Making socioeconomic health inequality comparisons when health concentration curves intersect

Abstract

Among the various methods adopted to compare health inequality, Makdissi and Yazbeck (J Health Econ 34:84–95, 2014) developed positional stochastic dominance conditions to identify an ordering. To reach a conclusion, their rules require that the (generalized) health concentration curve of the dominant distribution lie above that of the dominated one. However, it is frequently observed in practice that these curves intersect. Our paper proposes new criteria to cope with this problem by allowing a relatively small violation of the condition proposed by Makdissi and Yazbeck (2014). We characterize our conditions by linking them with some ethical constraints of the weight functions. We further use individual data for Côte d’Ivoire and Guinea from the Demographic and Health Survey to demonstrate the usefulness of our newly-proposed method.

Appendix

1.1 Proof of Theorem 2

• (1) “If” part: We show that if

  $$\begin{aligned} GC_{\widetilde{H}}^{2}\left( 1\right) \ge GC_{H}^{2}\left( 1\right) \end{aligned}$$

  (A.1)

  and

  $$\begin{aligned}&\underset{D}{\max }\left\{ \frac{\left( 1-2\varepsilon _{2}\right) \int _{D} \left[ GC_{H}^{2}\left( p\right) -GC_{\widetilde{H}}^{2}\left( p\right) \right] dp+\varepsilon _{2}\int _{0}^{1}\left[ GC_{H}^{2}\left( p\right) -GC_{ \widetilde{H}}^{2}\left( p\right) \right] dp}{\left( 1-2\varepsilon _{2}\right) \left| D\right| +\varepsilon _{2}}\right\} \nonumber \\&\quad \le \frac{\varepsilon _{1}}{1-2\varepsilon _{1}}\left[ GC_{\widetilde{H} }^{2}\left( 1\right) -GC_{H}^{2}\left( 1\right) \right] , \end{aligned}$$

  (A.2)

  then $A(\widetilde{H})-A\left( H\right) \ge 0$ $\forall w\in W_{2}\left( \varepsilon _{1},\varepsilon _{2}\right)$.

By integration by parts, we have

$$\begin{aligned}&A(\widetilde{H})-A\left( H\right) \nonumber \\&\quad =\int _{0}^{1}w\left( p\right) \left[ \widetilde{H}\left( p\right) -H\left( p\right) \right] dp \nonumber \\&\quad = w\left( 1\right) \left[ GC_{\widetilde{H}}^{2}\left( 1\right) -GC_{H}^{2}\left( 1\right) \right] -\int _{0}^{1}w^{\prime }\left( p\right) \left[ GC_{\widetilde{H}}^{2}\left( p\right) -GC_{H}^{2}\left( p\right) \right] dp \nonumber \\&\quad =w\left( 1\right) \left\{ \left[ GC_{\widetilde{H}}^{2}\left( 1\right) -GC_{H}^{2}\left( 1\right) \right] -\int _{0}^{1}\left[ -\frac{w^{\prime }\left( p\right) }{w\left( 1\right) }\right] \left[ GC_{H}^{2}\left( p\right) -GC_{\widetilde{H}}^{2}\left( p\right) \right] dp\right\} . \end{aligned}$$

(A.3)

By definition, $w^{\prime }\left( p\right) <0$ and $\frac{\sup \left\{ w\left( p\right) \right\} }{\inf \left\{ w\left( p\right) \right\} }\le \frac{1}{\varepsilon _{1}}-1$ for all $p\in \left[ 0,1\right]$. Since $\frac{w\left( 0\right) }{w\left( 1\right) }$ is bounded by $\frac{1}{ \varepsilon _{1}}-1$, it follows that $\int _{0}^{1}\left[ -\frac{w^{\prime }\left( p\right) }{w\left( 1\right) }\right] dp=\frac{w\left( 0\right) }{ w\left( 1\right) }-1\le \frac{1}{\varepsilon _{1}}-2$.

