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Maximum likelihood estimation for stationary point processes.

M L Puri1, P D Tuan

  • 1Department of Mathematics, Indiana University, Bloomington, IN 47405.

Proceedings of the National Academy of Sciences of the United States of America
|February 1, 1986
PubMed
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This study introduces a new log likelihood function for point processes using martingale theory. The derived approximate function ensures consistent and normal estimates, advancing statistical modeling.

Area of Science:

  • Statistics
  • Probability Theory
  • Stochastic Processes

Background:

  • Point processes are fundamental in modeling event occurrences.
  • Stochastic intensity is a key parameter in point process analysis.
  • Martingale methods offer a rigorous framework for stochastic modeling.

Purpose of the Study:

  • To derive the log likelihood function for point processes using stochastic intensities.
  • To develop a practical, approximate log likelihood function.
  • To establish the asymptotic properties of the derived estimators.

Main Methods:

  • Utilized the martingale approach to derive the log likelihood function.
  • Developed an approximate log likelihood function for practical applications.
  • Investigated the asymptotic properties of the resulting parameter estimates.

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Main Results:

  • Derived the log likelihood function in terms of stochastic intensities.
  • The approximate log likelihood function exhibits standard asymptotic properties.
  • Parameter estimates obtained are strongly consistent and asymptotically normal.
  • Derived a strong law of large numbers and a central limit theorem for continuous-time martingales.

Conclusions:

  • The martingale approach provides a robust method for point process log likelihood derivation.
  • The approximate log likelihood function is suitable for practical statistical inference.
  • The study contributes theoretical advancements in martingale theory for continuous-time processes.