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Completeness and self-decomposability of mixtures

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Summary

In this article, we first introduce the concept of strong completeness and then show that the mixture of every strongly complete distribution is complete if the mixing distribution is complete. This, in effect, reveals the completeness of several well-known mixtures. For instance, Xekalaki (1983,Ann. Inst. Statist. Math., to appear) showed that the Univariate Generalized Waring Distribution is boundedly complete only relative to one of its three parameters. Now, as a consequence of our result, it follows that this distribution is actually complete relative to any of its parameters.

Self-decomposability of mixtures is also discussed here. It is shown that a mixture of self-decomposable distributions is not necessarily self-decomposable when the mixing distribution is self-decomposable. For a special case of Poisson mixture, however, the result is valid when the mixing distribution is continuous self-decomposable, a result due to Forst (1979,Zeit. Wahrscheinlichkeitsth.,49, 349–352).

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  1. M. H. Alamatsaz

    Present address: University of Isfahan, Iran

Authors and Affiliations

  1. University of Sheffield, UK

    M. H. Alamatsaz

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  1. M. H. Alamatsaz
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Alamatsaz, M.H. Completeness and self-decomposability of mixtures. Ann Inst Stat Math 35, 355–363 (1983). https://doi.org/10.1007/BF02480991

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  • DOI: https://doi.org/10.1007/BF02480991

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