Supplement to Estimation of a Two-component Mixture Model with Applications to Multiple Testing

Remark 12.2. If Fs is N(μs, σ 2 s) and Fb ( 6= Fs) is N(μb, σ b ) then it can be easily shown that the problem is identifiable if and only if σs ≤ σb. When σs > σb, the model is not identifiable, an application of Lemma 2.4 gives α0 = α [ 1−(σb/σs) exp ( −σsσb(μb−μs)/2 )] . Thus, α0 increases to α as |μs − μb| tends to infinity. It should be noted that the problem is actually identifiable if we restrict ourselves to the parametric family of a two-component Gaussian mixture model.