Maximum Likelihood Estimation for Maximal Distribution under Sublinear Expectation

TL;DR

This paper extends maximum likelihood estimation to the framework of sublinear expectation, providing a theoretical foundation for estimating upper and lower variances relevant in financial risk models.

Contribution

It introduces a maximum likelihood estimator for maximal distributions within sublinear expectation, linking it to minimax problems and unbiased estimation.

Findings

Derived the MLE for maximal distribution parameters

Connected MLE to minimax and unbiased estimators

Supported applications in financial variance estimation

Abstract

Maximum likelihood estimation is a common method of estimating the parameters of the probability distribution from a given sample. This paper aims to introduce the maximum likelihood estimation in the framework of sublinear expectation. We find the maximum likelihood estimator for the parameters of the maximal distribution via the solution of the associated minimax problem, which coincides with the optimal unbiased estimation given by Jin and Peng \cite{JP21}. A general estimation method for samples with dependent structure is also provided. This result provides a theoretical foundation for the estimator of upper and lower variances, which is widely used in the G-VaR prediction model in finance.