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STUDIA MATHEMATICA  Issue no. 4 / 2007  
Article: 
THE ORTHOGONAL PRINCIPLE AND CONDITIONAL DENSITIES. Authors: ION MIHOC, CRISTINA IOANA FĂTU. 

Abstract: Let X, Y 2 L2( ,K, P) be a pair of random variables, where L2( ,K, P) is the space of random variables with finite second moments. If we suppose that X is an observable random variable but Y is not, than we wish to estimate the unobservable component Y from the knowledge of observations of X. Thus, if g = g(x) is a Borel function and if the random variable g(X) is an estimator of Y, then e = E{[Y − g(X)]2} is the mean square error of this estimator. Also, if bg(X) is an optimal estimator (in the meansquare sense) of Y, then we have the following relation emin = e(Y, bg(X)) = E{[Y − bg(X)]2} = inf g E{[Y − g(X)]2}, where inf is taken over all Borel functions g = g(x). In this paper we shall present some results relative to the meansquare estimation, conditional expectations and conditional densities.  