Let $k\left( p\right) =-\frac{w^{\prime }\left( p\right) }{w\left( 1\right) } \left( \frac{\varepsilon _{1}}{1-2\varepsilon _{1}}\right)$. Thus, $\int _{0}^{1}k\left( p\right) dp\le 1$ and $\frac{\sup \left\{ k\left( p\right) \right\} }{\inf \left\{ k\left( p\right) \right\} }\le \frac{1}{ \varepsilon _{2}}-1$. Equation (A.3) can be rewritten as

$$\begin{aligned} A(\widetilde{H})-A\left( H\right)= &\, w\left( 1\right) \left( \frac{ 1-2\varepsilon _{1}}{\varepsilon _{1}}\right) \left\{ \frac{\varepsilon _{1} }{1-2\varepsilon _{1}}\left[ GC_{\widetilde{H}}^{2}\left( 1\right) -GC_{H}^{2}\left( 1\right) \right] \right. \nonumber \\&\left. -\int _{0}^{1}k\left( p\right) \left[ GC_{H}^{2}\left( p\right) -GC_{\widetilde{H}}^{2}\left( p\right) \right] dp\right\} . \end{aligned}$$

(A.4)

According to Eq. (A.4), if

$$\begin{aligned} \int _{0}^{1}k\left( p\right) \left[ GC_{H}^{2}\left( p\right) -GC_{ \widetilde{H}}^{2}\left( p\right) \right] dp\le \frac{\varepsilon _{1}}{ 1-2\varepsilon _{1}}\left[ GC_{\widetilde{H}}^{2}\left( 1\right) -GC_{H}^{2}\left( 1\right) \right] , \end{aligned}$$

(A.5)

then $A(\widetilde{H})-A\left( H\right) \ge 0$.

If $\int _{0}^{1}k\left( p\right) \left[ GC_{H}^{2}\left( p\right) -GC_{ \widetilde{H}}^{2}\left( p\right) \right] dp\le 0$, then the above condition holds. On the other hand, if $\int _{0}^{1}k\left( p\right) \left[ GC_{H}^{2}\left( p\right) -GC_{\widetilde{H}}^{2}\left( p\right) \right] dp>0$ and $\int _{0}^{1}k\left( p\right) dp<1$, then

$$\begin{aligned} \int _{0}^{1}k\left( p\right) \left[ GC_{H}^{2}\left( p\right) -GC_{ \widetilde{H}}^{2}\left( p\right) \right] dp= & {} \int _{0}^{1}\left( \int _{0}^{1}k\left( p\right) dp\right) \frac{k\left( p\right) }{\left( \int _{0}^{1}k\left( p\right) dp\right) }\left[ GC_{H}^{2}\left( p\right) \right. \\&\left. -GC_{\widetilde{H}}^{2}\left( p\right) \right] dp \\\le & {} \int _{0}^{1}\frac{k\left( p\right) }{\left( \int _{0}^{1}k\left( p\right) dp\right) }\left[ GC_{H}^{2}\left( p\right) -GC_{\widetilde{H} }^{2}\left( p\right) \right] dp \\= & {} \int _{0}^{1}k^{*}\left( p\right) \left[ GC_{H}^{2}\left( p\right) -GC_{\widetilde{H}}^{2}\left( p\right) \right] dp, \end{aligned}$$

where $k^{*}\left( p\right) =\frac{k\left( p\right) }{ \int _{0}^{1}k\left( p\right) dp}\ge 0$, $\int _{0}^{1}k^{*}\left( p\right) dp=1$ and $\frac{\sup \left\{ k^{*}\left( p\right) \right\} }{ \inf \left\{ k^{*}\left( p\right) \right\} }\le \frac{1}{\varepsilon _{2}}-1$.

