Abstract
Consider a stationary first-order autoregressive process, with i.i.d. residuals following an unknown mean zero distribution. The customary estimator for the expectation of a bounded function under the residual distribution is the empirical estimator based on the estimated residuals. We show that this estimator is not efficient, and construct a simple efficient estimator. It is adaptive with respect to the autoregression parameter.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akahira, M. (1976). On the asymptotic efficiency of estimators in an autoregressive process,Ann. Inst. Statist. Math.,28, 35–48.
Akritas, M. G. and Johnson, R. A. (1982). Efficiencies of tests and estimators forp-order autoregressive processes when the error distribution is nonnormal,Ann. Inst. Statist. Math.,34, 579–589.
Bickel, P. J. (1982). On adaptive estimation,Ann. Statist.,10, 647–671.
Bickel, P. J., Klaassen, C. A. J., Ritov, Y. and Wellner, J. A. (1993).Efficient and Adaptive Estimation in Semiparametric Models, Johns Hopkins University Press, Baltimore.
Boldin, M. V. (1982). Estimation of the distribution of noise in an autoregression scheme,Theory Probab. Appl.,27, 866–871.
Boldin, M. V. (1983). Testing hypotheses in autoregression schemes by the Kolmogorov andω 2 criteria,Soviet Math. Dokl.,28, 550–553.
Boldin, M. V. (1989). On testing hypotheses in the sliding average scheme by the Kolmogorov-Smirnov andω 2 tests,Theory Probab. Appl.,27, 866–871.
Greenwood, P. E. and Wefelmeyer, W. (1990). Efficiency of estimators for partially specified filtered models,Stochastic Process. Appl.,36, 353–370.
Huang, W.-M. (1986). A characterization of limiting distributions of estimators in an autoregressive process,Ann. Inst. Statist. Math.,38, 137–144.
Koshevnik, Y. (1992). Efficient estimation for restricted nonparametric models, preprint.
Koul, H. L. (1991). A weak convergence result useful in robust autoregression,J. Statist. Plann. Inference,29, 291–308.
Koul, H. L. (1992).Weighted Empiricals and Linear Models, IMS Lecture Notes—Monograph Series, Vol. 21, Hayward, California.
Koul, H. L. and Leventhal, S. (1989). Weak convergence of the residual empirical process in explosive autoregression,Ann. Statist.,17, 1784–1794.
Kreiss, J.-P. (1987). On adaptive estimation in autoregressive models when there are nuisance parameters,Statist. Decisions,5, 59–76.
Kreiss, J.-P. (1991). Estimation of the distribution function of noise in stationary processes,Metrika,38, 285–297.
Pfanzagl, J. and Wefelmeyer, W. (1982).Contributions to a General Asymptotic Statistical Theory, Lecture Notes in Statistics,13, Springer, New York.
Schick, A. (1987). A note on the construction of asymptotically linear estimators,J. Statist. Plann. Inference,16, 89–105 (correction:ibid. (1989).22, 269–270).
Wefelmeyer, W. (1992). Quasi-likelihood models and optimal inference, preprint.
Author information
Authors and Affiliations
About this article
Cite this article
Wefelmeyer, W. An efficient estimator for the expectation of a bounded function under the residual distribution of an autoregressive process. Ann Inst Stat Math 46, 309–315 (1994). https://doi.org/10.1007/BF01720587
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01720587