Sample splitting with Markov chains

Anton Schick

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Sample splitting techniques play an important role in constructing estimates with prescribed influence functions in semi-parametric and nonparametric models when the observations are independent and identically distributed. This paper shows how a contiguity result can be used to modify these techniques to the case when the observations come from a stationary and ergodic Markov chain. As a consequence, sufficient conditions for the construction of efficient estimates in semi-parametric Markov chain models are obtained. The applicability of the resulting theory is demonstrated by constructing an estimate of the innovation variance in a nonparametric autoregression model, by constructing a weighted least-squares estimate with estimated weights in an autoregressive model with martingale innovations, and by constructing an efficient and adaptive estimate of the autoregression parameter in a heteroscedastic autoregressive model with symmetric errors.

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Bernoulli, Volume 7, Number 1 (2001), 33-61.

First available in Project Euclid: 29 March 2004

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contiguity efficient estimation ergodicity heteroscedastic autoregressive model nonparametric autoregressive model semi-parametric models stationary Markov chains V-uniform ergodicity weighted least-squares estimation


Schick, Anton. Sample splitting with Markov chains. Bernoulli 7 (2001), no. 1, 33--61.

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  • [1] Bickel, P.J. (1982) On adaptive estimation. Ann. Statist., 10, 647-671.
  • [2] Bhattacharya, R.N. and Lee, C. (1995) Ergodicity of nonlinear first order autoregressive models. J. Theoret. Probab., 8, 207-219. Abstract can also be found in the ISI/STMA publication
  • [3] Drost, F.C., Klaassen, C.A.J. and Werker, B.J.M. (1997) Adaptive estimation in time-series models. Ann. Statist., 25, 786-817. Abstract can also be found in the ISI/STMA publication
  • [4] Fabian, V. and Hannan, J. (1982) On estimation and adaptive estimation for locally asymptotically normal families. Z. Wahrscheinlichkeitstheorie Verw. Geb., 59, 459-478.
  • [5] Gordin, M.I. (1969) The central limit theorem for stationary processes. Soviet Math. Dokl., 10, 1174- 1176.
  • [6] Hájek, J. (1962) Asymptotically most powerful rank-order tests. Ann. Math. Statist., 33, 1124-1147.
  • [7] Jeganathan, P. (1995) Some aspects of asymptotic theory with applications to time series models. Econometric Theory, 11, 818-887. Abstract can also be found in the ISI/STMA publication
  • [8] Kessler, M., Schick, A. and Wefelmeyer, W. (1999) The information in the marginal law of a Markov chain. Preprint, Department of Mathematics, University of Binghamton. Abstract can also be found in the ISI/STMA publication
  • [9] Klaassen, C.A.J. (1987) Consistent estimation of the influence function of locally asymptotically linear estimators. Ann. Statist., 15, 1548-1563.
  • [10] Koul, H.L. and Schick, A. (1997) Efficient estimation in nonlinear autoregressive time-series models. Bernoulli, 3, 247-277. Abstract can also be found in the ISI/STMA publication
  • [11] Kreiss, J.-P. (1987a) On adaptive estimation in stationary ARMA processes. Ann. Statist., 15, 112- 133.
  • [12] Kreiss, J.-P. (1987b) On adaptive estimation in autoregressive models when there are nuisance functions. Statist. Decisions, 5, 59-76.
  • [13] LeCam, L. (1960) Locally asymptotically normal families of distributions. Certain approximations to families of distributions and their use in the theory of estimation and testing hypotheses. Univ. California Publ. Statist., 3, 37-98.
  • [14] Maercker, G. (1997) Statistical Inference in Conditional Heteroskedastic Autoregressive Models. Shaker Verlag, Aachen.
  • [15] Meyn, S.P. and Tweedie, R.L. (1993) Markov Chains and Stochastic Stability. Berlin: Springer-Verlag.
  • [16] Meyn, S.P. and Tweedie, R.L. (1994) Computable bounds for geometric convergence rates of Markov chains. Ann. Appl. Probab., 4, 981-1011. Abstract can also be found in the ISI/STMA publication
  • [17] Roberts, G.O. and Rosenthal, J.S. (1997) Geometric ergodicity and hybrid Markov chains. Electron. Comm. Probab., 2, 13-25.
  • [18] Rudin, W. (1974) Real and Complex Analysis (2nd edition). New York: McGraw-Hill.
  • [19] Schick, A. (1986) On asymptotically efficient estimation in semiparametric models. Ann. Statist., 14, 1139-1151.
  • [20] Schick, A. (1987) A note on the construction of asymptotically linear estimators. J. Statist. Plann.
  • [21] Inference, 16, 89-105. Correction 22 (1989), 269-270.
  • [22] Schick, A. (1993) On efficient estimation in regression models. Ann. Statist., 21, 1486-1521. Abstract can also be found in the ISI/STMA publication
  • [23] Correction and addendum 23 (1995), 1862-1863.
  • [24] Schick, A. (1999) Efficient estimation in a semiparametric additive autoregressive model. Stat. Inference Stoch. Process., 2, 69-98. Abstract can also be found in the ISI/STMA publication
  • [25] Schick, A. and Wefelmeyer, W. (1999) Efficient estimation of invariant distributions of some semiparametric Markov chain models. Math. Methods Statist., 8, 426-440. Abstract can also be found in the ISI/STMA publication
  • [26] Tong, H. (1990) Nonlinear Time Series: A Dynamical Approach. New York: Oxford University Press.
  • [27] van Eeden, C. (1970) Efficiency robust estimation of location. Ann. Math. Statist., 41, 172-181.
  • [28] Wefelmeyer, W. (1997) Adaptive estimators for parameters of the autoregressive function of a Markov chain. J. Statist. Plann. Inference, 58, 389-398.
  • [29] Wefelmeyer, W. (1999) Efficient estimation in Markov chain models: an introduction. In S. Ghosh (ed.), Asymptotics, Nonparametrics, and Time Series, Statist. Textbooks Monogr. 158, pp. 427- 459. New York: Dekker. Abstract can also be found in the ISI/STMA publication