The maximum of $\int _{0}^{1}k^{*}\left( p\right) \left[ GC_{H}^{2}\left( p\right) -GC_{\widetilde{H}}^{2}\left( p\right) \right] dp$ can be written as

$$\begin{aligned} \underset{D}{\max }\left\{ \frac{\left( 1-2\varepsilon _{2}\right) \int _{D}^{{}}\left[ GC_{H}^{2}\left( p\right) -GC_{\widetilde{H}}^{2}\left( p\right) \right] dp+\varepsilon _{2}\int _{0}^{1}\left[ GC_{H}^{2}\left( p\right) -GC_{\widetilde{H}}^{2}\left( p\right) \right] dp}{\left( 1-2\varepsilon _{2}\right) \left| D\right| +\varepsilon _{2}} \right\} , \end{aligned}$$

(A.6)

where $D\subset \left[ 0,1\right]$ and $\left| D\right| =\int _{D}dp$ . To see it, let

$$\begin{aligned} k^{*}\left( p\right) =\left\{ \begin{array}{ll} 1-\varepsilon _{2}, &{}\quad \text {if }p\in D \\ \varepsilon _{2}, &{} \quad \text {if }p\notin D \end{array} \right. \end{aligned}$$

and therefore the term $\left( 1-2\varepsilon _{2}\right) \left| D\right| +\varepsilon _{2}$ in the denominator of Eq. (A.6) is the normalization factor which ensures that $\int _{0}^{1}k^{*}\left( p\right) dp=1$.

Thus, according to Eq. (A.6), if

$$\begin{aligned}&\underset{D}{\max }\left\{ \frac{\left( 1-2\varepsilon _{2}\right) \int _{D} \left[ GC_{H}^{2}\left( p\right) -GC_{\widetilde{H}}^{2}\left( p\right) \right] dp+\varepsilon _{2}\int _{0}^{1}\left[ GC_{H}^{2}\left( p\right) -GC_{ \widetilde{H}}^{2}\left( p\right) \right] dp}{\left( 1-2\varepsilon _{2}\right) \left| D\right| +\varepsilon _{2}}\right\} \\&\quad \le \frac{\varepsilon _{1}}{1-2\varepsilon _{1}}\left[ GC_{\widetilde{H} }^{2}\left( 1\right) -GC_{H}^{2}\left( 1\right) \right] , \end{aligned}$$

then $A(\widetilde{H})-A\left( H\right) \ge 0$. The sufficient condition for the Theorem is proven.

• (2) “Only if” part: We show that if

  $$\begin{aligned} GC_{\widetilde{H}}^{2}\left( 1\right) <GC_{H}^{2}\left( 1\right) \end{aligned}$$

  (A.7)

  or

  $$\begin{aligned}&\underset{D}{\max }\left\{ \frac{\left( 1-2\varepsilon _{2}\right) \int _{D} \left[ GC_{H}^{2}\left( p\right) -GC_{\widetilde{H}}^{2}\left( p\right) \right] dp+\varepsilon _{2}\int _{0}^{1}\left[ GC_{H}^{2}\left( p\right) -GC_{ \widetilde{H}}^{2}\left( p\right) \right] dp}{\left( 1-2\varepsilon _{2}\right) \left| D\right| +\varepsilon _{2}}\right\} \nonumber \\&\quad >\frac{\varepsilon _{1}}{1-2\varepsilon _{1}}\left[ GC_{\widetilde{H} }^{2}\left( 1\right) -GC_{H}^{2}\left( 1\right) \right] , \end{aligned}$$

  (A.8)

  then there exists a $w\in W_{2}\left( \varepsilon _{1},\varepsilon _{2}\right)$ such that $A(\widetilde{H})-A\left( H\right) <0$.

