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Statistical distributions and the moment problem

Research Project
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01.10.2010
 - 31.12.2025

The moment problem asks whether or not a given probability distribution is uniquely determined by its moments. The existence of moment-indeterminate distributions has been known since the late 19th century; however, first examples were often considered as mere mathematical curiosities. Recent research has shown that the problem is more widespread than previously thought and also that it is relevant in more practical contexts. Specifically, distributions that are not determined by their moments arise in applied economics, notably the modelling of economic size distributions, and also in mathematical finance. The project investigates determinacy issues for selected distributions from these fields. It will also consider the multivariate case, for which current knowledge is much more fragmentary than in the classical univariate setting. The moment problem asks, for a given distribution with distribution function (d.f.) $F$ with finite moments $m_k (F) = E[X^k] = int x^k F(dx)$, $k = 1, 2, dots, $ of all orders, whether or not $F$ is uniquely determined by the sequence ${m_k(F), k geq 1}$for a given distribution with distribution function (d.f.) $F$ with finite moments $m_k (F) = E[X^k] = int x^k F(dx)$, $k = 1, 2, dots, $ of all orders, whether or not $F$ is uniquely determined by the sequence ${m_k(F), k geq 1}$.

Collaborations & Cooperations

2025 - Participation or Organization of Collaborations on an international level
Stoyanov, Jordan, Bulgarian Academy of Sciences, Research cooperation

Publications

Kleiber, Christian (2014) ‘The generalized lognormal distribution and the Stieltjes moment problem’, Journal of Theoretical Probability, 27(4), pp. 1167–1177. Available at: https://doi.org/10.1007/s10959-013-0477-0.

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Kleiber, Christian (2013) ‘On moment indeterminacy of the Benini income distribution’, WWZ Discussion Papers. WWZ (WWZ Discussion Papers, 08). Available at: http://wwz.unibas.ch/uploads/tx_x4epublication/2013-08_Kleiber_inkl._Titelblatt_03.pdf.

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Kleiber, Christian (2013) ‘On moment indeterminacy of the Benini income distribution’, Statistical Papers, 54(4), pp. 1121–1130. Available at: https://doi.org/10.1007/s00362-013-0535-9.

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Kleiber, Christian and Stoyanov, Jordan (2013) ‘Multivariate distributions and the moment problem’, Journal of multivariate analysis, 113, pp. 7–18. Available at: https://doi.org/10.1016/j.jmva.2011.06.001.

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Members (1)

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Christian Kleiber

Principal Investigator