Bernoulli

  • Bernoulli
  • Volume 10, Number 5 (2004), 919-938.

Nonparametric methods of inference for finite-state, inhomogeneous Markov processes

Peter Hall and Efstathia Bura

Full-text: Open access

Abstract

In some inferential problems involving Markov process data, the inhomogeneity of the process is of central interest. One example is of a binary time series of data on the presence or absence of a species at a particular site over time. Here the two states correspond to `presence' or `absence', respectively, of the species, and the main topic of interest is temporal variation in the process. In principle this variation can be modelled parametrically, but in the absence of information about the physical mechanism causing species numbers to fluctuate, it is usually very difficult to suggest a plausible model that explains the data, at least until a more adaptive analysis is conducted. These issues argue in favour of nonparametric methods for estimating probabilities of transition, and for estimating probabilities of the process being in a given state at a given time. Such techniques, which in practice might be a prelude to parametric modelling, will be introduced and explored, under assumptions motivated by characteristics of the data set mentioned above. These assumptions will be shown to lead to consistent estimation of probabilities, and so to imply that nonparametric methodology gives accurate information about properties of the process.

Article information

Source
Bernoulli, Volume 10, Number 5 (2004), 919-938.

Dates
First available in Project Euclid: 4 November 2004

Permanent link to this document
https://projecteuclid.org/euclid.bj/1099579162

Digital Object Identifier
doi:10.3150/bj/1099579162

Mathematical Reviews number (MathSciNet)
MR2093617

Zentralblatt MATH identifier
1058.62066

Keywords
bandwidth binary time series kernel methods local linear methods nonparametric regression state probability transition probability

Citation

Hall, Peter; Bura, Efstathia. Nonparametric methods of inference for finite-state, inhomogeneous Markov processes. Bernoulli 10 (2004), no. 5, 919--938. doi:10.3150/bj/1099579162. https://projecteuclid.org/euclid.bj/1099579162


Export citation

References

  • [1] Cheetham, A.H. (1986) Tempo of evolution in a Neogene bryozoan: rates of morphologic change within and across species boundaries. Paleobiology, 12, 190-202.
  • [2] Cheetham, A.H. (1987) Tempo of evolution in a Neogene bryozoan: are trends in single morphologic characters misleading? Paleobiology, 13, 286-296.
  • [3] Cox, D.R. and Miller, H.D. (1965) The Theory of Stochastic Processes. London: Methuen.
  • [4] Fan, J. and Gijbels, I. (1996) Local Polynomial Modelling and its Applications. London: Chapman and Hall.
  • [5] Gould, S.J. (1991) Opus 200. Natural History, 100, 12-18.
  • [6] Greenwood, P.E. and Wefelmeyer, W. (1994) Nonparametric estimators for Markov step processes. Stochastic Process. Appl., 52, 1-16. Abstract can also be found in the ISI/STMA publication
  • [7] Hart, J.D. (1991) Kernel regression estimation with time series errors. J. Roy. Statist. Soc. Ser. B, 53, 173-187. Abstract can also be found in the ISI/STMA publication
  • [8] Jackson, J.B.C. and Cheetham, A.H. (1990) Evolutionary significance of morphospecies: A test with cheilostome Bryozoa. Science, 248, 579-583.
  • [9] Karlsen, H.A. and Tjøstheim, D. (2001) Nonparametric estimation in null recurrent time series. Ann. Statist., 29, 372-416. Abstract can also be found in the ISI/STMA publication
  • [10] Komlo´s, J., Major, P. and Tusnády, G. (1976) An approximation of partial sums of independent RV´s, and the sample DF. II. Z. Wahrscheinlichkeitstheorie Verw. Geb., 34, 33-58.
  • [11] Masry, E. (1996) Multivariate local polynomial regression for time series: uniform strong consistency and rates. J. Time Ser. Anal., 17, 571-599. Abstract can also be found in the ISI/STMA publication
  • [12] Masry, E. (2001) Local linear regression estimation under long-range dependence: strong consistency and rates. IEEE Trans. Inform. Theory, 47, 2863-2875. Abstract can also be found in the ISI/STMA publication
  • [13] Masry, E. and Mielniczuk, J. (2001) Local linear regression estimation for time series with long-range dependence. Stochastic Process. Appl., 82, 173-193. Abstract can also be found in the ISI/STMA publication
  • [14] Robinson, P.M. (1983) Nonparametric estimators for time series. J. Time Ser. Anal., 4, 185-207.
  • [15] Robinson, P.M. (1997) Large sample inference for nonparametric regression with dependent errors. Ann. Statist., 25, 2054-2083. Abstract can also be found in the ISI/STMA publication
  • [16] Robinson, P.M. and Hidalgo, F.J. (1997) Time series regression with long-range dependence. Ann. Statist., 25, 77-104. Abstract can also be found in the ISI/STMA publication
  • [17] Roussas, G. (1990) Nonparametric regression estimation under mixing conditions. Stochastic Process. Appl., 36, 107-116. Abstract can also be found in the ISI/STMA publication
  • [18] Roussas, G., Tran, L. and Ioannides, D.A. (1992) Fixed design regression for time series: asymptotic normality. J. Multivariate Anal., 40, 262-291. Abstract can also be found in the ISI/STMA publication
  • [19] Simonoff, J. S. (1996) Smoothing Methods in Statistics. New York: Springer-Verlag.
  • [20] Tran, L. (1993) Nonparametric function estimation for time series by local average estimators. Ann. Statist., 21, 1040-1057. Abstract can also be found in the ISI/STMA publication
  • [21] Tran, L., Roussas, G., Yakowitz, S. and Truong Van, B. (1996) Fixed-design regression for linear time series. Ann. Statist., 24, 975-991. Abstract can also be found in the ISI/STMA publication
  • [22] Truong, Y.K. and Stone, C.J. (1992) Nonparametric function estimation involving time series. Ann. Statist., 20, 77-97. Abstract can also be found in the ISI/STMA publication
  • [23] Utikal, K.J. (1997) Nonparametric inference for Markovian interval processes. Stochastic Process. Appl., 67, 1-23. Abstract can also be found in the ISI/STMA publication
  • [24] Wand, M.P. and Jones, M.C. (1995) Kernel Smoothing. London: Chapman and Hall. Abstract can also be found in the ISI/STMA publication
  • [25] Wu, J.S. and Chu, C.K. (1994) Nonparametric estimation of a regression function with dependent observations. Stochastic Process. Appl., 50, 149-160. Abstract can also be found in the ISI/STMA publication
  • [26] Yakowitz, S. (1993) Nearest neighbor regression estimation for null-recurrent Markov times series. Stochastic Process. Appl., 48, 311-318. Abstract can also be found in the ISI/STMA publication