On adaptive estimation in stationary ARMA processes. (English) Zbl 0616.62042

Kreiss, Jens-Peter

Ann. Stat. 15, 112-133 (1987).

The paper deals with adaptive construction of locally asymptotically minimax (LAM) estimators for stationary ARMA processes with independent and identically, but not necessarily normally distributed innovations. First the local asymptotic normality (LAN) for this model is proved using the sufficient conditions for LAN given by G. G. Roussas [Z. Wahrscheinlichkeitstheor. Verw. Geb. 47, 31-46 (1979; Zbl 0377.62011)]. Then a construction of LAM estimators is suggested if only \(\sqrt{n}\)- consistent initial estimators are available. As these estimates can depend on the distribution of the innovations strongly adaptive estimators are suggested finally which are optimal in LAM sense for a wide class of symmetric innovation distributions. The kernel estimators for the score function -f’/2f (f is the density of the innovation distribution) are used for this purpose. The suggested construction of the adaptive estimators generalizes the results of R. Beran [Ann. Inst. Stat. Math. 28, 77-89 (1976; Zbl 0362.62093)] and P. J. Bickel [Ann. Stat. 10, 647-671 (1982; Zbl 0489.62033)]. A comparative simulation study concludes the paper.

Reviewer: T.Cipra

MSC:

62F35	Robustness and adaptive procedures (parametric inference)
62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F12	Asymptotic properties of parametric estimators
62E20	Asymptotic distribution theory in statistics

Keywords:

autoregressive moving average; optimality criterion; log-likelihood ratio; locally asymptotically minimax; LAM; stationary ARMA processes; local asymptotic normality; LAN; strongly adaptive estimators; kernel estimators; score function; innovation distribution; simulation study

Citations:

Zbl 0377.62011; Zbl 0362.62093; Zbl 0489.62033