Arctangent M Estimation For Linear Models With Outliers

Linear model has been concerned by many statistical researchers.A large number of models are transformed on the basis of linear model,and its application is quite extensive.If the data is improperly collected in the linear model,or the stored procedure is attacked,there will be outliers in the data which may affect the result of parameter estimation.In the early days,list of statisticians tried to reduce the impact of outliers by locating and removing outliers.But for multidimensional data,it is very difficult to detect the position of outliers,and using robust estimation can reduce the influence of outliers on the estimation without detecting the position of outliers.With the coming of big data,it is not difficult to collect data.A large amount of data needs more than one machine to be stored completely.This kind of distributed data can also be contaminated,in general,the estimation under the distribution is inevitably affected by the existence of outliers.Therefore,it is important to study the robust estimation of linear models with outliers in single-machine and distributed systems.Based on the idea of penalty weighted least square M estimation(PWLS),this paper constructs a robust estimation of Arctangent M estimation(ATAN)under outliers,considering the two cases of single-machine data and distributed data.Based on the idea of M estimation,the ATAN penalty estimation is constructed to further reduce the influence of the data above the threshold on the estimation,the strong consistency of the two methods is explained,and they are further used to detect outliers in the data.By means of numerical simulation,the results of PWLS and ATAN are compared,and the effectiveness of ATAN method is illustrated.Then we consider the construction of ATAN robust estimation for distributed data,using the traditional one-shot(OS)method and CSL method,the properties of ATAN estimation under CSL are proved theoretically.In the numerical simulation,the communication cost of OS method is lower,but for the non-linear model,the effect is worse.The cost of the CSL method is slightly higher than that of the OS method due to the need for multiple iterations,but it is also suitable for nonlinear models.