# American Institute of Mathematical Sciences

September  2021, 6(3): 189-198. doi: 10.3934/puqr.2021009

## Optimal unbiased estimation for maximal distribution

 1 Mathematical Institute, Oxford University, Oxford, OX1 2JD, United Kingdom 2 School of Mathematics, Shandong University, Jinan 250100, Shandong, China

E-mail: peng@sdu.edu.cn

Received  April 16, 2021 Accepted  August 10, 2021 Published  September 2021

Fund Project: We thank Dr. Wang Hanchao who provided some very useful suggestions to improve the first draft of this paper. This research is partially supported by Zhongtai Institute of Finance, Shandong University, Tian Yuan Fund of the National Natural Science Foundation of China (Grant Nos. L1624032. and 11526205) and Chinese SAFEA (111 Project) (Grant No. B12023).

Unbiased estimation for parameters of maximal distribution is a fundamental problem in the statistical theory of sublinear expectations. In this paper, we proved that the maximum estimator is the largest unbiased estimator for the upper mean and the minimum estimator is the smallest unbiased estimator for the lower mean.

Citation: Hanqing Jin, Shige Peng. Optimal unbiased estimation for maximal distribution. Probability, Uncertainty and Quantitative Risk, 2021, 6 (3) : 189-198. doi: 10.3934/puqr.2021009
##### References:
 [1] Bayraktar, E. and Munk, A., α-Stable Limit Theorem Under Sublinear Expectation, Bernoulli, 2016, 22(4): 2548-2578. [2] Chen, Z., Strong laws of large numbers for sub-linear expectations, Science China Mathematics, 2016, 59: 945-954. doi: 10.1007/s11425-015-5095-0. [3] Hu, M., The independence under sublinear expectations, arXiv: 1107.0361, 2011. [4] Hu, Z. and Zhou, L., Multi-dimensional central limit theorems and laws of large numbers under sublinear expectations, Acta Math. Sin. (English Series), 2015, 31: 305-318. doi: 10.1007/s10114-015-3212-1. [5] Li, X., A central limit theorem for m-dependent random variables under sublinear expectations, Acta Mathematicae Applicatae Sinica (English Series), 2015, 31(2): 435-444. doi: 10.1007/s10255-015-0477-1. [6] Lin, L., Shi, Y., Wang, X. and Yang, S., Sublinear expectation linear regression, Statistics, 2013. [7] Peng, S., Backward SDE and related g-expectation, In: Nicole El Karoui and Laurent Mazliak (ed.), Backward Stochastic Differential Equations, 1997, 364: 141-159, MR1752680. [8] Peng, S., Nonlinear expectations and nonlinear Markov chains, Chin, Ann. Math., 2006, 26B(2): 159-184. [9] Peng, S., G-expectation, G-Brownian motion and related stochastic calculus of Itô type. Stochastic analysis and applications, Stochastic analysis and applications, Abel Symp., 2007, 2(2): 541-567. [10] Peng, S., A new central limit theorem under sublinear expectations, arXiv: 0803.2656v1, 2008. [11] Peng, S., Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation, Stochastic Process. Appl., 2008, 118(12): 2223-2253. [12] Peng, S., Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations, Sci. China Ser. A, 2009, 52(7): 1391-1411. doi: 10.1007/s11425-009-0121-8. [13] Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty, Springer, 2019. [14] Peng, S., Yang, S. and Yao, J., Improving value-at-risk prediction under model uncertainty, Journal of Financial Econometrics, 2021. [15] Peng, S. and Yang, S., Distributional uncertainty of the financial time series measured by G-expectation, arXiv: 2011.09226v2, 2021. [16] Rokhlin, D., Asymptotic sequential rademacher complexity of a finite function class, arXiv: 1605.03843v1, 2016. [17] Wu, P. and Chen, Z., Invariance principles for the law of the iterated logarithm under G-framework, Science China Mathematics, 2015, 58: 1251-1264. doi: 10.1007/s11425-015-5002-8. [18] Song, Y. and Lin, L., Sublinear Expectation Nonlinear Regression for the Financial Risk Measurement and Management, In: Discrete Dynamics in Nature and Society, 2013, 2013: 398750. [19] Zhang, L., Exponential inequalities under sublinear expectations with applications to laws of the iterated logarithm, arXiv: 1409.0285, 2014. [20] Zhang, L., Donsker’s invariance principle under the sub-linear expectation with an application to Chung’s law of the iterated logarithm, Communications in Mathematics and Statistics, 2015, 3: 187-214. doi: 10.1007/s40304-015-0055-0. [21] Zhang, L., Rosenthal’s inequalities for independent and negatively dependent random variables under sub-linear expectations with applications, Science China Mathematics, 2016, 59: 751-768. doi: 10.1007/s11425-015-5105-2.

