Distributional Transformations Without Orthogonality Relations

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)-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 (k1997. doi: 10.1214/aoap/1043862419). Further applications include the derivation of Stein-type characterizations without needing to solve any Stein equation and the presentation of a general framework for estimating the distance from the distribution of a given real random variable X to that of a random variable Z, whose distribution is characterized by some mth-order linear differential operator. We also explain the fact that, in general, the biased distribution depends on the choice of the sign change points, if these are ambiguous. This new phenomenon does not appear in the framework from Goldstein and Reinert (2005).