# 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:

show all references

##### References:
 [1] H.T. Banks, Jimena L. Davis. Quantifying uncertainty in the estimation of probability distributions. Mathematical Biosciences & Engineering, 2008, 5 (4) : 647-667. doi: 10.3934/mbe.2008.5.647 [2] Mingshang Hu, Shige Peng. G-Lévy processes under sublinear expectations. Probability, Uncertainty and Quantitative Risk, 2021, 6 (1) : 1-22. doi: 10.3934/puqr.2021001 [3] Mingshang Hu, Xiaojuan Li, Xinpeng Li. Convergence rate of Peng’s law of large numbers under sublinear expectations. Probability, Uncertainty and Quantitative Risk, 2021, 6 (3) : 261-266. doi: 10.3934/puqr.2021013 [4] Yongsheng Song. Stein’s method for the law of large numbers under sublinear expectations. Probability, Uncertainty and Quantitative Risk, 2021, 6 (3) : 199-212. doi: 10.3934/puqr.2021010 [5] Kaitlyn Muller. The relationship between backprojection and best linear unbiased estimation in synthetic-aperture radar imaging. Inverse Problems & Imaging, 2016, 10 (2) : 549-561. doi: 10.3934/ipi.2016011 [6] Evelyn Herberg, Michael Hinze, Henrik Schumacher. Maximal discrete sparsity in parabolic optimal control with measures. Mathematical Control & Related Fields, 2020, 10 (4) : 735-759. doi: 10.3934/mcrf.2020018 [7] Claude Stolz. On estimation of internal state by an optimal control approach for elastoplastic material. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 599-611. doi: 10.3934/dcdss.2016014 [8] Mrinal Kanti Roychowdhury. Least upper bound of the exact formula for optimal quantization of some uniform Cantor distributions. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4555-4570. doi: 10.3934/dcds.2018199 [9] Sel Ly, Nicolas Privault. Stochastic ordering by g-expectations. Probability, Uncertainty and Quantitative Risk, 2021, 6 (1) : 61-98. doi: 10.3934/puqr.2021004 [10] Hui Huang, Eldad Haber, Lior Horesh. Optimal estimation of $\ell_1$-regularization prior from a regularized empirical Bayesian risk standpoint. Inverse Problems & Imaging, 2012, 6 (3) : 447-464. doi: 10.3934/ipi.2012.6.447 [11] H. T. Banks, D. Rubio, N. Saintier, M. I. Troparevsky. Optimal design for parameter estimation in EEG problems in a 3D multilayered domain. Mathematical Biosciences & Engineering, 2015, 12 (4) : 739-760. doi: 10.3934/mbe.2015.12.739 [12] Dawan Mustafa, Bernt Wennberg. Chaotic distributions for relativistic particles. Kinetic & Related Models, 2016, 9 (4) : 749-766. doi: 10.3934/krm.2016014 [13] Axel Heim, Vladimir Sidorenko, Uli Sorger. Computation of distributions and their moments in the trellis. Advances in Mathematics of Communications, 2008, 2 (4) : 373-391. doi: 10.3934/amc.2008.2.373 [14] Fengwei Li, Qin Yue, Fengmei Liu. The weight distributions of constacyclic codes. Advances in Mathematics of Communications, 2017, 11 (3) : 471-480. doi: 10.3934/amc.2017039 [15] Emiliano Alvarez, Silvia London. Emerging patterns in inflation expectations with multiple agents. Journal of Dynamics & Games, 2020, 7 (3) : 175-184. doi: 10.3934/jdg.2020012 [16] Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu. A curve of positive solutions for an indefinite sublinear Dirichlet problem. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 817-845. doi: 10.3934/dcds.2020063 [17] Zengjing Chen, Qingyang Liu, Gaofeng Zong. Weak laws of large numbers for sublinear expectation. Mathematical Control & Related Fields, 2018, 8 (3&4) : 637-651. doi: 10.3934/mcrf.2018027 [18] BronisŁaw Jakubczyk, Wojciech Kryński. Vector fields with distributions and invariants of ODEs. Journal of Geometric Mechanics, 2013, 5 (1) : 85-129. doi: 10.3934/jgm.2013.5.85 [19] Noriaki Kawaguchi. Maximal chain continuous factor. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5915-5942. doi: 10.3934/dcds.2021101 [20] Shige Peng. Law of large numbers and central limit theorem under nonlinear expectations. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 4-. doi: 10.1186/s41546-019-0038-2

Impact Factor: