Exact Bernoulli Huber modulus and indistinguishable mixtures

The weights in $P_{\theta }=(1-\theta ){\delta }_0+\theta {\delta }_1$ are nonnegative and sum to one. By [F4] and [F5], each $P_{\theta }$ is a probability law. On the full power set of $\{0,1\}$ the four events are $\mathrm{\varnothing },\{0\},\{1\}$, and $\{0,1\}$.

For the four events in step 1.1, the probability differences are either zero or $\pm ({\theta }_0-{\theta }_1)$. Thus [F3] gives $TV(P_{{\theta }_0},P_{{\theta }_1})=\mathrm{\mid }{\theta }_0-{\theta }_1\mathrm{\mid }$. Under absolute loss, the allowed parameter separation is at most $\min \{ϵ\mathrm{/}(1-ϵ),1\}$ when $ϵ<1$. If the fraction is at most $1$, the pair $0,ϵ\mathrm{/}(1-ϵ)$ attains it; if it is at least $1$, the endpoints attain separation $1$. The endpoint definition gives $\omega (1,\mathrm{\Theta })=1$.

For $0<ϵ\le 1\mathrm{/}2$, both chosen parameters lie in $[0,1]$. Under $({\theta }_0,Q_0)$ the success probability is $ϵ$; under $({\theta }_1,Q_1)$ it is $(1-ϵ)ϵ\mathrm{/}(1-ϵ)=ϵ$. Hence both mixtures are Bernoulli$(ϵ)$.

For $1\mathrm{/}2\le ϵ\le 1$, take ${\theta }_0=0,{\theta }_1=1$, set $q_0=1\mathrm{/}(2ϵ)$ and $q_1=1-1\mathrm{/}(2ϵ)$, and define $Q_j=(1-q_j){\delta }_0+q_j{\delta }_1$. The $q_j$ lie in $[0,1]$, so [F4] and [F5] make these probability laws. The mixture success probabilities are $ϵq_0=1\mathrm{/}2$ and $(1-ϵ)+ϵq_1=1\mathrm{/}2$. At $ϵ=1$ the clean part vanishes and both mixtures still equal Bernoulli$(1\mathrm{/}2)$.