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Application in Statistical hypothesis testing (due to LeCam).

Let $\mathcal{P}_1$ and $\mathcal{P}_0$ be to set of probability measures on the same space $(X,\mathcal{A})$, dominated by a sigma finite measure $\lambda$. Then, if $\Psi$ is the set of $[0,1]$ valued measurable functions from $X$, minimax theorem implies:

$$\inf_{\psi \in \Psi} \sup_{P_1\in \mathcal{P}_1, P_0\in \mathcal{P}_0} \int\psi dP_0 +\int1-\psi dp_1= 1-\frac{1}{2} \sup_{P_1\in conv(\mathcal{P}_1), P_0\in conv(\mathcal{P}_0)} |P_1-P_0|_1$$

where $conv(\mathcal{P}_1)$ is the set of convex combinations of elements in $\mathcal{P}_1$ and $|P_1-P_0|_1=\int|dP_1-dP_0|$ ($L_1$ distance).

The left side of the above equation is the minimal worst case sum of type I and type II error, and it is directly connected to $L_1$ distance between the sets of distributions (null and alternative hypothesis).

Additionally, if the suppremum on the right is obtained for $P_0^*$ and $P_1^*$ then the problem of finding a minimax test between $\mathcal{P}_1$ and $\mathcal{P}_0$ reduces to the problem of finding a test between $P_1^*$ and $P_0^*$. This last simple testing problem is easely solved using Neyman-Pearson lemma.

Let $\mathcal{P}_1$ and $\mathcal{P}_0$ be to set of probability measures on the same space $(X,\mathcal{A})$, dominated by a sigma finite measure $\lambda$. Then, if $\Psi$ is the set of $[0,1]$ valued measurable functions from $X$, minimax theorem implies:
$$\inf_{\psi \in \Psi} \sup_{P_1\in \mathcal{P}_1, P_0\in \mathcal{P}_0} \int\psi dP_0 +\int1-\psi dp_1= 1-\frac{1}{2} \sup_{P_1\in conv(\mathcal{P}_1), P_0\in conv(\mathcal{P}_0)} |P_1-P_0|_1$$
where $conv(\mathcal{P}_1)$ is the set of convex combinations of elements in $\mathcal{P}_1$ and $|P_1-P_0|_1=\int|dP_1-dP_0|$ ($L_1$ distance).
The left side of the above equation is the minimal worst case sum of type I and type II error, and it is directly connected to $L_1$ distance between the sets of distributions (null and alternative hypothesis).