On Tests for a Normal Mean with Known Coefficient of Variation

Abstract

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.

Confidence Intervals for the Normal Mean With Known Coefficient of Variation

“…Their simulation results shown that, for the two-sided alternatives the likelihood ratio test or the Wald test is the best test. This paper extends the recent work of Bhat and Rao [1] to the confidence interval for the normal population mean with known coefficient of variation. We propose two new confidence intervals based on the estimator of the mean with known coefficient of variation of Searls [5] where he showed that the mean squares error of his proposed Suparat Niwitpong is with the Department of Applied Statistics, King Mongkut's University of Technology North Bangkok, Bangkok, THAILAND, 10800, e-mail: suparatn@kmutnb.ac.th estimator is lower than the unbiased estimator (the sample mean).…”

“…However, in practice, there are situations in area of agricultural, biological, environmental and physical sciences that a coefficients of variation are known. For example, in environmental studies, Bhat and Rao [1] argued that there are some situations that show the standard deviation of a pollutant is directly related to the mean, that means the τ is known. Furthermore in clinical chemistry, " when the batches of some substance (chemicals) are to be analyzed, if sufficient batches of the substances are analyzed, their coefficients of variation will be known".…”

“…Khan [3] , Searls ([5], [6]) and the references cited in the mentioned papers. Recent paper of Bhat and Rao [1] extended the mentioned papers to the test for the normal mean, when its coefficient of variation is known. Their simulation results shown that, for the two-sided alternatives the likelihood ratio test or the Wald test is the best test.…”

Confidence Intervals for Mean and Difference between Means of Normal Distributions with Unknown Coefficients of Variation

“…Gleser and Healy [4] proposed the minimum quadratic risk scale-invariant estimator for the normal mean with known CV. Bhat and Rao [5] investigated the tests for a normal mean with known CV. Niwitpong et al [6] provided confidence intervals for the difference between normal population means with known CVs.…”

Statistical Tests for the Reciprocal of a Normal Mean with a Known Coefficient of Variation

“…This phenomenon arises in area of agricultural, biological, environmental, and physical sciences. For instance, in environmental science, Bhat and Rao [5] explain that there are some situations that show the standard deviation of a pollutant is directly related to the mean, which means is known. In clinical chemistry, Bhat and Rao [5] also state that "when the batches of some substance (chemicals) are to be analyzed, if sufficient batches of the substances are analyzed, their coefficients of variation will be known."…”