We first show that if Eq. (A.7) holds, then $\exists w\in W_{2}\left( \varepsilon _{1},\varepsilon _{2}\right)$ such that $A(\widetilde{H})-A\left( H\right) <0$. Let $\theta$ be a constant, and define a weight function $w\in W_{2}\left( \varepsilon _{1},\varepsilon _{2}\right)$ such that $w\left( 0\right) =\frac{1}{1-\frac{\theta }{2}}$, $w\left( 1\right) =\frac{1-\theta }{1-\frac{\theta }{2}}$, and $w^{\prime }\left( p\right) =-\theta$. To guarantee $w\left( 0\right)>w\left( 1\right) >0$ and $w^{\prime }\left( p\right) <0$, we require that $\theta$ lie between 0 and 1. By integration by parts, we have

$$\begin{aligned} A(\widetilde{H})-A\left( H\right)= & {} w\left( 1\right) \left[ GC_{\widetilde{H }}^{2}\left( 1\right) -GC_{H}^{2}\left( 1\right) \right] -\int _{0}^{1}w^{ \prime }\left( p\right) \left[ GC_{\widetilde{H}}^{2}\left( p\right) -GC_{H}^{2}\left( p\right) \right] dp \\= & {} \left( \frac{1-\theta }{1-\frac{\theta }{2}}\right) \left[ GC_{\widetilde{ H}}^{2}\left( 1\right) -GC_{H}^{2}\left( 1\right) \right] +\theta \int _{0}^{1}\left[ GC_{\widetilde{H}}^{2}\left( p\right) -GC_{H}^{2}\left( p\right) \right] dp \\= & {} \left( \frac{1}{1-\frac{\theta }{2}}\right) \left[ GC_{\widetilde{H} }^{2}\left( 1\right) -GC_{H}^{2}\left( 1\right) \right] \\&+\theta \left\{ \int _{0}^{1}\left[ GC_{\widetilde{H}}^{2}\left( p\right) -GC_{H}^{2}\left( p\right) \right] dp-\left( \frac{1}{1-\frac{\theta }{2}} \right) \left[ GC_{\widetilde{H}}^{2}\left( 1\right) -GC_{H}^{2}\left( 1\right) \right] \right\} . \end{aligned}$$

We assume that $GC_{\widetilde{H}}^{2}\left( 1\right) -GC_{H}^{2}\left( 1\right) <0$, and thus we have

$$\begin{aligned} A(\widetilde{H})-A\left( H\right) <\int _{0}^{1}\left[ GC_{\widetilde{H} }^{2}\left( p\right) -GC_{H}^{2}\left( p\right) \right] dp-\left( \frac{1}{1- \frac{\theta }{2}}\right) \left[ GC_{\widetilde{H}}^{2}\left( 1\right) -GC_{H}^{2}\left( 1\right) \right] . \end{aligned}$$

Since $GC_{\widetilde{H}}^{2}\left( 1\right) -GC_{H}^{2}\left( 1\right) <0$, if

$$\begin{aligned} \theta >2\left\{ 1-\frac{\left[ GC_{\widetilde{H}}^{2}\left( 1\right) -GC_{H}^{2}\left( 1\right) \right] }{\int _{0}^{1}\left[ GC_{\widetilde{H} }^{2}\left( p\right) -GC_{H}^{2}\left( p\right) \right] dp}\right\} , \end{aligned}$$

then $A(\widetilde{H})-A\left( H\right) <0$.

Next, we show that if Eq. (A.8) holds, then $\exists w\in W_{2}\left( \varepsilon _{1},\varepsilon _{2}\right)$ such that $A(\widetilde{H})-A\left( H\right) <0$. If the LHS of Eq. (A.8) is positive, then Eq. (A.8) holds since Eq. (A.7) holds. On the other hand, if the LHS of Eq. (A.8) is nonpositive, let

$$\begin{aligned} D^{*}=\underset{D}{\arg \max }\left\{ \frac{\left( 1-2\varepsilon _{2}\right) \int _{D}\left[ GC_{H}^{2}\left( p\right) -GC_{\widetilde{H} }^{2}\left( p\right) \right] dp+\varepsilon _{2}\int _{0}^{1}\left[ GC_{H}^{2}\left( p\right) -GC_{\widetilde{H}}^{2}\left( p\right) \right] dp}{ \left( 1-2\varepsilon _{2}\right) \left| D\right| +\varepsilon _{2}} \right\} , \end{aligned}$$

