Efficient likelihood estimation in state space models

Annals of Statistics

Volumn 34, Issue 4, 2006, Pages 2026-2068

  Fuh, Cheng Der ^a,b  

^a NATIONAL CENTRAL UNIVERSITY   (Taiwan)

^b INSTITUTE OF STATISTICAL SCIENCE   (Taiwan)

Author keywords

(g)ARCH models; ARMA models; Asymptotic expansion; Asymptotic normality; Consistency; Efficiency; Incomplete data; Iterated random functions; Markov switching models; Maximum likelihood; Stochastic volatility models

Indexed keywords

EID: 33845328063     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/009053606000000614     Document Type: Article

References (53)

1. BALL, F. G. and RICE, J. A. (1992). Stochastic models for ion channels: Introduction and bibliography. Mart. Biosci. 112 189-206.
2. BARNSLEY, M. F., ELTON, J. H. and HARDIN, D. P. (1989). Recurrent iterated functions systems. Fractal approximation. Constr. Approx. 5 3-31. MR0982722
3. BAUM, L. E. and PETRIE, T. (1966). Statistical inference for probabilistic functions of finite state Markov chains. Ann. Math. Statist. 37 1554-1563. MR0202264
4. BHATTACHARYA, R. N. and GHOSH, J. K. (1978). On the validity of the formal Edgeworth expansion. Ann. Statist. 6 434-451. MR0471142
5. BHATTACHARYA, R. N. and RANGA RAO, R. (1976). Normal Approximation and Asymptotic Expansions. Wiley, New York. MR0436272
6. BICKEL, P. and RITOV, Y. (1996). Inference in hidden Markov models. I. Local asymptotic normality in the stationary case. Bernoulli 2 199-228. MR1416863
7. BICKEL, P., RITOV, Y. and RYDÉN, T. (1998). Asymptotic normality of the maximum likelihood estimator for general hidden Markov models. Ann. Statist. 26 1614-1635. MR1647705
8. BOLLERSLEV, T. (1986). Generalized autoregressive conditional heteroscedasticity. J. Econometrics 31 307-327. MR0853051
9. BOLLERSLEV, T., ENGLE, R. F. and NELSON, D. B. (1994). ARCH models in finance. In Handbook of Econometrics 4 (R. F. Engle and D. L. McFadden, eds.) 2959-3038. North-Holland, Amsterdam. MR1315984
10. BOUGEROL, P. and PICARD, N. (1992). Strict stationary of generalized autoregressive processes. Ann. Probab. 20 1714-1730. MR1188039
11. CAINES, P. E. (1988). Linear Stochastic Systems. Wiley, New York. MR0944080
12. CLARK, P. K. (1973). A subordinated stochastic process model with finite variance for speculative prices. Econometrica 41 135-155. MR0415944
13. DIACONIS, P. and FREEDMAN, D. (1999). Iterated random functions. SIAM Rev. 41 45-76. MR1669737
14. DOUC, R. and MATIAS, C. (2001). Asymptotics of the maximum likelihood estimator for general hidden Markov models. Bernoulli 7 381-420. MR1836737
15. DOUC, R., MOULINES, É. and RYDÉN, T. (2004). Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime. Ann. Statist. 32 2254-2304. MR2102510
16. DUNSMUIR, W. (1979). A central limit theorem for parameter estimation in stationary vector time series and its application to models for a signal observed with noise. Ann. Statist. 7 490-506. MR0527485
17. ELLIOTT, R., AGGOUN, L. and MOORE, J. (1995). Hidden Markov Models: Estimation and Control. Springer, New York. MR1323178
18. ELTON, J. H. (1990). A multiplicative ergodic theorem for Lipschitz maps. Stochastic Process. Appl. 34 39-47. MR1039561
19. ENGLE, R. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50 987-1007. MR0666121
20. FAN, J. and YAO, Q. (2003). Nonlinear Time Series. Nonparametric and Parametric Methods. Springer, New York. MR1964455
21. FRANCQ, C. and ROUSSIGNOL, M. (1998). Ergodicity of autoregressive processes with Markov-switching and consistency of the maximum likelihood estimator. Statistics 32 151-173. MR1708120
22. FUH, C.-D. (2003). SPRT and CUSUM in hidden Markov models. Ann. Statist. 31 942-977. MR1994736
23. FUH, C.-D. (2004). On Bahadur efficiency of the maximum likelihood estimator in hidden Markov models. Statist. Sinica 14 127-154. MR2036765
24. FUH, C.-D. (2004). Uniform Markov renewal theory and ruin probabilities in Markov random walks. Ann. Appl. Probab. 14 1202-1241. MR2071421
25. FUH, C.-D. (2004). Asymptotic operating characteristics of an optimal change point detection in hidden Markov models. Ann. Statist. 32 2305-2339. MR2102511
