Summary: This paper presents some results on asymptotic optimality of the test procedure introduced recently by Y. Baraud, S. Huet and B. Laurent [Adaptive tests of linear hypotheses by model selection. Ann. Stat. 31, 225–251 (2003; Zbl 1018.62037)]. The optimality relies on some comparisons of the ability of the new statistic to discriminate between the null hypothesis and convergent alternatives with capability of the respective Neyman–Pearson test.