Hypothesis testing for normal distributions: A unified framework and new developments

Hypothesis testing for normal distributions is one important problem in statistics and related fields including management science, engineering science and medical science. In this paper, from a very unique perspective, we propose a unified framework to comprehensively review the existing literature on the one- and two-sample testing problems of normal distributions. The unified framework has integrated the literature in a way that it includes most commonly used tests as special cases, including the one-sample mean test, the one-sample variance test, the two-sample mean test, the two-sample variance test, and the Behrens-Fisher test. The unified framework has also put forward two new hypothesis tests that are rarely studied in the literature. To complete the puzzle, we propose two likelihood ratio test statistics to solve those new testing problems. Simulation studies and real data examples are also provided to demonstrate that our proposed test statistics are appropriate for practical implementation.