Robust Estimation With Equality Constraint

Statisticians point out that the occurrence of gross error can never be completely eliminated,and robust estimation can reduce or eliminate the impact of gross error on parameter estimation as far as possible,so the study of robust estimation is very important.Currently,the commonly used robust estimation criteria include Maximum Likelihood estimation(M estimation),Ordered Linear estimation(L estimation)and Rank estimation(R estimation).Based on M estimation,this paper studies M estimation under stochastic equality constraint and fixed equality constraint.The principle of equivalent weight is used to transform M estimation into Robust Least Square estimation.The main work and contributions of this paper are as follows:(1)Deduce the expression of the estimated value of the general M estimation parameter and the expression of the influence function of the observed value.Based on the one-dimensional M estimation criterion and the definition of the onedimensional M estimation influence curve,the M estimation and the influence function of multi-dimensional parameter estimation are derived in detail.Taking the indirect adjustment model as an example,the influence functions are derived from the independent observations,and the influence functions of the relevant observations are further derived.(2)Deduce the expression of parameter estimation value in M estimation with equality constraint.The equality constraint is divided into random equality constraint and fixed equality constraint.Under the stochastic equality constraint without pollution distribution,the estimated values of parameters and the influence functions of the corresponding observations and prior information under the Least Square criterion are derived firstly.Then,according to different pollution distribution situation,robust estimation criterion is divided into M-LS estimation(the observations follow pollution distribution,the priori information follows normal distribution),LS-M estimation(the observations follow normal distribution,the priori information follows pollution distribution)and M-M estimation(both the observations and the priori information follows pollution distribution).The robust estimation of the parameters under three estimation criteria are derived.Under the constraint of fixed equation,the robust estimation of the parameters under the pollution distribution of the observations values are derived.(3)Deduce the expression of the influence function in the robust estimation of equality constraints.Under the constraint of random equation,the expressions of the influence functions of the observations and the prior information in the M-LS,LS-M and M-M estimation models are derived.Under the constraint of fixed equation,the expression of the influence function of the observations with pollution distribution is derived.(4)Random equality constraint and fixed equality constraint example calculation.The results of Least Squares estimation are obtained when no pollution distribution is calculated under stochastic equality constraints.Under different pollution distributions,the estimated parameters of the Least Square criterion and the robust estimation criterion are compared with the estimated values of the influence function.The results of the Least Squares estimation when the pollution distribution is calculated in the fixed constraint.When the observations are contaminated,the estimated values of the parameters and the estimated values of the influence function are compared between the Least Square estimation and the M estimation.