Abstract
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.
Highlights
The sample mean (x) and standard deviation (s) are usually used as estimators for the population mean (μ) and standard deviation (σ ), respectively, in various applications of statistical inference
We propose new asymptotically unbiased estimators for the true mean and standard deviation of normally distributed populations when outlying errors exist in the sample
UPPER AND LOWER BOUNDS FOR THE TRUE MEAN AND STANDARD DEVIATION In this subsection, we derive upper and lower bounds for the true mean and standard deviation of normal distribution in the presence of errors. i.e. we find μU, μL, σU, σL such that μL ≤ μ ≤ μU and σL ≤ σ ≤ σU given that 0 ≤ α ≤ αU < 0.5
Summary
The sample mean (x) and standard deviation (s) are usually used as estimators for the population mean (μ) and standard deviation (σ ), respectively, in various applications of statistical inference. These estimators are used in hypotheses testing, statistical process control, and outlier detection. Version of the Z-score method that depends on robust estimators for outlier detection in normally distributed samples. These estimators are the median and the median absolute deviation from the median
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