Simultaneous Tests with Combining Functions under Normality

We propose simultaneous tests for mean and variance under the normality assumption. After formulating the null hypothesis and its alternative, we construct test statistics based on the individual p-values for the partial tests with combining functions and derive the null distributions for the combining functions. We then illustrate our procedure with industrial data and compare the efficiency among the combining functions with individual partial ones by obtaining empirical powers through a simulation study. A discussion then follows on the intersectionunion test with a combining function and simultaneous confidence region as a simultaneous inference; in addition, we discuss weighted functions and applications to the statistical quality control. Finally we comment on nonparametric simultaneous tests.