Maximum Likelihood Estimation Of Autoregressive And Moving Average Coefficients In ARMA(1,1)Model

Time series analysis is a branch of probability theory and mathematical statistics and a kind of dynamic data processing statistical method.Its main purpose is according to the existing historical data to predict the future.The current science in statistics,finance,trade,signal processing and measurement and other fields are widely applied to time series analysis.Time series analysis on the related fields of research can provide an effective reference method,find some unknown value from it.ARM A model?autoregressive moving average model?is an important model in time series and has an important value in practical application.In the parameter estimation,maximum likelihood estimation is usually a relatively accurate estimation method.But because of the complexity of the ARMA model of the covariance,the likelihood function is more complicated the nature of the study.ARM A?1,1?is a relatively simple model in ARMA model.In this paper,we hope to provide some reference for the related ARMA?p,q?model,and provide some theoretical support for the practical application.This paper mainly includes the following aspects:First of all,this paper introduces the background and significance of the research,and then,from the parameter estimation of the moving average model,this paper makes a review of the relevant research literature at home and abroad,and then puts forward the content of this paper.The second chapter and the third chapter are the main content of this paperThe second chapter first presents the ARMA?1,1?model,through the transformation of the random variable X calculate the likelihood function.After the parameters ?² of the log likelihood function for processing,and find that the existence and uniqueness of the maximum likelihood estimation of parameter??₁,?₁,?x,?²?is equivalent to the existence and uniqueness of the maximum value of ?????₁,?₁,?_x|X?In the third chapter,maximum likelihood problem of ARMA?1,1?model are studied in the case of ?x known.Firstly,in order to study the problem of convenience,gives the application of the theory,and gives a two element function supremum lemma existence problem.After that,the properties of ?????₁,?₁,?_x|X?are studied,and it is proved that ?????₁,?₁,?_x|X?has supremum.Finally,the promotion of the conclusions,and gives two propositions,using these two propositions can be perfectly proved that ARMA?1,1?the existence of the maximum likelihood estimation of the model.The fourth chapter is the summary of this paper and puts forward the problems that can be further studied.