubey and A. Sen). The data set used for illustration in Section 5 was kindly provided by B. St-Onge and U. Vepakomma of the UQAM. The simulation study in the last section showing comparison with the local linear estimator was carried out by Baohua He, graduate student at Concordia University. The authors are also grateful to the two referees and the Associate Editor for constructive comments and suggestions.