Linear estimators and measurable linear transformations on a Hilbert space

  • Avi Mandelbaum
Article

Summary

We consider the problem of estimating the mean of a Gaussian random vector with values in a Hilbert space. We argue that the natural class of linear estimators for the mean is the class of measurable linear transformations. We give a simple description of all measurable linear transformations with respect to a Gaussian measure. If X and θ are jointly Gaussian then E[θ¦X] is a measurable linear transformation. As an application of the general theory we describe all measurable linear transformations with respect to the Wiener measure in terms of Wiener integrals.

Keywords

Hilbert Space Conditional Expectation Gaussian Measure Trace Class Continuous Linear Operator 

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References

  1. 1.
    Cohen, A.: All admissible linear estimates of the mean vector. Ann. Math. Statist. 37, 458–463 (1966)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Doob, J.L.: Stochastic Processes. New York: Wiley 1953MATHGoogle Scholar
  3. 3.
    Kuo, H.: Gaussian Measures in Banach Spaces. Lecture Notes in Mathematics 403. BerlinHeidelberg-New York: Springer 1975Google Scholar
  4. 4.
    Ito, K.: The topological support of Gauss measure on Hilbert space. Nagoya Math. J. 38, 181–183 (1970)MathSciNetMATHGoogle Scholar
  5. 5.
    Neveu, J.: Discrete Parameter Martingales. Amsterdam: North-Holland 1975MATHGoogle Scholar
  6. 6.
    Lipster, R.S., Shiryayev, A.N.: Statistics of Random Processes I, General Theory. Berlin-Heidelberg-New York: Springer 1977Google Scholar
  7. 7.
    Reed, M., Simon, B.: Functional Analysis I (Revised and Enlarged Edition). New York: Academic Press 1980Google Scholar
  8. 8.
    Rozanov, J.A.: Infinite-dimensional Gaussian distribution. American Mathematical Society (Trans. Proceedings of the Steklov Institute of Mathematics 108) (1971)Google Scholar
  9. 9.
    Shepp, L.A.: Radon-Nikodym derivatives of Gaussian measures. Ann. Math. Statist. 37, 321–354 (1966)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Šilov, G.E., Fan Dyk Tin: Integral, Measure and Derivative on Linear Spaces. In Russian. Moscow: Nauka 1967Google Scholar
  11. 11.
    Skorohod, A.V.: Integration in Hilbert Space. Berlin-Heidelberg-New York: Springer 1974CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Avi Mandelbaum
    • 1
  1. 1.Graduate School of BusinessStanford UniversityStanfordUSA

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