Summary
Bounds are obtained for the Kullback-Leibler discrimination distance between two random vectorsX andY. IfX is a sequence of independent random variables whose densities have similar tail behavior andY=AX, whereA is an invertible matrix, then the bounds are a product of terms depending onA andX separately. We apply these bounds to obtain the best possible rate of convergence for any estimator of the parameters of an autoregressive process with innovations in the domain of attraction of a stable law. We provide a general theorem establishing the link between total variation proximity of measures and the rate of convergence of statistical estimates to complete the exposition for this application.
Article PDF
Similar content being viewed by others
References
Akahira, M., Takeuchi, K.: The concept of asymptotic efficiency and higher order efficiency in statistical estimation theory. (Lect. Notes Stat., vol. 7). Berlin Heidelberg New York: Springer 1981
Basawa, I.V., Prakasa Rao, B.L.S.: Statistical inference for stochastic processes. New York: Academic Press 1980
Birgé, L.: Vitesses optimales de convergence des estimateurs, in grandes deviations et applications statistiques. Société Math. de France. Astérisque 6, Paris 1979
Davis, R., Resnick, S.: Limit theory for the sample covariance and correlation functions of moving averages. Ann. Stat.14, 532–558 (1986)
Edwards, R.: Fourier series: a modern introduction, vol. II. Berlin Heidelberg New York: Springer 1982
Hannan, E.J., Hesse, C.H.: Rates of convergence for the quantile function of a linear process. Aust. J. Stat.30 A, 283–295 (1988)
Hannan, E.J., Kanter, M.: Autoregressive processes with infinite variance. J. Appl. Probab.14, 411–415 (1977)
Ibragimov, I.A., Has'Minskii, R.Z.: Statistical estimation, asymptotic theory. (Applications of math. series vol. 16.) Berlin Heidelberg New York: Springer 1981
Kanter, M., Steiger, W.L.: Regression and autoregression with infinite variance. Adv. Appl. Probab.6, 768–783 (1974)
Kanter, M.: Entropy bounds for stochastic processes and statistical applications. (1982 unpublished)
Kullback, S.: Information theory and statistics. New York: Wiley 1959
Kuratowski, K.: Introduction to set theory and topology. Reading, Mass: Pergamon Press/Addison Wesley 1962
Le Cam, L.: Notes on asymptotic methods in statistical decision theory. Centre de recherches mathématiques, Université de Montréal, Montréal
Le Cam, L.: Asymptotic methods in statistical decision theory. Berlin Heidelberg New York: Springer 1986
Loéve, M.: Probability theory, vol. 1. Berlin Heidelberg New York: Springer 1977
Roussas, G.C.: Contiguity of probability measures. Cambridge: Cambridge University Press 1972
Simon, B.: Trace ideals and their applications. London mathematical lecture notes series 35. Cambridge: Cambridge University Press 1979
Vostrikova, L.J.: On criteria forc n -consistency of estimators. Stochastics11, 265–290 (1984)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kanter, M. Discrimination distance bounds and statistical applications. Probab. Th. Rel. Fields 86, 403–422 (1990). https://doi.org/10.1007/BF01208258
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01208258