Abstract
Suppose we observe a stationary Markov chain with unknown transition distribution. The empirical estimator for the expectation of a function of two successive observations is known to be efficient. For reversible Markov chains, an appropriate symmetrization is efficient. For functions of more than two arguments, these estimators cease to be efficient. We determine the influence function of efficient estimators of expectations of functions of several observations, both for completely unknown and for reversible Markov chains. We construct simple efficient estimators in both cases.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bickel, P. J.: Estimation in semiparametric models, In: C. R. Rao (ed), Multivariate Analysis: Future Directions, North-Holland, Amsterdam, (1993) pp. 55-73.
Gordin, M. I.: The central limit theorem for stationary processes, Soviet Math. Dokl. 10 (1969), 1174-1176.
Greenwood, P. E. and Wefelmeyer, W.: Efficiency of empirical estimators for Markov chains, Ann. Statist. 23 (1995), 132-143.
Greenwood, P. E. and Wefelmeyer, W.: Maximum likelihood estimator and Kull-back-Leibler information in misspecified Markov chain models, Theory Probab. Appl. 42 (1998), 103-111.
Greenwood, P. E. and Wefelmeyer, W.: Reversible Markov chains and optimality of symmetrized empirical estimators, Bernoulli. 9 (1999), 109-123.
Greenwood, P. E., McKeague, I. W. and Wefelmeyer, W.: Splitting Markov fields and combining empirical estimators, (1999) (in preparation).
Györfi, L., Härdle W., Sarda, P. and Vieu, P.: Nonparametric Curve Estimation from Time Series, Lecture Notes in Statistics, vol. 60, Springer-Verlag, Berlin, (1989).
Hájek, J.: A characterization of limiting distributions of regular estimates, Z. Wahrsch. Verw. Gebiete. 14 (1970), 323-330.
Höpfner, R.: On statistics of Markov step processes: representation of log-likelihood ratio processes in filtered local models, Probab. Theory Related Fields, 94 (1993), 375-398.
Kartashov, N. V.: Criteria for uniform ergodicity and strong stability of Markov chains with a common phase space, Theory Probab. Math. Statist. 30 (1985), 71-89.
Kartashov, N. V.: Strong Stable Markov Chains, VSP, Utrecht, (1996).
Koshevnik, Y. A. and Levit, B. Y.: On a non-parametric analogue of the information matrix, Theory Probab. Appl. 21 (1976), 738-753.
Levit, B. Y.: On optimality of some statistical estimates, In: J. Hájek, (ed.), Proceedings of the Prague Symposium on Asymptotic Statistics, vol. 2, Charles University, Prague, (1974) pp. 215-238.
Meyn, S. P. and Tweedie, R. L.: Markov Chains and Stochastic Stability, Springer, London, (1993).
Penev, S.: Efficient estimation of the stationary distribution for exponentially ergodic Markov chains, J. Statist. Plann. Infer. 27 (1991), 105-123.
Roberts, G. O. and Rosenthal, J. S.: Geometric ergodicity and hybrid Markov chains, Electron. Comm. Probab. 2 (1997), 13-25.
Roussas, G. G.: Asymptotic inference in Markov processes, Ann. Math. Statist. 36 (1965), 987-992.
Roussas, G. G.: Contiguous Probability Measures: Some Applications in Statistics, Cambridge University Press, London, (1972).
Schick, A.: Sample splitting with Markov chains, Bernoulli 7 (2001), 33-61.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Schick, A., Wefelmeyer, W. Estimating Joint Distributions of Markov Chains. Statistical Inference for Stochastic Processes 5, 1–22 (2002). https://doi.org/10.1023/A:1013754529425
Issue Date:
DOI: https://doi.org/10.1023/A:1013754529425