<p>Here, we consider the problem of testing hypotheses regarding whether the distribution of an observed random variable belongs to the family of Gaussian mixture distributions. We apply the general theory of the Cramér–von Mises test to the complex hypothesis that the observed random variable belongs to a five-parameter family of two-component Gaussian mixtures. This type of distribution is particularly studied in cluster analysis. The limiting distribution of such a statistic depends on unknown parameters. A formula for the limiting covariance function of the empirical process has been derived. Additionally, a new general and effective method for computing the limiting distribution of the considered statistic is proposed.</p>