Modeling and analysis of two Normal populations based on an unlabeled paired sample

Conducting hypothesis tests on two populations is essential in various applications. However, the data quality cannot always be guaranteed in practice. It is common for a data analyst to handle unlabeled data due to missing information or for the protection of privacy. This paper derives the sampling distribution of the ratio of the minimum to the maximum of two random numbers drawn from two normally distributed populations. The Likelihood Ratio Test conducted based on the Min-Max ratio distribution is able to detect the differences or shifts of means or variances based on the poor quality or confidential data consisting of unlabeled paired samples where the conventional hypothesis tests are malfunctioning. Numerical examples are provided to illustrate how to conduct the related hypothesis tests based on the derived distribution and to demonstrate the effectiveness of the proposed test by comparing it with the conventional tests on the population means or variances. In addition, to further demonstrate the practical value of the proposed methods, the potential applications related to rapid quality inspection in the mass production of symmetric mechanical pieces and applications requiring privacy protection are also discussed.