Asymptotically Unbiased Estimation of Mean and Standard Deviation in the Presence of Outlying Errors

In this work, we develop a new method for estimating the mean and standard deviation of normally distributed populations when errors exist in the sample. The proposed estimators are asymptotically unbiased if all errors are outlying errors (i.e. all errors lie outside a region defined by the parameters of normal distribution). However, the proposed method requires having an upper bound for the percentage of errors in the sample. The proposed method is compared with the already existing and commonly used estimators in terms of bias, mean square error, and Pitman closeness criterion. The proposed method is found to be superior to the others when the sample size is not too small. A simulation study is designed to test the performance of proposed estimators when most but not all errors are outlying errors. The findings of the simulation study also indicate the superiority of the proposed method when the sample size is moderately large. As an application, we use our method in Phase I of designing a control chart to improve its performance. We apply the method on a dataset where the robustness of our proposed method is tested (and compared with the other estimators) against the presence of outlying errors in Phase I data. In the findings of the application, we notice that proposed estimators were the only ones that identified out of control data points in Phase II when Phase I samples are contaminated with the errors.