## 3 Theoretical Limit Theorem  
**Theorem 1**. : _Let \(\mathbf{X}=(X_{1},X_{2},\cdots,X_{n})^{T}\) and \(\mathbf{Y}=(Y_{1},Y_{2},\cdots,Y_{n})^{T}\) be random vectors following the joint distributions \(f(\mathbf{x})\) and \(g(\mathbf{x})\), respectively, where \(m\{\mathbf{x}:f(\mathbf{x})/g(\mathbf{x})=C,\ g(\mathbf{x})\
eq 0,\ \forall C\in\mathbb{R}\} =0\). (\(m\) represents the measure of a set.) Then the following conditions are equivalent._  
1. _A monotonically increasing function_ \(r(x)\) _exists, such that_ \(l(\mathbf{x})=r[f(\mathbf{x})/g(\mathbf{x})],\ g(\mathbf{x})\
eq 0\)_, a.e._ \(\mathbf{x}\in\mathbb{R}^{n}\)_._
2. _Transformation function_ \(l(\mathbf{x})\)_produces maximum AUC. If we add a condition in Theorem that prior probability of existing and nonexistent links be_ \(P(\omega_{1})\) _and_ \(P(\omega_{2})\)_, respectively. Then the following conditions are equivalent to (a) and (b):_
3. _for any_ \(\alpha\)_, there exists the corresponding threshold_ \(\mu_{l}\) _for transformation_ \(l(\mathbf{x})\)_, and satisfies_ \(\alpha=P(\omega_{1})\int_{\mu_{l}}^{+\infty}f_{X}(x)dx+P(\omega_{2})\int_{\mu _{l}}^{+\infty}g_{Y}(x)dx\)_, such that transformation function_ \(l(\mathbf{x})\) _produces maximum Precision._  
2. For any \(\mu^{*}_{\text{FFR}}>\mu_{\text{FFR}}\), if \(E(l(\mathbf{x})>\mu^{*}_{\text{FPR}})=E(l_{2}(\mathbf{x})>\mu^{*}_{\text{2 FFR}})\) and \(E(l(\mathbf{x})>\mu_{\text{FPR}})=E(l_{2}(\mathbf{x})>\mu_{\text{2 FFR}})\), then \(\mu^{*}_{\text{2 FFR}}>\mu_{\text{2 FFR}}\). Let \(y_{1}=l(\mathbf{x})\), then a set of \(y_{1}\) exist with nonzero measure, such that \(l_{2}(\mathbf{x})\
eq r[l(\mathbf{x})]\), i.e., \(m\{y_{1}:l_{2}(\mathbf{x})\
eq r[l(\mathbf{x})]\}\
eq 0\). Let \(\sigma=\{y_{1}:l_{2}(\mathbf{x})\
eq r[l(\mathbf{x})]\}\). If \(y_{1}\in\sigma\), \(l_{2}(\mathbf{x}),l_{1}(\mathbf{x})\) satisfies function relation \(l_{2}(\mathbf{x})=s[l(\mathbf{x})]\), but \(s(x)\) is not increasing, then for any \(\mu_{1}\in\sigma\), condition (ii) does not hold. If \(y_{1}\in\sigma\), \(l_{2}(\mathbf{x})\) and \(l(\mathbf{x})\) are not functionally related, then neither condition (i) or (ii) hold. Thus \((\mathsf{b})\Rightarrow(\mathsf{a})\) is established. \((\mathsf{c})\Leftrightarrow(\mathsf{b})\): Let \(k=\text{TPR}/\text{FPR}\) be the slope of the secant for any point on the ROC curve to the origin, then \(\text{Precision}=k/(k+\lambda)\), \(\lambda=P(\omega_{2})/P(\omega_{1})\). For any \(\alpha\), that \(l(\mathbf{x})\) produces maximum Precision is equivalent that \(k\) reaches maximum. And equivalent that for any \(\alpha\), \(\alpha=P(\omega_{1})\int_{\mu_{1}}^{+\infty}f_{X}(x)dx+P(\omega_{2})\int_{\mu _{1}}^{+\infty}g_{Y}(x)dx\), \(\text{TPR}=\int_{\mu_{1}}^{+\infty}f_{X}(x)dx\) is maximum. Since this condition is established for any \(\alpha\), then it is equivalent that for any \(\text{FPR}\in[0,1]\), the corresponding TPR reaches maximum, and equivalent to \(l(\mathbf{x})\) produces maximum AUC. \(\square\)  
**Note 1:** the condition \(m\{\mathbf{x}:f(\mathbf{x})/g(\mathbf{x})=C\), \(g(\mathbf{x})\
eq 0\), \(\forall C\in\mathbb{R}\}=0\) is for exclusion that when \(f(\mathbf{x})/g(\mathbf{x})=C\), (\(C\) is a constant), transformation function can be defined randomly on set \(\sigma=\{\mathbf{x}:f(\mathbf{x})/g(\mathbf{x})=C\), \(g(\mathbf{x})\
eq 0\), \(\forall C\in\mathbb{R}\}\cap\mathbb{R}^{n}\). For example, let us construct the pdf of some random vector \(\widehat{\mathbf{X}}\) as  
\[\widetilde{f}(\mathbf{x})=\left\{ \begin{array}{l}f(\mathbf{x}),\ \mathbf{x}\in\mathbb{R}^{n} \backslash\sigma\\ kg(\mathbf{x}),\ \mathbf{x}\in\sigma\end{array} \right.,\ k\in\mathbb{R},\ k<\frac{f(\mathbf{x})}{g( \mathbf{x})}, \tag{12}\]  
\(\int_{\mathbb{R}^{n}}\widetilde{f}(\mathbf{x})d\mathbf{x}=1\). Let the transformation function be  
\[l(\mathbf{x})=\left\{ \begin{array}{l}\frac{f(\mathbf{x})}{g(\mathbf{x})},\ \mathbf{x}\in\mathbb{R}^{n} \backslash\sigma\\ l^{*}(\mathbf{x}),\ \mathbf{x}\in\sigma\end{array} \right.