# Outlier separability analysis with a multiple alternative hypotheses test

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## Abstract

Outlier separability analysis is a fundamental component of modern geodetic measurement analysis, positioning, navigation, and many other applications. The current theory of outlier separability is based on using two alternative hypotheses—an assumption that may not necessarily be valid. In this paper, the current theory of outlier separability is statistically analysed and then extended to the general case, where there are multiple alternative hypotheses. Taking into consideration the complexity of the critical region and the probability density function of the outlier test, the bounds of the associated statistical decision probabilities are then developed. With this theory, the probabilities of committing type I, II, and III errors can be controlled so that the probability of successful identification of an outlier can be guaranteed when performing data snooping. The theoretical findings are then demonstrated using a simulated GPS point positioning example. Detailed analysis shows that the larger the correlation coefficient, between the outlier statistics, the smaller the probability of committing a type II error and the greater the probability of committing a type III error. When the correlation coefficient is greater than 0.8, there is a far greater chance of committing a type III error than committing a type II error. In addition, to guarantee successful identification of an outlier with a set probability, the minimal detectable size of the outlier (often called the Minimal Detectable Bias or MDB) should dramatically increase with the correlation coefficient.

## Keywords

Reliability Separability Multiple alternative hypotheses Correlation coefficient## Notes

### Acknowledgments

The authors are grateful to the three reviewers for their very helpful comments, which led a major improvement in this paper. The first author is sponsored by the China Scholarship Council (CSC) for her PhD studies at the University of New South Wales, Australia. The research was partially supported by the National Natural Science Foundation of China (Project No. 41074018). The authors thank Prof. Chris Rizos for his comments on the draft of this paper. Appreciation should also be given to Dr. Yong Li, Dr. Zebo Zhou and Dr. Bofeng Li for their suggestions on the revision of this paper.

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