(A.9)

then we consider a weight function $w\in W_{2}\left( \varepsilon _{1},\varepsilon _{2}\right)$ such that $w\left( 0\right) =\frac{ 1-\varepsilon _{1}}{4}$, $w\left( 1\right) =\frac{\varepsilon _{1}}{4}$, and

$$\begin{aligned} w^{\prime }\left( p\right) =-\frac{1-2\varepsilon _{1}}{4}\frac{1}{\left( 1-2\varepsilon _{2}\right) D^{*}+\varepsilon _{2}}\cdot \left\{ \begin{array}{ll} \left( 1-\varepsilon _{2}\right) , &{}\quad \text {if }p\in D^{*} \\ \varepsilon _{2}, &{}\quad \text {if }p\notin D^{*} \end{array} \right. . \end{aligned}$$

This weight function belongs to $W_{2}\left( \varepsilon _{1},\varepsilon _{2}\right)$. By integration by parts, we have

$$\begin{aligned} A(\widetilde{H})-A\left( H\right)= & {} w\left( 1\right) \left[ GC_{\widetilde{H }}^{2}\left( 1\right) -GC_{H}^{2}\left( 1\right) \right] -\int _{0}^{1}w^{ \prime }\left( p\right) \left[ GC_{\widetilde{H}}^{2}\left( p\right) -GC_{H}^{2}\left( p\right) \right] dp \\= & {} \frac{\varepsilon _{1}}{4}\left[ GC_{\widetilde{H}}^{2}\left( 1\right) -GC_{H}^{2}\left( 1\right) \right] \\&-\frac{1-2\varepsilon _{1}}{4}\frac{1}{\left( 1-2\varepsilon _{2}\right) \left| D^{*}\right| +\varepsilon _{2}}\left\{ \left( 1-2\varepsilon _{2}\right) \int _{D^{*}}^{{}}\left[ GC_{H}^{2}\left( p\right) -GC_{\widetilde{H}}^{2}\left( p\right) \right] dp\right. \\&\left. +\varepsilon _{2}\int _{0}^{1}\left[ GC_{H}^{2}\left( p\right) -GC_{ \widetilde{H}}^{2}\left( p\right) \right] dp\right\} , \end{aligned}$$

where $\left| D^{*}\right| =\int _{D^{*}}dp$. Thus, by the definition of $D^{*}$, if

$$\begin{aligned}&\underset{D}{\max }\left\{ \frac{\left( 1-2\varepsilon _{2}\right) \int _{D} \left[ GC_{H}^{2}\left( p\right) -GC_{\widetilde{H}}^{2}\left( p\right) \right] dp+\varepsilon _{2}\int _{0}^{1}\left[ GC_{H}^{2}\left( p\right) -GC_{ \widetilde{H}}^{2}\left( p\right) \right] dp}{\left( 1-2\varepsilon _{2}\right) \left| D\right| +\varepsilon _{2}}\right\} \\&\quad >\frac{\varepsilon _{1}}{1-2\varepsilon _{1}}\left[ GC_{\widetilde{H} }^{2}\left( 1\right) -GC_{H}^{2}\left( 1\right) \right] , \end{aligned}$$

then $A(\widetilde{H})-A\left( H\right) <0$. The necessary condition for the Theorem is proven.

1.2 Proof of Corollary 1

If $\varepsilon _{2}=0$, by Theorem 2, the numerator of the LHS of Eq. (8) becomes

$$\begin{aligned} \underset{D}{\max }\left\{ \frac{\int _{D}\left[ GC_{H}^{2}\left( p\right) -GC_{\widetilde{H}}^{2}\left( p\right) \right] dp}{\left| D\right| } \right\} . \end{aligned}$$

By attaching all the weight to max $\left\{ GC_{H}^{2}\left( p\right) -GC_{ \widetilde{H}}^{2}\left( p\right) \right\}$, we obtain the results.

