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Asymptotic Behavior of the Log-Likelihood Function in Stochastic Processes when Based on a Random Number of Random Variables

In: Semi-Markov Models and Applications

Author

Listed:
  • George G. Roussas

    (University of California)

  • Debasis Bhattacharya

    (University of California)

Abstract

A general discrete time stochastic process is considered, and it is assumed that its probability laws are of known functioned form, except that they depend on a k-dimensional parameter θ. On the basis of a random number of random variables vn, defined on the underlying process, the log-likelihood function Λ vn is formed, and its asymptotic behavior, as n tends to (Infinity), is studied. It is shown that the asymptotic distribution of Λvn is normal, both under a fixed probability measure, as well as under certain contiguous probability measures. The same is shown to be true for a k-dimensional random vector, defined in terms of a certain quadratic mean derivative. The presence of the random variable Vn introduces some technical difficulties. These difficulties are resolved by means of certain results discussed in an appendix. The underlying idea involved is that, when in a sequence of random variables the deterministic index is replaced by Vn, the resulting sequence and the original sequence do not differ substantially in the probability sense. The paper is concluded with the suggestion of some statistical applications.

Suggested Citation

  • George G. Roussas & Debasis Bhattacharya, 1999. "Asymptotic Behavior of the Log-Likelihood Function in Stochastic Processes when Based on a Random Number of Random Variables," Springer Books, in: Jacques Janssen & Nikolaos Limnios (ed.), Semi-Markov Models and Applications, chapter 0, pages 119-147, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4613-3288-6_7
    DOI: 10.1007/978-1-4613-3288-6_7
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