Existence And Uniqueness Of Restricted Maximum Likelihood Estimation Of Parameters In Reversible MA (1) Model

The time series analysis is a significant branch of probability statistics.The MA model is an extremely important and basic model in the time series analysis, which is widely used. And parameter is a part of model. Then the parameter estimation is always an important issue. Maximum Likelihood Estimation is one of most important and widely-used methods of parameter analysis.In the time series analysis, although the general serial distribution is usually unknown, we can assume that it is in line with the multivariate normal distribution.Then write the log-likelihood function and by its unknown parameter get the partial derivative. Thus we can get an log-likelihood equation set. Theoretically, solving the equation set can lead us to obtain the maximum likelihood estimation of the unknown parameter. However, in the MA model, the log-likelihood equation set is formed by the nonlinear equations, which can be solved under the help of computer and complicated iterative algorithm.This paper is to prove that through the mathematical analysis method, there exists the limited maximum likelihood estimation in the reversible MA model and to prove its existences and uniqueness. In the course of deduction, the mathematical analysis and related conclusion of linear model and the Matlab software would be made full use to undertake simulated estimation on the unknown outcome. Finally, the partial conclusion would be drawn.