• Bernoulli
  • Volume 5, Number 1 (1999), 109-123.

Reversible Markov chains and optimality of symmetrized empirical estimators

Priscilla E. Greenwood and Wolfgang Wefelmeyer

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Suppose that we want to estimate the expectation of a function of two arguments under the stationary distribution of two successive observations of a reversible Markov chain. Then the usual empirical estimator can be improved by symmetrizing. We show that the symmetrized estimator is efficient. We point out applications to discretely observed continuous-time processes. The proof is based on a result for general Markov chain models which can be used to characterize efficient estimators in any model defined by restrictions on the stationary distribution of a single or two successive observations.

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Bernoulli, Volume 5, Number 1 (1999), 109-123.

First available in Project Euclid: 12 March 2007

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discretely observed diffusions efficient estimation inference for stochastic processes martingale approximation


Greenwood, Priscilla E.; Wefelmeyer, Wolfgang. Reversible Markov chains and optimality of symmetrized empirical estimators. Bernoulli 5 (1999), no. 1, 109--123.

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  • [1] Aït-Sahalia, Y. (1996) Nonparametric pricing of interest rate derivative securities. Econometrica, 64, 527-560.
  • [2] Bibby, B.M. and Sørensen, M. (1995) Martingale estimating functions for discretely observed diffusion processes. Bernoulli, 1, 17-39.
  • [3] Bickel, P.J. (1993) Estimation in semiparametric models. In C.R. Rao (ed.), Multivariate Analysis: Future Directions, pp. 55-73. Amsterdam: North-Holland.
  • [4] Drost, F.C. and Klaassen, C.A.J. (1996) Efficient estimation in semiparametric GARCH models. J. Econometrics., 81, 193-221.
  • [5] Fill, J.A. (1991) Eigenvalue bounds on convergence to stationarity for nonreversible Markov chains, with an application to the exclusion process. Ann. Appl. Probab., 1, 62-87.
  • [6] Gordin, M.I. and Lifsic, B.A. (1978) The central limit theorem for stationary Markov processes. Sov. Math. Dokl., 19, 392-394.
  • [7] Greenwood, P.E. and Wefelmeyer, W. (1990) Efficiency of estimators for partially specified filtered models. Stochastic Processes Applic., 36, 353-370.
  • [8] Greenwood, P.E. and Wefelmeyer, W. (1995) Efficiency of empirical estimators for Markov chains. Ann. Statist., 23, 132-143.
  • [9] Greenwood, P.E. and Wefelmeyer, W. (1997) Maximum likelihood estimator and Kullback-Leibler information in misspecified Markov chain models. Teor. Veroyatnost. Primenen, 42, 169-178.
  • [10] Hajek, J. (1970) A characterization of limiting distributions of regular estimates. Z. Wahrscheinlichkeitstheorie Verw. Geb., 14, 323-330.
  • [11] Hansen, L.P. and Scheinkman, J.A. (1995) Back to the future: generating moment implications for continuous-time Markov processes. Econometrica, 63, 767-804.
  • [12] Höpfner, R. (1993) On statistics of Markov step processes: representation of log-likelihood ratio processes in filtered local models. Probab. Theory Related Fields, 94, 375-398.
  • [13] Höpfner, R., Jacod, J. and Ladelli, L. (1990) Local asymptotic normality and mixed normality for Markov statistical models. Probab. Theory Related Fields, 86, 105-129.
  • [14] Jeganathan, P. (1995) Some aspects of asymptotic theory with applications to time series models. Econometric Theory, 11, 818-887.
  • [15] Kartashov, N.V. (1985a) Criteria for uniform ergodicity and strong stability of Markov chains with a common phase space. Theory Probab. Math. Statist., 30, 71-89.
  • [16] Kartashov, N.V. (1985b) Inequalities in theorems of ergodicity and stability for Markov chains with common phase space. I. Theory Probab. Applic., 30, 247-259.
  • [17] Kessler, M., Schick, A. and Wefelmeyer, W. (1997) The information in the marginal law of a Markov chain. Preprint.
  • [18] Koul, H.L. and Schick, A. (1997) Efficient estimation in nonlinear autoregressive time series models. Bernoulli, 3, 247-277.
  • [19] Kreiss, J.-P. (1987) On adaptive estimation in stationary ARMA processes. Ann. Statist., 15, 112-133.
  • [20] Pedersen, A.R. (1995) Consistency and asymptotic normality of an approximate maximum likelihood estimator for discretely observed diffusion processes. Bernoulli, 1, 257-279.
  • [21] Penev, S. (1991) Efficient estimation of the stationary distribution for exponentially ergodic Markov chains. J. Statist. Plann. Inference, 27, 105-123.
  • [22] Wefelmeyer, W. (1996) Quasi-likelihood models and optimal inference. Ann. Statist., 24, 405-422.