Abstract
When a testing problem has nuisance parameters, the uniformly most powerful (UMP) tests do not generally exist. Exceptional examples were given by Dubey (1962, Skand. Aktuarietidskr., 45, 25–38; 1963, Skand. Aktuarietidskr., 46, 1–24) and Takeuchi (1968, Ann. Math. Statist., 40, 1838–1839) for the exponential distributions. What is essential for proving the existence of UMP tests lies in a special relationship between null hypothesis and the alternative. Assuming a similar relationship between them, a similar kind of result can be shown under more general situation.
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References
Dubey, S. D. (1962). A simple test function for guarantee time associated with the exponential failure law, Skand. Aktuarietidskr., 45, 25–38.
Dubey, S. D. (1963). A generalization of a simple test function for guarantee time associated with the exponential failure law, Skand. Aktuarietidskr., 46, 1–24.
Kabe, D. G. and Laurent, A. G. (1981). On some nuisance parameter free uniformly most powerful tests, Biometrical J., 23, 245–250.
Lehmann, E. L. (1986). Testing Statistical Hypothesis, Wiley, New York.
Takeuchi, K. (1968). A note on the test for the location parameter of an exponential distribution, Ann. Math. Statist., 40, 1838–1839.
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Nomakuchi, K. A note on the uniformly most powerful tests in the presence of nuisance parameters. Ann Inst Stat Math 44, 141–145 (1992). https://doi.org/10.1007/BF00048676
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DOI: https://doi.org/10.1007/BF00048676