The Annals of Statistics

On sequential estimation of parameters in semimartingale regression models with continuous time parameter

L. Galtchouk and V. Konev

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Abstract

We consider the problem of parameter estimation for multidimensional continuous-time linear stochastic regression models with an arbitrary finite number of unknown parameters and with martingale noise. The main result of the paper claims that the unknown parameters can be estimated with prescribed mean-square precision in this general model providing a unified description of both discrete and continuous time process. Among the conditions on the regressors there is one bounding the growth of the maximal eigenvalue of the design matrix with respect to its minimal eigenvalue. This condition is slightly stronger as compared with the corresponding conditions usually imposed on the regressors in asymptotic investigations but still it enables one to consider models with different behavior of the eigenvalues. The construction makes use of a two-step procedure based on the modified least-squares estimators and special stopping rules.

Article information

Source
Ann. Statist. Volume 29, Number 5 (2001), 1508-1536.

Dates
First available in Project Euclid: 8 February 2002

Permanent link to this document
http://projecteuclid.org/euclid.aos/1013203463

Digital Object Identifier
doi:10.1214/aos/1013203463

Mathematical Reviews number (MathSciNet)
MR1873340

Zentralblatt MATH identifier
01829064

Subjects
Primary: 62L12: Sequential estimation 62M09: Non-Markovian processes: estimation

Keywords
Weighted least-squares estimators sequential procedure estimators with prescribed precision stochastic regression semimartingales stopping times

Citation

Galtchouk, L.; Konev, V. On sequential estimation of parameters in semimartingale regression models with continuous time parameter. Ann. Statist. 29 (2001), no. 5, 1508--1536. doi:10.1214/aos/1013203463. http://projecteuclid.org/euclid.aos/1013203463.


