Bernoulli

  • Bernoulli
  • Volume 7, Number 2 (2001), 243-266.

The information in the marginal law of a Markov chain

Mathieu Kessler, Anton Schick, and Wolfgang Wefelmeyer

Full-text: Open access

Abstract

If we have a parametric model for the invariant distribution of a Markov chain but cannot or do not want to use any information about the transition distribution (except, perhaps, that the chain is reversible), what is the best use we can make of the observations? We determine a lower bound for the asymptotic variance of regular estimators and show constructively that the bound is attainable. The results apply to discretely observed diffusions.

Article information

Source
Bernoulli, Volume 7, Number 2 (2001), 243-266.

Dates
First available in Project Euclid: 25 March 2004

Permanent link to this document
https://projecteuclid.org/euclid.bj/1080222086

Mathematical Reviews number (MathSciNet)
MR1828505

Zentralblatt MATH identifier
1036.62066

Keywords
discretely observed diffusion efficient estimator ergodic Markov chain

Citation

Kessler, Mathieu; Schick, Anton; Wefelmeyer, Wolfgang. The information in the marginal law of a Markov chain. Bernoulli 7 (2001), no. 2, 243--266. https://projecteuclid.org/euclid.bj/1080222086


Export citation

References

  • [1] Bibby, B.M. and S¢rensen, M. (1995) Martingale estimating functions for discretely observed diffusion processes. Bernoulli, 1, 17-39.
  • [2] Bibby, B.M. and S¢rensen, M. (1996) On estimation for discretely observed diffusions: a review. Theory Stochastic Process., 18(2), 49-56.
  • [3] Bibby, B.M. and S¢rensen, M. (1997) A hyperbolic diffusion model for stock prices. Finance Stochastics, 1, 25-41. Abstract can also be found in the ISI/STMA publication
  • [4] Bickel, P.J., Klaassen, C.A.J., Ritov, Y. and Wellner, J.A. (1993) Efficient and Adaptive Estimation for Semiparametric Models. Baltimore, MD: Johns Hopkins University Press.
  • [5] Florens-Zmirou, D. (1989) Approximate discrete-time schemes for statistics of diffusion processes. Statistics, 20, 547-557.
  • [6] Genon-Catalot, V. and Jacod, J. (1993) On the estimation of the diffusion coefficient for multidimensional diffusion processes. Ann. Inst. H. Poincarè Probab. Statist., 29, 119-151.
  • [7] Gordin, M.I. (1969) The central limit theorem for stationary processes. Soviet Math. Dokl., 10, 1174- 1176.
  • [8] Gordin, M.I. and Lifsic, B.A. (1978) The central limit theorem for stationary Markov processes. Soviet Math. Dokl., 19, 392-394.
  • [9] Greenwood, P.E. and Wefelmeyer, W. (1999) Reversible Markov chains and optimality of symmetrized empirical estimators. Bernoulli, 5, 109-123. Abstract can also be found in the ISI/STMA publication
  • [10] Hájek, J. (1970) A characterization of limiting distributions of regular estimates. Z. Wahrscheinlichkeitstheorie Verw. Geb., 14, 323-330.
  • [11] Höpfner, R. (1993a) On statistics of Markov step processes: representation of log-likelihood ratio processes in filtered local models. Probab. Theory Related Fields, 94, 375-398.
  • [12] Höpfner, R. (1993b) Asymptotic inference for Markov step processes: observation up to a random time. Stochastic Process. Appl., 48, 295-310.
  • [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] 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.
  • [15] Kartashov, N.V. (1985b) Inequalities in theorems of ergodicity and stability for Markov chains with common phase space. I. Theory Probab. Appl., 30, 247-259.
  • [16] Kartashov, N.V. (1996) Strong Stable Markov Chains. Utrecht: VSP.
  • [17] Kessler, M. (1995) Martingale estimating functions for a Markov chain. Preprint.
  • [18] Kessler, M. (1997) Estimation of an ergodic diffusion from discrete observations. Scand. J. Statist., 24, 211-229. Abstract can also be found in the ISI/STMA publication
  • [19] Kessler, M. (2000) Simple and explicit estimating functions for a discretely observed diffusion process. Scand. J. Statist., 27, 65-82. Abstract can also be found in the ISI/STMA publication
  • [20] Kessler, M. and S¢rensen, M. (1999) Estimating equations based on eigenfunctions for a discretely observed diffusion process. Bernoulli, 5, 299-314. Abstract can also be found in the ISI/STMA publication
  • [21] Le Breton, A. (1976) On continuous and discrete sampling for parameter estimation in diffusion type processes. Math. Programming Stud., 5, 124-144.
  • [22] Meyn, S.P. and Tweedie, R.L. (1993) Markov Chains and Stochastic Stability. Berlin: Springer-Verlag.
  • [23] Pedersen, A.R. (1995a) A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scand. J. Statist., 22, 55-71. Abstract can also be found in the ISI/STMA publication
  • [24] Pedersen, A.R. (1995b) Consistency and asymptotic normality of an approximate maximum likelihood estimator for discretely observed diffusion processes. Bernoulli, 1, 257-279. Abstract can also be found in the ISI/STMA publication
  • [25] Penev, S. (1991) Efficient estimation of the stationary distribution for exponentially ergodic Markov chains. J. Statist. Plann. Inference, 27, 105-123. Abstract can also be found in the ISI/STMA publication
  • [26] Penev, S. (1993) Stability of nonparametric procedures against dependence. Theory Probab. Appl., 37, 353-355.
  • [27] Pfanzagl, J. and Wefelmeyer, W. (1982) Contributions to a General Asymptotic Statistical Theory, Lecture Notes in Statist. 13. New York: Springer-Verlag.
  • [28] Roussas, G.G. (1965) Asymptotic inference in Markov processes. Ann. Math. Statist., 36, 987-992.
  • [29] Schick, A. (2001) Sample splitting with Markov chains. Bernoulli, 7, 33-61. Abstract can also be found in the ISI/STMA publication
  • [30] S¢rensen, M. (1997) Estimating functions for discretely observed diffusions: a review. In I.V. Basawa, V.P. Godambe and R.L. Taylor (eds), Selected Proceedings of the Symposium on Estimating Functions, IMS, Lecture Notes Monogr. Ser. 32, pp. 305-325. Hayward, CA: Institute of Mathematical Statistics.
  • [31] S¢rensen, M. (1999) On asymptotics of estimating functions. Brazil. J. Probab. Statist., 13, 111-136.
  • [32] Wefelmeyer, W. (1996) Quasi-likelihood models and optimal inference. Ann. Statist., 24, 405-422. Abstract can also be found in the ISI/STMA publication