Abstract
A model of nonlinear regression is studied in infinite-dimensional space. Observation errors are equally distributed and have the identity correlation operator. A projective estimator of a parameter is constructed, and the conditions under which it is true are established. For a parameter that belongs to an ellipsoid in a Hilbert space, we prove that the estimators are asymptotically normal; for this purpose, the representation of the estimator in terms of the Lagrange factor is used and the asymptotics of this factor are studied. An example of the nonparametric estimator of a signal is examined for iterated observations under an additive noise.
Similar content being viewed by others
References
A. G. Kukush, “The convergence in distribution of a normalized projective estimator of an infinite-dimensional parameter of linear regression,”Teor. Ver. Mat. Statist., Issue 48, 1–10 (1993).
A. S. Nemirovskii and R. Z. Khas'minskii, “The nonparametric estimation of the functionals of derivatives of a signal observed under white noise,”Probl. Peredachi Inf.,23, No. 3, 27–38. (1987).
R. I. Jennrich, “Asymptotic properties of nonlinear square estimators,”Ann. Math. Stat.,40, No. 2, 633–643 (1969).
A. V. Ivanov,The Theory of Estimation of the Parameters of Nonlinear Regression Models [in Russian], Author's Abstract of the Doctoral Degree Thesis (Physics and Mathematics), Kiev, 1991.
A.G. Kukush, “The convergence in distribution of a normalized estimator of an infinite-dimensional regression parameter,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 5, 11–14 (1988).
Author information
Authors and Affiliations
Additional information
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 9, pp. 1205–1214, September, 1993.
Rights and permissions
About this article
Cite this article
Kukush, A.G. Asymptotic normality of a projective estimator of an infinite-dimensional parameter of nonlinear regression. Ukr Math J 45, 1348–1359 (1993). https://doi.org/10.1007/BF01058633
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01058633