Threshold inference for a mixture proportion ($\hat{q}=(\hat{p}-p_0)/(p_1-p_0)$)

For a two-component mixture model with known component tail probabilities $p_0,p_1$ above a threshold, the overall probability is $p(q)=(1-q)p_0+q{p}_1$. An empirical exceedance fraction $\hat{p}$ gives the displayed method-of-moments estimate when $p_1\ne p_0$. Under independent observations, the exceedance count has a binomial distribution, and its constrained maximum-likelihood estimate is the same expression clipped to $[0,1]$. Equal component tail probabilities make this summary uninformative about $q$.