The conditional expectation as estimator of normally distributed random variables with values in infinitely dimensional Banach spaces. (English) Zbl 0732.62068
The present paper deals with an infinite-dimensional linear model \(\hat Z=Z_ 0(\hat f)+Y\) where \(\hat f: \Omega \to B_ 1\) and \(Y: \Omega \to B_ 2\) denote symmetric Gaussian random variables with values in separable Banach spaces \(B_ i\). Let \(Z_ 0 : B_ 1\to B_ 2\) denote a linear operator. Based on the observations \(\hat Z\) the author gives an explicit formula for the conditional expectation of \(\hat f\) given \(\hat Z\). This result is related to the generalized least squares estimator of \(\hat f\) to \(\hat Z\). Both expressions are asymptotically the same if Y is replaced by \(\sigma\) Y with \(\sigma \downarrow 0\). The result seems to be of some importance in nonparametrics.
Reviewer: A.Janssen (Siegen)
MSC:
| 62J99 | Linear inference, regression |
| 60B11 | Probability theory on linear topological spaces |
| 28C20 | Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) |
Keywords:
infinite-dimensional linear model; symmetric Gaussian random variables; separable Banach spaces; linear operator; conditional expectation; generalized least squares estimatorReferences:
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