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Statistical Inference for Irregularly Observed Processes

  • David R. Brillinger
Part of the Lecture Notes in Statistics book series (LNS, volume 25)

Abstract

1. Statistical Inference. Statistics is part of the methodology of science—pure and applied. It is pertinent to the various goals of science proper: explanation and understanding, prediction and control, discovery and application, justification, classification. Various writers have set down block diagrams illustrating how scientific enquiry proceeds and how statistics impinges on that process. We mention Bartlett (1967), Box (1976), Mohr (1977) and Parzen (1980). An early writerwas Kempthorne (1952) who set down (essentially) the following diagram.

Keywords

Point Process Statistical Inference Spike Train Stationary Time Series Partial Coherency 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Akaike, H. (1980). Likelihood and the Bayes procedure. In Bayesian Statistics (Eds. J. M. Bernado, M. H. DeGroot, D. V. Lindley and A. F. M. Smith), University Press, Valencia, pp. 141–166.Google Scholar
  2. Allison, H. (1979). Inverse unstable problems and some of their applications. Math. Scientist 4, 9–30.Google Scholar
  3. Bartlett, M. S. (1966). Stochastic Processes. University Press, Cambridge.MATHGoogle Scholar
  4. Bartlett, M.S. (1967). Inference and stochastic processes. J. R. Statist. Soc. A 130, 457–477.CrossRefGoogle Scholar
  5. Bloomfield, P. (1970). Spectral analysis with randomly missing observations. J. R. Statist. Soc. B 32, 369–380.MathSciNetMATHGoogle Scholar
  6. Box, G. E. P. (1976). Science and statistics. J. Amer. Statist. Assoc. 71, 791–799.MathSciNetMATHGoogle Scholar
  7. Breiman, L., Dixon, W., Azen, S. and Hill, M. (1981). Missing value problems in multiple regression. Proc. Amer. Statist. Assoc. Annual Meeting.Google Scholar
  8. Bretherton, F. P. and McWilliams, J. C. (1980). Estimations from irregular arrays. Rev. Geophysics Space Physics 18, 789–812.CrossRefGoogle Scholar
  9. Brillinger, D. R. (1966). Discussion. J. R. Statist. Soc. B 28, 294.Google Scholar
  10. Brillinger, D. R. (1972). The spectral analysis of stationary interval functions. In Proc. Sixth Berkeley Symp. Math. Stat. Prob. (Eds. L. M. Le Cam, J. Neyman, E. L. Scott), University of California Press, Berkeley, pp. 483–513.Google Scholar
  11. Brillinger, D. R. (1973). Estimation of the mean of a stationary time series by sampling. J. Appl. Prob. 10, 419–431.MathSciNetMATHCrossRefGoogle Scholar
  12. Brillinger, D. R. (1975). Statistical inference for stationary point processes. In Stochastic Processes and Related Topics, Vol. 1 (Ed. M. L. Puri), Academic Press, New York, pp. 55–99.Google Scholar
  13. Brillinger, D. R. (1979). Analyzing point processes subjected to random deletions. Canadian J. Stat. 7, 21–27.MathSciNetMATHCrossRefGoogle Scholar
  14. Brillinger, D. R. (1981). Time Series: Data Analysis and Theory. Holden-Day, San Francisco.MATHGoogle Scholar
  15. Brillinger, D. R. (1982). Asymptotic normality of finite Fourier transforms of stationary generalized processes. J. Multivariate Anal. 12, 64–71.MathSciNetMATHCrossRefGoogle Scholar
  16. Brillinger, D. R., Bryant, H. L. Jr. and Segundo, J. P. (1976). Identification of synaptic interactions. Biol. Cybernetics 22, 213–228.MATHCrossRefGoogle Scholar
  17. Cox, D. R. and Lewis, P. A. W. (1966). The Statistical Analysis of Series of Events. Methuen, London.MATHGoogle Scholar
  18. Dunsmuir, W. and Robinson, P. M. (1981). Estimation of time series models in the presence of missing data. J. Amer. Statist. Assoc. 76, 560–568.MATHGoogle Scholar
  19. Good, I. J. and Gaskins, R. A. (1972). Global nonparametric estimation of probability densities. Virginia J. Science 23, 171–193.MathSciNetGoogle Scholar
  20. Grandell, J. (1976). Doubly Stochastic Poisson Processes. Springer, Berlin.MATHGoogle Scholar
  21. Grenander, U. (1981). Abstract Inference. Wiley, New York.MATHGoogle Scholar
  22. Hannan, E. J. (1971). Non-linear time series regression. J. Appl. Prob. 8, 767–780.MathSciNetMATHCrossRefGoogle Scholar
  23. Hinich, M. J, (1982). Estimating signal and noise using a random array. J. Acoust. Soc. Am. 71, 97–99.MATHCrossRefGoogle Scholar
  24. Jones, R. H. (1971). Spectrum estimation with missing observations. Ann. Inst. Stat. Math. 23, 387–398.MATHCrossRefGoogle Scholar
  25. Kempthorne, O. (1962). The Design and Analysis of Experiments. Wiley, New York.Google Scholar
  26. Kimeldorf, G. and Wahba, G. (1971). Some results on Tchebycheffian spline functions. J. Math. Anal. Applic. 33, 82–95.MathSciNetMATHCrossRefGoogle Scholar
  27. Lee, W. H. K. and Brillinger, D. R. (1979). On Chinese earthquake history—an attempt to model an incomplete data set by point process analysis. Pageoph. 117, 1229–1257.CrossRefGoogle Scholar
  28. Masry, E. (1978). Poisson sampling and spectral estimation of continuous-time processes. IEEE Trans. Inf. Theory IT-24, 173–183.MathSciNetMATHCrossRefGoogle Scholar
  29. Mohr, H. (1977). Structure and Significance of Science. Springer, New York.CrossRefGoogle Scholar
  30. Niemi, H. (1978). Stationary vector measures and positive definite translation invariant bimeasures. Ann. Acad. Sci. Fenn. Ser. A 4, 209–226.MathSciNetGoogle Scholar
  31. Parzen, E. (1970). Statistical inference on time series by RKHS methods. In Proc. 12th Biennial Seminar Canadian Math. Congress (Ed. R. Pyke), Canadian Math. Cong., Montreal, pp. 1–38.Google Scholar
  32. Parzen, E. (1980). Comment. Amer. Stat. 34, 78–79.CrossRefGoogle Scholar
  33. Roberts, J. B. and Gaster, M. (1980). On the estimation of spectra from randomly sampled signals: a method of reducing variability. Proc. Roy. Soc. London 371, 235–258.MathSciNetCrossRefGoogle Scholar
  34. Schwartzchild, M. (1979). New observation-outlier-resistant methods of spectrum estimation. Ph.D. Thesis (Statistics), Princeton University.Google Scholar
  35. Tukey, J. W. (1980). Can we predict where “time series” should go next? In Directions in. Times Series (Eds. D. R. Brillinger and G. C. Tiao), Inst. Math. Stat., Hayward, pp. 1-31.Google Scholar
  36. Vapnik, V. (1982). Estimation of Dependencies Based on Empirical Data. Springer, New York.Google Scholar
  37. Wahba, G. (1981). Constrained regularization for ill posed linear operator equations, with applications in meteorology and medicine. Proc. Third Purdue Symp. on Stat. Decision Theory (Eds. S. S. Gupta and J. O. Berger).Google Scholar
  38. Whittle, P. (1953). Estimation and information in stationary time series. Ark. Math. Astron. Fys. 2, 423–434.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • David R. Brillinger
    • 1
  1. 1.University of CaliforniaBerkeleyUSA

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