1.3 Proof of Theorem 4

• (1) “If” part: We show that if $\frac{\int _{C_{\widetilde{H}}^{2}\left( p\right) <C_{H}^{2}\left( p\right) } \left[ C_{H}^{2}\left( p\right) -C_{\widetilde{H}}^{2}\left( p\right) \right] dp}{\int \left| C_{\widetilde{H}}^{2}\left( p\right) -C_{H}^{2}\left( p\right) \right| dp}\le \varepsilon _{2}$, then $I(\widetilde{H} )-I\left( H\right) \le 0$ $\forall w\in W_{2}\left( \varepsilon _{1},\varepsilon _{2}\right)$. By integration by parts, we have

  $$\begin{aligned} I(\widetilde{H})-I\left( H\right)= & {} \int _{0}^{1}\left[ 1-w\left( p\right) \right] \left[ \frac{\widetilde{H}\left( p\right) }{\mu _{\widetilde{H}}}- \frac{H\left( p\right) }{\mu _{H}}\right] dp \nonumber \\= & {} \left[ 1-w\left( 1\right) \right] \left[ C_{\widetilde{H}}^{2}\left( 1\right) -C_{H}^{2}\left( 1\right) \right] -\int _{0}^{1}\left[ -w^{\prime }\left( p\right) \right] \left[ C_{\widetilde{H}}^{2}\left( p\right) -C_{H}^{2}\left( p\right) \right] dp \nonumber \\= & {} \int _{0}^{1}\left[ -w^{\prime }\left( p\right) \right] \left[ C_{H}^{2}\left( p\right) -C_{\widetilde{H}}^{2}\left( p\right) \right] dp. \end{aligned}$$

  (A.10)

  We then divide the integral into two sets. The first set is defined over ranges where $C_{\widetilde{H}}^{2}\left( p\right) <C_{H}^{2}\left( p\right)$. The second set is defined over ranges where $C_{\widetilde{H}}^{2}\left( p\right) \ge C_{H}^{2}\left( p\right)$. Equation (A.10 ) can be written as

  $$\begin{aligned} I(\widetilde{H})-I\left( H\right)= & {} \int _{C_{\widetilde{H}}^{2}\left( p\right)<C_{H}^{2}\left( p\right) }\left[ -w^{\prime }\left( p\right) \right] \left[ C_{H}^{2}\left( p\right) -C_{\widetilde{H}}^{2}\left( p\right) \right] dp \nonumber \\&+\int _{C_{\widetilde{H}}^{2}\left( p\right) \ge C_{H}^{2}\left( p\right) } \left[ -w^{\prime }\left( p\right) \right] \left[ C_{H}^{2}\left( p\right) -C_{\widetilde{H}}^{2}\left( p\right) \right] dp \nonumber \\\le & {} \sup \left\{ -w^{\prime }\left( p\right) \right\} \int _{C_{\widetilde{ H}}^{2}\left( p\right)<C_{H}^{2}\left( p\right) }\left[ C_{H}^{2}\left( p\right) -C_{\widetilde{H}}^{2}\left( p\right) \right] dp \nonumber \\&+\inf \left\{ -w^{\prime }\left( p\right) \right\} \int _{C_{\widetilde{H} }^{2}\left( p\right) \ge C_{H}^{2}\left( p\right) }\left[ C_{H}^{2}\left( p\right) -C_{\widetilde{H}}^{2}\left( p\right) \right] dp \nonumber \\= & {} \inf \left\{ -w^{\prime }\left( p\right) \right\} \left\{ \frac{\sup \left\{ -w^{\prime }\left( p\right) \right\} }{\inf \left\{ -w^{\prime }\left( p\right) \right\} }\int _{C_{\widetilde{H}}^{2}\left( p\right) <C_{H}^{2}\left( p\right) }\left[ C_{H}^{2}\left( p\right) -C_{\widetilde{H} }^{2}\left( p\right) \right] dp\right. \nonumber \\&\left. +\int _{C_{\widetilde{H}}^{2}\left( p\right) \ge C_{H}^{2}\left( p\right) }\left[ C_{H}^{2}\left( p\right) -C_{\widetilde{H}}^{2}\left( p\right) \right] dp\right\} . \end{aligned}$$

