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When Do the Moments Uniquely Identify a Distribution

In: Directional and Multivariate Statistics

Author

Listed:
  • Carlos A. Coelho

    (NOVA University of Lisbon, Mathematics Department and NOVA Math, NOVA School of Science and Technology)

  • Rui P. Alberto

    (NOVA University of Lisbon, NOVA Math, NOVA School of Science and Technology)

  • Luís M. Grilo

    (NOVA University of Lisbon, NOVA Math, NOVA School of Science and Technology
    Évora University, Mathematics Department and CIMA (Research Center in Mathematics and Applications))

Abstract

The authors establish when do the moments $$E(X^h)$$ E ( X h ) , for h in some subset C of $$\mathbb R$$ R , uniquely identify the distribution of any positive random variable X, that is, when is $$x^h$$ x h a separating function. The simple necessary and sufficient condition is shown to be related with the existence of the moment generating function of the random variable $$Y=log X$$ Y = l o g X . The subset C of $$\mathbb R$$ R is thus the set of values of h for which the moment generating function of Y is defined. Examples of random variables characterized in this way by the set of their h-th moments are given.

Suggested Citation

  • Carlos A. Coelho & Rui P. Alberto & Luís M. Grilo, 2025. "When Do the Moments Uniquely Identify a Distribution," Springer Books, in: Somesh Kumar & Barry C. Arnold & Kunio Shimizu & Arnab Kumar Laha (ed.), Directional and Multivariate Statistics, pages 239-256, Springer.
    • Handle: RePEc:spr:sprchp:978-981-96-2004-3_13
    DOI: 10.1007/978-981-96-2004-3_13
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