Maximum Likelihood Estimation For Maximal Distribution And Its Application Under Sublinear Expectation

The traditional maximum likelihood estimation is one of the commonly used methods for parameter estimation under the classical probability system.With the deepening of research,the classical probability system can no longer meet the needs of modeling when the distribution is uncertain.Academician Peng Shige proposed a nonlinear expectation theory system.which can still carry out robust quantitative analysis and calculation of probability and statistics problems on the basis that the probability distribution cannot be determined.The maximum distribution is a commonly used distribut,ion in the nonlinear expectation theory system,and its broad applicability is derived from the law of large numbers under nonlinear expectation.This paper provides a new perspective to study the parameter estimation problem of the maximum distribution under sublinear expectation.draws on the idea of classical maximum likelihood estimation,and puts forward the minimax optimization problem for the parameter estimation of the maximum distribution based on the nonlinear expectation theory,that is,the "probability" of the sample is maximized under the condition of the minimum uncertainty of the model.By solving the solution of the minimax optimization problem,the maximum likelihood estimation of the sample parameters obeying the maximum distribution under independent samples of the same distribution is obtained,which is consistent with the existing conclusion of the optimal unbiased estimation of the maximum distribution.This paper innovatively finds that the maximum likelihood estimation is still applicable to non-independent samples,which provides a solid theoretical basis for the existing method of estimating the upper and lower difference parameters using the moving average window.