The total median statistic to monitor contaminated normal data

Abstract

Despite the advantages of the use of the normal distribution in Statistical Quality Control, the normality assumption is too restrictive for modelling real data sets, which usually exhibit asymmetry or tails heavier than the normal tails. But even in potential normal situations, there is often a small to moderate percentage of contamination in the data. In this paper, we analyze the efficiency and robustness of the total median statistic in comparison with the sample mean and the sample median to estimate the mean value of symmetric contaminated normal distributions, close to the normal, but with heavier-than-normal tails. We also compare the performance of the total median and the sample mean charts to monitor the mean value of such processes. The simulation results lead us to suggest the use of the total median statistic due to its efficiency and degree of robustness.

Keywords:

• Control charts
• Monte Carlo simulations
• order statistics
• robust estimators
• statistical process control

Acknowledgements

The authors would like to acknowledge the helpful suggestions of the anonymous reviewers.

Disclosure statement

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of this article.

Funding information

Research partially supported by National Funds through FCT – Fundação para a Ciência e a Tecnologia, projects PEst-OE/MAT/UI0006/2011, 2014 (CEA/UL).

Notes on contributors

Fernanda Otilia Figueiredo, is graduated in Mathematics from Porto University (FCUP), and obtained her Master and PhD degrees in Probability and Statistics from Lisbon University (FCUL). She is currently an Assistant Professor at Faculty of Economics, Porto University (FEP.UP) and her research activity is developed as a member of the Centre for Statistics and Applications, University of Lisbon (CEAUL). Her areas of research are Statistical Quality Control (main area), Extreme Value Theory, Data Analysis, Sampling, Robust Estimation and Applications of Statistics in Economics. She has several scientific international publications in journals/books and scientific/teaching books, and is reviewer/referee for Mathematical Reviews (MR) and for several scientific journals. She is a member of the following scientific societies: Portuguese Statistical Society (SPE), Classification and Data Analysis Portuguese Society (CLAD), International Society for Business and Industrial Statistics (ISBIS) and European Network for Business and Industrial Statistics (ENBIS).

M. Ivette Gomes, got a degree in Pure Mathematics at FCUL – Faculty of Science of Lisbon, Portugal, a PhD in Statistics, University of Sheffield, United Kingdom and a Habilitation degree in Applied Mathematics, University of Lisbon. She was a Full Professor at the Department of Statistics and Operations Research until her retirement in 2011. Her research activity has been mainly developed at CEAUL – Centre for Statistics and Applications, University of Lisbon, where she has directed a research team on Order Statistics, Extremes and Applications. She has more than a hundred scientific international publications, and has been reviewer for several scientific journals. She is a member of several scientific societies, Associate Editor of several international journals and Editor-in-Chief of Revstat, since 2003. Her main areas of scientific activity are Order Statistics and Extreme Value Theory, Computational Statistics, Simulation, Resampling Methodologies, Non-parametric Statistics and Statistical Quality Control.