On Efficiency and Exponential Families in Stochastic Process Estimation

  • C. C. Heyde
  • P. D. Feigin
Part of the NATO Advanced Study Institutes Series book series (ASIC, volume 17)


A general definition of efficiency for stochastic process estimation is proposed and some of its ramifications are explored. Of particular importance in the definition is the form of the derivative of the logarithm of the likelihood. The question of the simplest possible form for this leads on to a discussion of extensions of the concepts of sufficiency and exponential families, the latter in a Markov process context. The paper concludes with several illustrative examples.

Key Words

Stochastic process estimation efficiency sufficiency exponential family maximum likelihood Martingale limit theory branching process first order autoregression power series distributions 


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  1. [1]
    Anderson, T. W. (1959). Ann. Math. Statist. 30, 676–687.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Billingsley, P. (1961). Statistical Inference for Markov Processes. University of Chicago Press, Chicago.zbMATHGoogle Scholar
  3. [3]
    Hall, P. G. and Heyde, C. C. (1974). Manuscript under preparation.Google Scholar
  4. [4]
    Harris, T. E. (1948). Ann. Math. Statist. 19, 474–494.zbMATHCrossRefGoogle Scholar
  5. [5]
    Heyde, C. C. (1974). Remarks on efficiency in estimation for branching processes. To appear in Biometrika.Google Scholar
  6. [6]
    Johnson, N. L. and Kotz, S. (1969). Discrete Distributions. Houghton Mifflin, Boston.zbMATHGoogle Scholar
  7. [7]
    Neveu, J. (1970). Calcul des Probabilites. 2nd ed. Masson et Cie, Paris.zbMATHGoogle Scholar
  8. [8]
    Ord, J. K. (1972). Families of Frequency Distributions. Griffin, London.zbMATHGoogle Scholar
  9. [9]
    Rao, C. R. (1965). Linear Statistical Inference and Its Applications. Wiley, New York.zbMATHGoogle Scholar
  10. [10]
    Roussas, G. G. (1972). Contiguity of Probability Measures. Cambridge University Press, Cambridge.zbMATHCrossRefGoogle Scholar

Copyright information

© D. Reidel Publishing Company, Dordrecht-Holland 1975

Authors and Affiliations

  • C. C. Heyde
    • 1
  • P. D. Feigin
    • 1
  1. 1.Department of StatisticsAustralian National UniversityCanberraAustralia

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