AIMS Mathematics, 2020, 5(4): 3423-3433. doi: 10.3934/math.2020221

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Stein’s lemma for truncated generalized skew-elliptical random vectors

School of Statistics, Qufu Normal University, Qufu 273165, Shandong, China

Inspired by Shushi [1] and Adcock et al. [2], we consider Stein’s lemma for truncated generalized skew-elliptical random vectors. We provide two Stein’s lemmas. One is Stein’s lemma for truncated generalized skew-elliptical random vectors, the other is a special form of Stein’s lemma for truncated generalized skew-elliptical random vectors. Finally, the conditional tail expectation allocation, the lower-orthant conditional tail expectation at probability level q, the upper-orthant conditional tail expectation at probability level q, the truncated version of Wang’s premium, the multivariate tail conditional expectation and the multivariate tail covariance matrix as applications are given.
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1. T. Shushi, Stein's lemma for truncated elliptical random vectors, Stat. Probabil. Lett., 137 (2018), 297-303.    

2. C. Adcock, Z. Landsman, T. Shushi, Stein's lemma for generalized skew-elliptical random vectors, Commun. Stat-Theor. M., 2019.

3. C. M. Stein, Estimation of the mean of a multivariate normal distribution, The Annals of Statistics, 9 (1981), 1135-1151.    

4. Z. Landsman, On the generalization of Stein's lemma for elliptical class of distributions, Stat. Probabil. Lett., 76 (2006), 1012-1016.    

5. Z. Landsman, J. Nešlehová, Stein's lemma for elliptical random vectors, J. Multivariate Anal., 99 (2008), 912-927.    

6. Z. Landsman, S. Vanduffel, J. Yao, A note on Stein's lemma for multivariate elliptical distributions, J. Stat. Plan. Infer., 143 (2013), 2016-2022.    

7. C. J. Adcock, K. Shutes, On the multivariate extended skew-normal, normal-exponential, and normal-gamma distributions, Journal of Statistical Theory and Practice, 6 (2012), 636-664.    

8. C. J. Adcock, Mean-variance-skewness efficient surfaces, Stein's lemma and the multivariate extended skew-Student distribution, Eur. J. Oper. Res., 234 (2014), 392-401.    

9. J. S. Liu, Siegel's formula via Stein's identities, Stat. Probabil. Lett., 21 (1994), 247-251.    

10. K. C. Li, On principal Hessian directions for data visualization and dimension reduction: Another application of Stein's lemma, J. Am. Stat. Assoc., 87 (1992), 1025-1039.    

11. Z. Landsman, U. Makov, T. Shushi, A multivariate tail covariance measure for elliptical distributions, Insurance: Mathematics and Economics, 81 (2018), 27-35.    

12. K. T. Fang, S. Kotz, K. W. Ng, Symmetric Multivariate and Related Distributions, CRC Press, New York, 1990.

13. Z. M. Landsman, E. A. Valdez, Tail conditional expectations for elliptical distributions, North American Actuarial Journal, 7 (2003), 55-71.

14. N. Loperfido, Skewness-based projection pursuit: A computational approach, Comput. Stat. Data An., 120 (2018), 42-57.    

15. J. H. T. Kim, S. Y. Kim, Tail risk measures and risk allocation for the class of multivariate normal mean-variance mixture distributions, Insurance: Mathematics and Economics, 86 (2019), 145-157.    

16. A. Cousin, E. D. Bernardino, On multivariate extensions of conditional-tail-expectation, Insurance: Mathematics and Economics, 55 (2014), 272-282.    

17. G. De Luca, M. Genton, N. Loperfido, A multivariate skew-GARCH model, In: Econometric Analysis of Financial and Economic Time Series, Emerald Group Publishing Limited, Bingley, 2006, 33-57.

18. G. De Luca, N. Loperfido, Modelling multivariate skewness in financial returns: a SGARCH approach, The European Journal of Finance, 21 (2015), 1113-1131.    

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

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