26. FUH, C.-D. and LAI, T. L. (2001). Asymptotic expansions in multidimensional Markov renewal theory and first passage times for Markov random walks. Adv. in Appl. Probab. 33 652-673. MR1860094
27. GENON-CATALOT, V., JEANTHEAU, T. and LARÉDO, C. (2000). Stochastic volatility models as hidden Markov models and statistical applications. Bernoulli 6 1051-1079. MR1809735
28. GHOSH, J. K. (1994). Higher Order Asymptotics. IMS, Hayward, CA.
29. GHYSELS, E., HARVEY, A. C. and RENAULT, E. (1996). Stochastic volatility. In Statistical Methods in Finance (C. S. Maddala and C. R. Rao, eds.) 119-191. North-Holland, Amsterdam. MR1602124
30. GOLDFELD, S. M. and QUANDT, R. E. (1973). A Markov model for switching regressions. J. Econometrics 1 3-15.
31. GÖTZE, F. and HIPP, C. (1983). Asymptotic expansions for sums of weakly dependent random vectors. Z. Wahrsch. Verw. Gebiete 64 211-239. MR0714144
32. HALL, P. and YAO, Q. (2003). Inference in ARCH and GARCH models with heavy-tailed errors. Econometrica 71 285-317. MR1956860
33. HAMILTON, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57 357-384. MR0996941
34. HAMILTON, J. D. (1994). Time Series Analysis. Princeton Univ. Press. MR1278033
35. HANNAN, E. J. (1973). The asymptotic theory of linear time-series models. J. Appl. Probab. 10130-145. MR0365960
36. HARVEY, A. C., RUIZ, E. and SHEPHARD, N. (1994). Multivariate stochastic variance models. Rev. Econom. Stud. 61 247-264.
37. HENNION, H. and HERVÉ, L. (2001). Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Lecture Notes in Math. 1766. Springer, Berlin. MR1862393
38. ITÔ, H., AMARI, S.-I. and KOBAYASHI, K. (1992). Identifiability of hidden Markov information sources and their minimum degrees of freedom. IEEE Trans. Inform. Theory 38 324-333. MR1162206
39. JENSEN, J. L. and PETERSEN, N. V. (1999). Asymptotic normality of the maximum likelihood estimator in state space models. Ann. Statist. 27 514-535. MR1714719
40. KROGH, A., BROWN, M., MIAN, I. S., SJÖLANDER, K. and HAUSSLER, D. (1994). Hidden Markov models in computational biology: Applications to protein modeling. J. Molecular Biology 235 1501-1531.
41. KÜNSCH, H. R. (2001). State space and hidden Markov models. In Complex Stochastic Systems (O. E. Barndorff-Nielsen. D. R. Cox and C. Klüppelberg, eds.) 109-173. Chapman and Hall/CRC, London. MR1893412
42. LAHIRI, S. N. (1993). Refinements in asymptotic expansions for sums of weakly dependent random vectors. Ann. Probab. 21 791-799. MR1217565
43. LEE, S.-W. and HANSEN, B. E. (1994). Asymptotic theory for the GARCH(1.1) quasimaximum likelihood estimator. Econometric Theory 10 29-52. MR1279689
44. LEROUX, B. O. (1992). Maximum likelihood estimation for hidden Markov models. Stochastic Process. Appl. 40 127-143. MR1145463
45. LUMSDAINE, R. L. (1996). Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1, 1) and covariance stationary GARCH(1, 1) models. Econometrica 64 575-596. MR1385558
46. MEYN, S. P. and TWEEDIE, R. L. (1993). Markov Chains and Stochastic Stability. Springer. London. MR1287609
47. NAGAEV, S. V. (1961). More exact statement of limit theorems for homogeneous Markov chains. Theory Probab. Appl. 6 62-81. MR0131291
48. NIEMI, S. and NUMMELIN, E. (1986). On non-singular renewal kernels with an application to a semigroup of transition kernels. Stochastic Process. Appl. 22 177-202. MR0860932
49. RABINER, L. R. and JUANG, B.-H. (1993). Fundamentals of Speech Recognition. Prentice Hall, Englewood Cliffs, NJ.
50. SHEPHARD, N. (1996). Statistical aspects of ARCH and stochastic volatility. In Time Series Models in Econometrics, Finance and Other Fields (D. R. Cox, D. V. Hinkley and O. E. Barndorff-Nielsen, eds.) 1-67. Chapman and Hall, London.
51. TAYLOR, S. J. (1986). Modelling Financial Time Series. Wiley, Chichester.
52. TAYLOR, S. J. (1994). Modeling stochastic volatility: A review and comprehensive study. Math. Finance 4 183-204.
53. YAO, Q. and BROCKWELL, P. J. (2001). Gaussian maximum likelihood estimation for ARMA models. I. Time series. Unpublished manuscript.