Optimal Detection of Exponential Component in Autoregressive Models

Abstract

Abstract. In this article, the problem of detecting the eventual existence of an exponential component in an AR(1) model, that is, the problem of testing ordinary AR(1) dependence against the alternative of an exponential autoregression [EXPAR(1)] model, was considered. A local asymptotic normality property was established for EXPAR(1) models in the vicinity of AR(1) ones. Two problems arose in this context, which were quite typical in the study of nonlinear time‐series models. The first was a problem of parameter identification in the EXPAR(1) model. A special parameterization was developed so as to overcome this technical problem. The second problem was related to the fact that the underlying innovation density had to be treated as a nuisance. The problem at hand, indeed, appeared to be nonadaptive. These problems were solved using semi‐parametrically efficient pseudo‐Gaussian methods (which did not require Gaussian observations).