= Threshold inference for a mixture proportion
{title2=$\widehat q=(\widehat p-p_0)/(p_1-p_0)$}
For a two-component <mixture model> with known component tail probabilities $p_0,p_1$ above a threshold, the overall probability is $p(q)=(1-q)p_0+qp_1$. An empirical exceedance fraction $\widehat p$ 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 $q$.
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