Adaptive Estimation of Causal Periodic Autoregressive Model

Abstract

This article deals with the adaptive estimation of a periodic autoregressive model, with unspecified innovation density satisfying only some general technical assumptions. We first establish, while verifying the adapted sufficient conditions of Swensen (1985) to our model, the Local Asymptotic Normality (LAN), the Local Asymptotic Quadratic (LAQ), and the Local Asymptotic properties satisfied by its central sequence. Secondly, the Locally Asymptotically Minimax (LAM) estimators are constructed. Using these results, we construct the adaptive estimators of the unknown autoregressive parameters. The performances of the established estimators are shown, via simulation studies.

Keywords:

• Adaptive estimation
• Asymptotic linearity
• Central sequence
• Local asymptotic normality
• Locally asymptotic minimax estimator
• Periodic autoregressive model

Acknowledgments

The authors express their most sincere thanks and grateful acknowledgments to Professor N. Balakrishnan, Editor in Chief of the Journal of Communications in Statistics – Simulation and Computation and to the anonymous referees for their unlimited assistance, valuable remarks and helpful suggestions which enabled us to improve the quality and the readability of the paper. The first author expresses his sincere thanks to Brahim Bentarzi, Legata, for his encouragements.

Notes

The simulation results, reported in the different tables, show empirically that the AE always performs better than the LSE for the densities f ₂ and f ₃, particularly for f ₃ whose density shape (a symmetric bimodal density) is significantly different of the normal density for which the LSE is equivalent to the MLE. More precisely, the RMSE of the adaptive estimators are always smaller than those of their corresponding LSE, particularly for f ₃. Moreover, the consistency of the adaptive estimators can be deduced empirically from the fact their bias and empirical variances are decreasing as the size of the time series increases.

On the other hand, one can clearly see in the tables that the RMSE of f ₁ is slightly in favor of the LSE. This is well awaited result because the least squares estimator, which is obtained by a nonparametric procedure, under the normality assumption, is equivalent to the maximum likelihood estimator. However, for the density f ₄, whose shape is not so different than the standard normal density, the RMSE criterion favors the AE. Moreover, simulation results show that even when the number of parameters increases to include unknown periodic variances, the AE maintains its good demonstrated finite sample behavior.