  (A.11)

  Since $w\in W_{2}\left( \varepsilon _{1},\varepsilon _{2}\right)$, by definition, we have $\frac{\sup \left\{ -w^{\prime }\left( p\right) \right\} }{\inf \left\{ -w^{\prime }\left( p\right) \right\} }\le \frac{1}{ \varepsilon _{2}}-1$. Therefore,

  $$\begin{aligned} I(\widetilde{H})-I\left( H\right)\le & {} \inf \left\{ -w^{\prime }\left( p\right) \right\} \left\{ \left( \frac{1}{\varepsilon _{2}}-1\right) \int _{C_{\widetilde{H}}^{2}\left( p\right) <C_{H}^{2}\left( p\right) }\left[ C_{H}^{2}\left( p\right) -C_{\widetilde{H}}^{2}\left( p\right) \right] dp\right. \nonumber \\&\left. +\int _{C_{\widetilde{H}}^{2}\left( p\right) \ge C_{H}^{2}\left( p\right) }\left[ C_{H}^{2}\left( p\right) -C_{\widetilde{H}}^{2}\left( p\right) \right] dp\right\} . \end{aligned}$$

  (A.12)

Thus, according to Eq. (A.12), if

$$\begin{aligned} \left( \frac{1}{\varepsilon _{2}}-1\right) \int _{C_{\widetilde{H}}^{2}\left( p\right) <C_{H}^{2}\left( p\right) }\left[ C_{H}^{2}\left( p\right) -C_{ \widetilde{H}}^{2}\left( p\right) \right] dp+\int _{C_{\widetilde{H} }^{2}\left( p\right) \ge C_{H}^{2}\left( p\right) }\left[ C_{H}^{2}\left( p\right) -C_{\widetilde{H}}^{2}\left( p\right) \right] dp\le 0, \end{aligned}$$

or,

$$\begin{aligned} \frac{\int _{C_{\widetilde{H}}^{2}\left( p\right) <C_{H}^{2}\left( p\right) } \left[ C_{H}^{2}\left( p\right) -C_{\widetilde{H}}^{2}\left( p\right) \right] dp}{\int \left| C_{\widetilde{H}}^{2}\left( p\right) -C_{H}^{2}\left( p\right) \right| dp}\le \varepsilon _{2}, \end{aligned}$$

then $I(\widetilde{H})-I\left( H\right) \le 0$. The sufficient condition for the Theorem is proven.

• (2) “Only if” part: We show that if $\frac{\int _{C_{\widetilde{H}}^{2}\left( p\right) <C_{H}^{2}\left( p\right) }\left[ C_{H}^{2}\left( p\right) -C_{\widetilde{H}}^{2}\left( p\right) \right] dp}{\int \left| C_{\widetilde{H}}^{2}\left( p\right) -C_{H}^{2}\left( p\right) \right| dp}>\varepsilon _{2}$, then there exists a $w\in W_{2}\left( \varepsilon _{1},\varepsilon _{2}\right)$ such that $I(\widetilde{H})-I\left( H\right) >0$. Take a weight function $w\in W_{2}\left( \varepsilon _{1},\varepsilon _{2}\right)$ such that $w\left( 0\right) =\frac{1-\varepsilon _{1}}{4}$, $w\left( 1\right) =\frac{ \varepsilon _{1}}{4}$, and

  $$\begin{aligned} w^{\prime }\left( p\right) =\left\{ \begin{array}{ll} -(1-\varepsilon _{2}), &{}\quad \text {if }C_{\widetilde{H}}^{2}\left( p\right) <C_{H}^{2}\left( p\right) \\ -\varepsilon _{2}, &{}\quad \text {if }C_{\widetilde{H}}^{2}\left( p\right) \ge C_{H}^{2}\left( p\right) \end{array} \right. . \end{aligned}$$

