Testing the means of subgroups in the varying mixing weight model

We consider two groups divided into several subgroups and we are interested in comparing the means of two subgroups, one from each group. The samples are drawn from the two groups and the subgroup label of each observation is not defined with certainty. We show that this problem is connected to the problem of testing the expected values of mixture components with two data samples. The underlying mixture model is associated with known varying mixing weights. We provide a testing procedure which takes into account this uncertainty and performs well. Then we compare the numerical performance of this testing procedure to that of Welch’s tt-test which would have been done if true labels had been available and we assess the loss of performance of our method due to the mixing effect.