Abstract
In this paper, we give an upper bound estimation about the probability of the event that the sample average of i.i.d. interval-valued random sets is included in a closed set. The main tool is Cramér theorem in the classic theory of large deviation principle about real-valued random variables.
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Acknowledgments
Authors first thank the anonymous reviewers for those comments and suggestions, and then would like to thank the support from National Natural Science Foundation of China (No. 11401016, 11301015, 11571024) and Collaborative Innovation Center on Capital Social Construction and Social Management.
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Appendix
Appendix
Cramér theorem: Let \(X_1, X_2,\cdots ,\) be i.i.d random variables and satisfy \(Ee^{\lambda |X_1|}<\infty \) for some \(\lambda >0.\) Then for any closed set \(F\subset \mathbb {R}\), we have
and for any open set \(G\subset \mathbb {R}\), we have
where
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Wang, X., Guan, L. (2017). An Upper Bound Estimation About the Sample Average of Interval-Valued Random Sets. In: Ferraro, M., et al. Soft Methods for Data Science. SMPS 2016. Advances in Intelligent Systems and Computing, vol 456. Springer, Cham. https://doi.org/10.1007/978-3-319-42972-4_64
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DOI: https://doi.org/10.1007/978-3-319-42972-4_64
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