The Adaptive Estimation Of Population Mean With Symmetric Errors

In this paper,we consider the issue of the adaptive estimation of population mean with symmetric errors.The commonly used methods are sample mean and sample me-dian.However,the sample mean is not robust when the present of outliers or the error follows a heavy-tail distribution;the alternative way is using sample median,but it may lose efficiency.Notice the objective functions of these two methods are both even functions.Based on these facts,in this paper we propose an adaptive estimation and establish its statistical theory merely using the symmetry property of regression errors and does not impose any conditions on their moments.Compared to sample mean,the new method is robust when the present of outliers or the error follows a heavy-tail distribution;on the other hand,it has a comparable performance to the sample mean when the error distri-bution is normal.We establish the consistency and asymptotic normality of the adaptive estimator to illustrate its rationality.In addition to theoretical research, we carry out a simulation study to compare these three methods in bias,standard deviation and the mean squared error,we can find the advantages of the adaptive estimation proposed in this paper.Simulation is mainly done by programming in Matlab.