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Distributional Transformations Without Orthogonality Relations

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  • Christian Döbler

    (Université du Luxembourg)

Abstract

Distributional transformations characterized by equations relating expectations of test functions weighted by a given biasing function on the original distribution to expectations of the test function’s higher derivatives with respect to the transformed distribution play a great role in Stein’s method and were, in great generality, first considered by Goldstein and Reinert (J Theoret Probab 18(1):237–260, 2005. doi: 10.1007/s10959-004-2602-6 ). We prove two abstract existence and uniqueness results for such distributional transformations, generalizing their $$X-P$$ X - P -bias transformation. On the one hand, we show how one can abandon previously necessary orthogonality relations by subtracting an explicitly known polynomial depending on the test function from the test function itself. On the other hand, we prove that for a given nonnegative integer m, it is possible to obtain the expectation of the m-th derivative of the test function with respect to the transformed distribution in the defining equation, even though the biasing function may have $$k

Suggested Citation

  • Christian Döbler, 2017. "Distributional Transformations Without Orthogonality Relations," Journal of Theoretical Probability, Springer, vol. 30(1), pages 85-116, March.
    • Handle: RePEc:spr:jotpro:v:30:y:2017:i:1:d:10.1007_s10959-015-0646-4
    DOI: 10.1007/s10959-015-0646-4
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    References listed on IDEAS

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    1. Larry Goldstein & Gesine Reinert, 2005. "Distributional Transformations, Orthogonal Polynomials, and Stein Characterizations," Journal of Theoretical Probability, Springer, vol. 18(1), pages 237-260, January.
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