Calculation of the Fisher Information Matrix for Periodic ARMA Models

ABSTRACT

The present article is mainly concerned with the calculation of the Fisher information matrix associated to a periodic autoregressive moving average model (P-ARMA). We provide a computation algorithm based on the conditional likelihood function expression. The established algorithm extends Klein and Mélard's algorithm (1989) elaborated for the classical ARMA models, to the case of periodic autoregressive moving average models. Moreover, for the application of this algorithm, we provide a procedure to compute the theoretical periodic autocovariance function in terms of the parameters of the periodic model. In addition, we give a necessary and sufficient condition for non singular Fisher information matrix of a periodic ARMA model.

Keywords:

• Fisher information matrix
• Periodically correlated autoregressive moving average process
• Periodic autocovariance function
• Recursive estimation

Mathematics Subject Classification: AMS 1991:

• Primary 62M10
• Secondary 62F15

Acknowledgments

The authors would like to express their most sincere thanks and grateful acknowledgments to professor N. Balakrishnan, Editor-in-Chief of the Journal of Communications in Statistics–-Theory and Methods, and to the anonymous referee for their valuable remarks, suggestions, and corrections that improved the quality and the readability of the article. Also, the authors present their thanks and acknowledgments to Professors Marc Hallin and Guy Mélard of the Institut de Statistique de l'UniversitéLibre de Bruxelles for their important help and encouragement.