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References

  • Anderson, T. W. and Taylor, J. B. (1976). Strong consistency of least squares estimates in normal linear regression. Ann. Statist. 4 788-790.
  • Aras, G. (1990). Sequential estimation of the mean of a first-order autoregressive process. Comm. Statist. Theory Methods 19 1639-1652.
  • Arato, M. (1982). Linear stochastic systems with constant coefficients. A statistical approach. Lecture Notes in Control Inform. Sci. 45 Springer, New York.
  • Arato, M., Kolmogorov, A. N. and Sinai, Y. G. (1962). On parameter estimation of a complex stationary gaussian process. Doklady Acad. USSR 146 747-750.
  • Borisov, V. Z. and Konev V. V. (1977). On sequential parameter estimation in discrete time processes. Automat. Remote Control 38 58-64.
  • Christopeit, N. (1986). Quasi-least-squares estimation in semimartingale regression models. Stochastics 16 255-278.
  • Darwich, A. R. and Le Breton, A. (1991). About asymptotic behavior of multidimensional Gaussian martingales in normal linear regression. Statist. Probab. Lett. 12 317-321.
  • Dellacherie, C. (1972). Capacites et processus stochastiques. Springer, New York.
  • Galtchouk, L. I. (1975). The structure of a class of martingales. Proceedings of the Seminar on Random Processes 7-32. Academy of Sciences of Lithuania, Vilnus.
  • Galtchouk, L. I. (1976). Representation of some martingales. Theory Probab. Appl. 21 613-620.
  • Galtchouk, L. I. (1980). Optional martingales. Math. Sbornik 112 483-521.
  • Galtchouk, L. I. and Maljutov, M.B. (1995). One bound for the mean duration of sequential testing homogeneity. Contributions to Statistics. Moda 4Advances in Model-Oriented Data Analysis 49-56. Springer, New York.
  • Galtchouk, L. I. and Konev, V. I. (1997). On sequential estimation of parameters in continuous time stochastic regression. Statistics and Control of Stochastic Processes. The R. Lipster Festschrift 123-138. World Scientific, Singapore.
  • Greenwood, P. and Shiryaev, A. N. (1992). Asymptotic minimaxity of a sequential estimator for a first order autoregressive model. Stochastics 38 49-65.
  • Jacod, J. (1979). Calcul stochastique et problemes de martingales. Lecture Notes in Math. 714. Springer, New York.
  • Jaukunas, A. and Khasminskii, R. Z. (1997). Estimation of parameters of linear homogeneous stochastic differential equations. Stochastic Processes Appl. 72 205-219.
  • Konev, V. V. (1985). Sequential Parameter Estimation of Stochastic Dynamical Systems. Tomsk Univ. Publishing House, Tomsk. (In Russian.)
  • Konev, V. V. and Lai, T. L. (1995). Estimators with prescribed precision in stochastic regression models. Sequential Anal. 14 179-192.
  • Konev, V. V. and Pergamenshchikov S. (1981). Sequential identification plans for dynamic systems. Automation Remote Control 42 917-924.
  • Konev, V. V. and Pergamenshchikov, S. (1992). Sequential estimation of parameters in unstable stochastic system with guaranteed mean-square accuracy. Problems Inform. Trans. 28 35-48.
  • Konev, V. V. and Pergamenshchikov,S. (1997). On guaranted estimation of the mean of an autoregressive process. Ann. Statist. 25 2127-2163.
  • Konev, V. V. and Vorobeichikov, S. E. (1980). On sequential identification of stochastic systems. Izvestia Acad. USSR Technical Cybern. N4 176-182.
  • Lai, T. L., Robbins, H. and Wei, C. Z. (1979). Strong consistency of least squares estimates in multiple regression II. J. Multivariate Anal. 9 343-361.
  • Lai, T. L. and Wei, C. Z. (1982). Least square estimates in stochastic regression regression models with application to identification and control of dynamic systems. Ann. Statist. 10 154- 166.
  • Lai, T. L. and Wei, C. Z. (1983). Asymptotic properties of general autoregressive models and strong consistency of least square estimates of their parameters. J. Multivariate Anal. 13 1-23.
  • Lai, T. L. and Ziegmund, D. (1983). Fixed-accuracy estimation of an autoregressive parameter. Ann. Statist. 11 478-485.
  • Le Breton, A. (1977). Parameter estimation in a linear stochastic differential equation. Proceedings of the 7th Prague Conference 15-32. Chechoslovak Academy of Science, Prague.
  • Le Breton, A. and Musiela, M. (1987). Strong consistency of LS-estimates in linear regression models driven by semimartingales. J. Multivariate Anal. 23 77-92.
  • Lipster, R. Sh. and Shiryaev, A. N. (1977). Statistics of Random Processes. I, II. Springer, New York.
  • Lipster, R. Sh. and Shiryaev, A. N. (1989). Theory of Martingales. Kluwer, Dordrecht.
  • Melnikov, A. V. (1996). Stochastic differential equations: nonsmoothness of coeffiicients, regression models and stochastic approximation. Russian Math. Surveys 51 43-136.
  • Melnikov, A. V. and Novikov, A. A. (1988). Sequential inference with fixed accuracy for semimartigales. Theory Probab. Appl. 33 480-494.
  • Meyer, P. A. (1975). Un cours sur les integrales stochastiques. Seminaire de Probabilites X. Lecture Notes in Math. 511. Springer, New York.
  • Novikov, A. A. (1984). Consistency of least squares estimates in regression models with martingale errors. Statistics and Control of Stochastic Processes (N. V. Krylov, R. S. Liptser and A. A. Novikov, eds.) 389-409. Springer, New York.
  • Rao, C. R. (1968). Linear Statistical Inferences and Its Applications. Wiley, New York.
  • Shiryaev, A. N. and Spokoiny, V. G. (1997). On sequential estimation of an autoregressive parameter. Stochastics Stochastic Reports 60 219-240.
  • Sriram, T. N. (1988). Sequential estimation of the autoregressive parameter in a first-order autoregressive process. Sequential Anal. 7 53-74.
  • Stricker, Ch. (1981). Les intervalles de constance de X X. Seminaire de Probabilites XVI. Lecture Notes in Math. 920 219-220. Springer, New York.