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Characterization of laws by balancing weighting against a binary operation

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Abstract

LetL(X) be the law of a positive random variableX, andZ be positive and independent ofX. Solution pairs (L(X), L(Z)) are sought for the in-law equation\(\hat X \cong X \circ Z\) where\(L(\hat X)\) is a weighted law constructed fromL(X), and ° is a binary operation which in some sense is increasing. The class of weights includes length biasing of arbitrary order. When ° is the maximum operation a complete solution in terms of a product integral is found for arbitrary weighting. Examples are given. An identity for the length biasing operator is used when ° is multiplication to establish a general solution in terms of an already solved inverse equation. Some examples are given.

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  1. Department of Mathematics, University of Western Australia, 6907, Nedlands, WA, Australia

    Anthony G. Pakes

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  1. Anthony G. Pakes
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Pakes, A.G. Characterization of laws by balancing weighting against a binary operation. Statistical Papers 37, 123–140 (1996). https://doi.org/10.1007/BF02926577

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  • DOI: https://doi.org/10.1007/BF02926577

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