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The expression inside the second pair of integrals only depends on o$_{1}$ through T ( o$_{1}$ ). Thus we can rewrite this term as E$_{P}$$_{0}$ [ f ( T ( O$_{1}$ ))] for a fixed function f that relies on P$_{0}$ , h$_{1}$ , h$_{2}$ , and U . Under H$_{0}$ , we can rewrite this term as E$_{P}$$_{0}$ [ f ( U ( O$_{1}$ ))]. That is, we can replace each T ( O$_{1}$ ) in the second pair of integrals with U ( O$_{1}$ ). This yields Π( U ). Switching the roles of o$_{1}$ and o$_{2}$ in the first pair of integrals above and applying Fubini’s theorem shows that 

<img src='content_image/313762.jpg'>

By the same arguments used to for the second pair of integrals, the above expression is equal to Π( T ) under H$_{0}$ . By (S3), the third pair of integrals can be rewritten as 

<img src='content_image/313761.jpg'>

Proof of Theorem 2. We start by noting that 1 2 d 2 dt $_{2}$ψ$_{t}$ ∣ ∣ t =0 is equal to 

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where the penultimate equality makes use of A.4. It is easy to verify that D Ψ 2 ( o$_{1}$, o$_{2}$ ) = 

