Publication: Characterizations involving conditional expectations based on a functional derivative approach
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2000-02
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Abstract
We introduce a notion of the derivative with respect to a distribution function, not relating necessarily to probability, which generalizes the concept of the derivative as proposed by Lebesgue (1973). The differential calculus required to solve the linear differential equation involved in this notion of the derivative is included in the paper. The definition given here may also be considered as the inverse operator of a modified notion of the Riemmann--Stieltjes integral. Both this unified approach and the results of differential calculus allow us to characterize distributions in terms of three different types of conditional expectations. In applying these results, a test of goodness-of-fit is also indicated. Finally, two characterizations of a general Poisson process are included, based on conditional expectations. Specifically, a useful result for the homogeneous Poisson process is generalized to a general context.
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Functional derivative, H-mean lifetime (deathtime) function, General Poisson processes, Characterization results