Advertisement

More on Stochastic Inference

  • M. M. Rao
Part of the Mathematics and Its Applications book series (MAIA, volume 508)

Abstract

Most of the work in Chapters IV and V on inference has focussed on a simple hypothesis versus a simple alternative. When either (or both) of the latter is composite, several new problems arise. In this chapter, we consider some of these questions in detail. The results again depend on likelihood ratios, and an extension of the Neyman-Pearson-Grenander theorem is once more of importance. Parameter estimation plays a key role, often involving uncountable sets of measures, and some of the work of Pitcher and his associates is presented. This utilizes certain abstract methods, and moreover includes some results appearing here for the first time. The Gaussian dichotomy in an alternative form (without separability restrictions) is considered. A general Girsanov theorem is also established because these assertions are needed in many important applications, including the Wiener-Itô chaos, white noise and the Wick products as complements. With further detailed analysis on the more tractable Gaussian processes, the following work substantially advances that of the earlier chapters.

Keywords

Likelihood Ratio Gaussian Process Gaussian Measure Compact Interval Volterra Kernel 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliographical notes

  1. [1]
    Grenander, U. “Stochastic processes and statistical inference,” Arkiv fur Mat., 1 (1950), 195–277.MathSciNetGoogle Scholar
  2. [1]
    Velman, J. R. Likelihood ratios determined by differentiable families of isometries,Hughes Aircraft Co., Research Report No. 35(1970), (USC Ph.D. thesis, 1969, AMS Notices, 17, p.899).Google Scholar
  3. [1]
    Rao, C. R., and Varadarajan, V. S. “Discrimination of Gaussian processes,” Sankhya Ser. A25 (1963), 303–330. Google Scholar
  4. [1]
    Guichardet, A. Representation theorems for Banach function spaces,“ Mem. Am. Math. Soc., 84 (1968), 1–56.Google Scholar
  5. [1]
    Vakhania, N. N., and Tarieladze, V. I. On singularity and equivalence of Gaussian measures,“ In Real and Stochastic Analysis: Recent Advances,CRC Press, Boca Raton, FL, (1997)Google Scholar
  6. [1]
    Hitsuda, M. Representation of Gaussian processes equivalent to Wiener processes,“ Osaka J. Math. 5 (1968), 299–312.MathSciNetzbMATHGoogle Scholar
  7. [2]
    Neveu, J. Processus Aléatoires Gaussiens, U. of Montréal, Montréal, Canada, 1968. Google Scholar
  8. [1]
    Green, M. L. “Planar stochastic integration relative to quasi-martingales,” In Real and Stochastic Analysis: Recent Advances,CRC Press, Boca Raton, FL, (1997), 65–157. Google Scholar
  9. [1]
    Berger, M. A., and Mizel, V. J. “An extension of the stochastic integral,” Ann. Prob., 10 (1982), 435–450.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • M. M. Rao
    • 1
  1. 1.University of CaliforniaRiversideUSA

Personalised recommendations