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##### References:
 [1] Bayraktar, E. and Munk, A., α-Stable Limit Theorem Under Sublinear Expectation, Bernoulli, 2016, 22(4): 2548-2578. [2] Chen, Z., Strong laws of large numbers for sub-linear expectations, Science China Mathematics, 2016, 59: 945-954. doi: 10.1007/s11425-015-5095-0. [3] Hu, M., The independence under sublinear expectations, arXiv: 1107.0361, 2011. [4] Hu, Z. and Zhou, L., Multi-dimensional central limit theorems and laws of large numbers under sublinear expectations, Acta Math. Sin. (English Series), 2015, 31: 305-318. doi: 10.1007/s10114-015-3212-1. [5] Li, X., A central limit theorem for m-dependent random variables under sublinear expectations, Acta Mathematicae Applicatae Sinica (English Series), 2015, 31(2): 435-444. doi: 10.1007/s10255-015-0477-1. [6] Lin, L., Shi, Y., Wang, X. and Yang, S., Sublinear expectation linear regression, Statistics, 2013. [7] Peng, S., Backward SDE and related g-expectation, In: Nicole El Karoui and Laurent Mazliak (ed.), Backward Stochastic Differential Equations, 1997, 364: 141-159, MR1752680. [8] Peng, S., Nonlinear expectations and nonlinear Markov chains, Chin, Ann. Math., 2006, 26B(2): 159-184. [9] Peng, S., G-expectation, G-Brownian motion and related stochastic calculus of Itô type. Stochastic analysis and applications, Stochastic analysis and applications, Abel Symp., 2007, 2(2): 541-567. [10] Peng, S., A new central limit theorem under sublinear expectations, arXiv: 0803.2656v1, 2008. [11] Peng, S., Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation, Stochastic Process. Appl., 2008, 118(12): 2223-2253. [12] Peng, S., Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations, Sci. China Ser. A, 2009, 52(7): 1391-1411. doi: 10.1007/s11425-009-0121-8. [13] Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty, Springer, 2019. [14] Peng, S., Yang, S. and Yao, J., Improving value-at-risk prediction under model uncertainty, Journal of Financial Econometrics, 2021. [15] Peng, S. and Yang, S., Distributional uncertainty of the financial time series measured by G-expectation, arXiv: 2011.09226v2, 2021. [16] Rokhlin, D., Asymptotic sequential rademacher complexity of a finite function class, arXiv: 1605.03843v1, 2016. [17] Wu, P. and Chen, Z., Invariance principles for the law of the iterated logarithm under G-framework, Science China Mathematics, 2015, 58: 1251-1264. doi: 10.1007/s11425-015-5002-8. [18] Song, Y. and Lin, L., Sublinear Expectation Nonlinear Regression for the Financial Risk Measurement and Management, In: Discrete Dynamics in Nature and Society, 2013, 2013: 398750. [19] Zhang, L., Exponential inequalities under sublinear expectations with applications to laws of the iterated logarithm, arXiv: 1409.0285, 2014. [20] Zhang, L., Donsker’s invariance principle under the sub-linear expectation with an application to Chung’s law of the iterated logarithm, Communications in Mathematics and Statistics, 2015, 3: 187-214. doi: 10.1007/s40304-015-0055-0. [21] Zhang, L., Rosenthal’s inequalities for independent and negatively dependent random variables under sub-linear expectations with applications, Science China Mathematics, 2016, 59: 751-768. doi: 10.1007/s11425-015-5105-2.
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