Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2025-01-05T16:21:41.149Z Has data issue: false hasContentIssue false
  • English
  • Français

Some comments concerning a curious singularity

Published online by Cambridge University Press:  14 July 2016

Paul D. Feigin
Paul D. Feigin*
Affiliation:
Technion — Israel Institute of Technology
*
∗Postal address: Faculty of Industrial and Management Engineering, Technion — Israel Institute of Technology, Haifa, Israel.
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider the maximum likelihood estimation of θ based on continuous observation of the process X, which satisfies dXt = θXtdt + dWt. Feigin (1976) showed that, when suitably normalized, the maximum likelihood estimate is asymptotically normally distributed when the true value of θ ≠ 0. The claim that this asymptotic normality also holds for θ = 0 is shown to be false. The parallel discrete-time model is mentioned and the ramifications of these singularities to martingale central limit theory is discussed.

Keywords

MAXIMUM LIKELIHOOD ESTIMATIONORNSTEIN-UHLENBECK PROCESSMARTINGALE CENTRAL LIMIT THEORY
Type
Short Communications
Information
Journal of Applied Probability , Volume 16 , Issue 2 , June 1979 , pp. 440 - 444
Copyright
Copyright © Applied Probability Trust 1979 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anderson, T. W. (1959) On asymptotic distributions of estimates of parameters of stochastic difference equations. Ann. Math. Statist. 30, 676–687.Google Scholar
David, H. A. (1968) Order Statistics. Wiley, New York.Google Scholar
Feigin, P. D. (1976) Maximum likelihood estimation for continuous-time stochastic processes. Adv. Appl. Prob. 8, 712–736.Google Scholar
Hall, P. G. (1977) Martingale invariance principles. Ann. Prob. 5, 875–887.Google Scholar
Rootzen, H. (1977) A note on convergence to mixtures of normal distributions. Z. Wahrscheinlichkeitsth. 38, 211–216.Google Scholar
White, J. S. (1958) The limiting distribution of the serial correlation coefficient in the explosive case. Ann. Math. Statist. 29, 1188–1197.Google Scholar