Relationship Between MLE And Roots Of Likelihood Equations

Maximum Likelihood Estimation,as also called MLE,was proposed by a Ger-man mathematician C.F.Gauss in1821,but it did not attract people’s attention un-til1922,an British statist R.A.Fisher,who proposed this estimation again and ex-plained its main thought,from then on,people started to accept MLE. After so many years study,MLE now is more sophisticated,it is widely used in many fields,for ex-ample statistics,economics,finance,and so on,to estimate parameters in model.Although these years, papers about MLE’s theoretical study are not so many,this dose not mean that people have already known everything about MLE.In practice,people al-ways take the roots of likelihood equations as MLE,or take the properties of the roots of likelihood equations as MLE’s,sometimes, this is uncorrect.This paper make a summarization of formers’achievements,explain the properties of MLE and roots of likelihood equations one by one,in the end,compare these two ways,point out the differences and connections.In Section1,we introduce relevant definition and theorems which will be used later.In Section2,we introduce MLE’s properties,include,sufficiency,consistency,best asymptotic normality, Cramer asymptotic efficiency,Bahadur asymptotic efficiency.In Section3,we introduce properties of the roots of likelihood equations,include con-sistency,best asymptotic normality.In Section4,we compare the MLE and the roots of likelihood equations,from existence,uniqueness, consistency,asymptotic efficiency.In Section5,we use the example of Normal Mixture to illustrate how to find the MLE.