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The Hilbert Space Theoretical Foundation of Semi-Nonparametric Modeling

Semi-nonparametric (SNP) models are models where only a part of the model is parametrized, and the non-specified part is an unknown function which is represented by an infinite series expansion. Therefore, SNP models are in essence models with infinitely many parameters. The theoretical foundation of series expansions of functions is Hilbert space theory, in particular the properties of Hilbert spaces of square integrable real functions. In Hilbert spaces of functions there exist sequences of orthonormal functions such that any function in this space can be represented by a linear combination of these orthonormal functions. Such orthonormal sequences are called complete. The main purpose of this chapter is to show how these orthonormal functions can be constructed and how to construct general series representations of density and distribution functions. Moreover, in order to explain why this can be done the necessary Hilbert space theory involved will be reviewed as well.