On Tests for a Normal Mean with Known Coefficient of Variation

Hinkley (1977) derived two tests for testing the mean of a normal distribution with known coefficient of variation (c.v.) for right alternatives. They are the locally most powerful (LMP) and the conditional tests based on the ancillary statistic for μ. In this paper, the likelihood ratio (LR) and Wald tests are derived for the one- and two-sided alternatives, as well as the two-sided version of the LMP test. The performances of these tests are compared with those of the classical "t", sign and Wilcoxon signed rank tests. The latter three tests do not use the information on c.v. Normal approximation is used to approximate the null distribution of the test statistics except for the "t" test. Simulation results indicate that all the tests maintain the type-I error rates, that is, the attained level is close to the nominal level of significance of the tests. The power functions of the tests are estimated through simulation. The power comparison indicates that for one-sided alternatives the LMP test is the best test whereas for the two-sided alternatives the LR or the Wald test is the best test. The "t", sign and Wilcoxon signed rank tests have lower power than the LMP, LR and Wald tests at various alternative values of μ. The power difference is quite large in several simulation configurations. Further, it is observed that the "t", sign and Wilcoxon signed rank tests have considerably lower power even for the alternatives which are far away from the null hypothesis when the c.v. is large. To study the sensitivity of the tests for the violation of the normality assumption, the type I error rates are estimated on the observations of lognormal, gamma and uniform distributions. The newly derived tests maintain the type I error rates for moderate values of c.v. Copyright 2007 The Authors. Journal compilation (c) 2007 International Statistical Institute.