Where is the likelihood ratio test powerful for detecting two component normal mixtures?

We compare the power of the likelihood ratio test, the Engelman-Hartigan test, two outlier tests, four goodness-of-fit tests, and eight tests of normality to detect a mixture consisting of two components that are normally distributed with different means but equal variances. We consider the entire range of mixing proportions pi, 0 < pi < 1. For pi > .85 or pi < .15, overall Fisher's skewness statistic is best with Filliben's probability plot correlation coefficient test somewhat less powerful. A combined skewness and kurtosis test, the Anderson-Darling test, and the likelihood ratio test are also competitive. For .35 < pi < .65, the Engelman-Hartigan test is best. For other mixing proportions, the likelihood ratio test is best. For situations in which the preferred test had power 50% or more, the power of the likelihood ratio test is also above 50% and within 15 percentage points of the preferred test.