ESTIMATING THE INNOVATION DISTRIBUTION IN NONLINEAR AUTOREGRESSIVE MODELS

ANTON SCHICK^{1} AND WOLFGANG WEFELMEYER^{2}

1Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902-6000,

U.S.A., e-mail: anton@math.binghamton.edu

2Fachbereich 6 Mathematik, Universität Siegen, Walter-Flex-Str. 3, 57068 Siegen, Germany, e-mail: wefelmeyer@mathematik.uni-siegen.de

(Received April 10, 2000; revised October 27, 2000)

Abstract. The usual estimator for the expectation of a function under the innovation distribution of a nonlinear autoregressive model is the empirical estimator based on estimated innovations. It can be improved by exploiting that the innovation distribution has mean zero. We show that the resulting estimator is efficient if the innovations are estimated with an efficient estimator for the autoregression parameter. Efficiency of this estimator is necessary except when the expectation of the function can be estimated adaptively. Analogous results hold for heteroscedastic models.

Key words and phrases: Constrained model, empirical estimator, influence function, Markov chain model, semiparametric model.