Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
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Two tests using more assumptions but lower power

  • Sang Kyu Lee (Department of Statistics and Probability, Michigan State University) ;
  • Hyoung-Moon Kim (Department of Applied Statistics, Konkuk University)
  • Received : 2022.07.15
  • Accepted : 2022.10.17
  • Published : 2023.01.31
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Abstract

Intuitively, a test with more assumptions has greater power than a test with fewer assumptions. This kind of examples are abundant in the nonparametric tests vs corresponding parametric ones. In general, the nonparametric tests are less efficient in terms of asymptotic relative efficiency (ARE) compared to corresponding parametric tests (Daniel, 1990). However, this is not always true. To test equal means under independent normal samples, the usual test involves using the t-distribution with the pooled estimator of the common variance. Adding the assumption of equal sample size, we may derive another test. In this case, two tests using more assumptions were performed for univariate (multivariate) cases. For these examples, it was found that the power function of a test with more assumptions is less than or equal to that of a test with fewer assumptions. This finding can be used as an expository example in master's mathematical statistics courses.

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Acknowledgement

The corresponding author's research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A 1B07045603), and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2021R1A4A5032622).

References

  1. Daniel WW (1990). Applied Nonparametric Statistics (2nd ed), PWS-KENT Publishing Company, Boston.
  2. Hogg RV, McKean JW, and Craig AT (2019). Introduction to Mathematical Statistics (8th ed), Pearson, London.
  3. Johnson RA and Wichern DW (2007). Applied Multivariate Statistical Analysis (6th ed), Pearson, London.
  4. Lehmann EL (1999). Elements of Large-Sample Theory, Springer-Verlag, New York.