Tests for normal mixtures based on the empirical characteristic function

A goodness–of–fit test for two–component homoscedastic and homothetic mixtures of normal distributions is proposed. The tests are based on a weighted L2–type distance between the empirical characteristic function and its population counterpart, where in the latter, parameters are replaced by consistent estimators. Consequently the resulting tests are consistent against general alternatives. When moment estimation is employed and as the decay of the weight function tends to infinity the test statistics approach limit values, which are related to the first nonvanishing moment equation. The new tests are compared via simulation to other omnibus tests for mixtures of normal distributions, and are applied to several real data sets.