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Sums of standard uniform random variables

Part of: Applications Distribution theory - Probability

Published online by Cambridge University Press:  01 October 2019

Tiantian Mao ,
Bin Wang  and
Ruodu Wang
Tiantian Mao*
Affiliation:
University of Science and Technology of China
Bin Wang*
Affiliation:
Chinese Academy of Sciences
Ruodu Wang*
Affiliation:
University of Waterloo
*
*Postal address: Department of Statistics and Finance, University of Science and Technology of China, Hefei, Anhui 230026, China. Email address: tmao@ustc.edu.cn
**Postal address: RCSDS, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China. Email address: wangbin@amss.ac.cn
***Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON N2L3G1, Canada. Email address: wang@uwaterloo.ca
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Abstract

In this paper, we analyse the set of all possible aggregate distributions of the sum of standard uniform random variables, a simply stated yet challenging problem in the literature of distributions with given margins. Our main results are obtained for two distinct cases. In the case of dimension two, we obtain four partial characterization results. For dimension greater than or equal to three, we obtain a full characterization of the set of aggregate distributions, which is the first complete characterization result of this type in the literature for any choice of continuous marginal distributions.

Keywords

Uniform distributionaggregationdependence uncertaintyjoint mixability

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 62P05: Applications to actuarial sciences and financial mathematics

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 3 , September 2019 , pp. 918 - 936
Copyright
© Applied Probability Trust 2019 

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References

Bernard, C., Bondarenko, O. and Vanduffel, S. (2018). Rearrangement algorithm and maximum entropy. Ann. Oper. Res. 261, 107–134.CrossRefGoogle Scholar
Bernard, C., Jiang, X. and Wang, R. (2014). Risk aggregation with dependence uncertainty. Insurance Math. Econom . 54, 93–108.CrossRefGoogle Scholar
Embrechts, P., Puccetti, G. and Rüschendorf, L. (2013). Model uncertainty and VaR aggregation. J. Bank. Finance 37 (8), 2750–2764.CrossRefGoogle Scholar
Mao, T. and Wang, R. (2015). On aggregation sets and lower-convex sets. J. Multivar. Anal. 136, 12–25.Google Scholar
McNeil, A. J., Frey, R. and Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools, revised edn. Princeton University Press.Google Scholar
Müller, A. and Stoyan, D. (2002). Comparison Methods for Statistical Models and Risks. Wiley, UK.Google Scholar
Puccetti, G. and Wang, R. (2015). Extremal dependence concepts. Statist. Sci. 30 (4), 485–517.CrossRefGoogle Scholar
Rüschendorf, L. (1982). Random variables with maximum sums. Adv. Appl. Prob. 14, 623–632.CrossRefGoogle Scholar
Rüschendorf, L. (2013). Mathematical Risk Analysis: Dependence, Risk Bounds, Optimal Allocations and Portfolios. Springer, Heidelberg.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders (Springer Series in Statistics). Springer.Google Scholar
Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist. 36 (2), 423–439.CrossRefGoogle Scholar
Vovk, V. and Wang, R. (2018). Combining p-values via averaging. Available from arXiv:1212.4966v4.Google Scholar
Wang, B. and Wang, R. (2016). Joint mixability. Math. Oper. Res. 41 (3), 808–826.CrossRefGoogle Scholar
Wang, R., Peng, L. and Yang, J. (2013). Bounds for the sum of dependent risks and worst Value-at-Risk with monotone marginal densities. Finance Stoch . 17 (2), 395–417.CrossRefGoogle Scholar