  From Eq. (A.10), we have

  $$\begin{aligned} I(\widetilde{H})-I\left( H\right)= & {} \int _{0}^{1}\left[ -w^{\prime }\left( p\right) \right] \left[ C_{H}^{2}\left( p\right) -C_{\widetilde{H} }^{2}\left( p\right) \right] dp \\= & {} (1-\varepsilon _{2})\int _{C_{\widetilde{H}}^{2}\left( p\right)<C_{H}^{2}\left( p\right) }^{{}}\left[ C_{H}^{2}\left( p\right) -C_{ \widetilde{H}}^{2}\left( p\right) \right] dp \\&+\varepsilon _{2}\int _{C_{\widetilde{H}}^{2}\left( p\right) \ge C_{H}^{2}\left( p\right) }^{{}}\left[ C_{H}^{2}\left( p\right) -C_{ \widetilde{H}}^{2}\left( p\right) \right] dp \\= & {} \int _{C_{\widetilde{H}}^{2}\left( p\right) <C_{H}^{2}\left( p\right) }^{{}}\left[ C_{H}^{2}\left( p\right) -C_{\widetilde{H}}^{2}\left( p\right) \right] dp \\&-\varepsilon _{2}\int _{0}^{1}\left| C_{H}^{2}\left( p\right) -C_{ \widetilde{H}}^{2}\left( p\right) \right| dp. \end{aligned}$$

We assume that $\int _{C_{\widetilde{H}}^{2}\left( p\right) <C_{H}^{2}\left( p\right) }\left[ C_{H}^{2}\left( p\right) -C_{\widetilde{H}}^{2}\left( p\right) \right] dp>\varepsilon _{2}\int \left| C_{\widetilde{H} }^{2}\left( p\right) -C_{H}^{2}\left( p\right) \right| dp$, and thus we have $I(\widetilde{H})-I\left( H\right) >0$. The necessary condition for the Theorem is proven.

1.4 Method of implementing Eq. (8)

We proceed to maximize the LHS of Eq. (8). To implement the scheme, we need to evaluate the integral numerically. The idea is to divide the integration interval [0, 1] into a large number of small intervals, calculate $GC_{H}^{2}\left( p\right) -GC_{\widetilde{H} }^{2}\left( p\right)$ for each, and determine which one (or combination), D, is optimal when the LHS reaches its maximum. We propose three main steps to implement this framework as follows.

First, the interval [0, 1] can be partitioned into N subintervals with increment $\Delta p=1/N$. Let us define $p_{i}=i/N$ for $i=0$, 1, ..., N . There are $N+1$ points between 0 and 1. Note that a good approximation is achieved by making N sufficiently large. In our paper, we assume that $N=10{,}000$.

Second, for each point $p_{i}$, we calculate the difference between two generalized concentration curves. From this we obtain $GC_{H}^{2}\left( p_{i}\right) -GC_{\widetilde{H}}^{2}\left( p_{i}\right)$ for each i. What kind of subset $D\in \left\{ p_{0}\text {, }p_{1} , \ldots , p_{N}\right\}$ should be chosen for optimization? Since there will be $2^{N+1}$ different combinations from $N+1$ points, we need to deal with this problem more effectively when N is extremely large.

Third, we develop an integer linear programming technique to solve for the optimal solution D. Let $x_{i}$ be a binary variable with value 1 if $p_{i}$ is selected, and 0 otherwise. The integer programming problem can be written as

$$\begin{aligned} \underset{x_{1},x_{2},\ldots ,x_{N}}{\max }\frac{\left( 1-2\varepsilon _{2}\right) \left\{ \sum \limits _{i=1}^{N}x_{i}\left[ GC_{H}^{2}\left( p_{i}\right) -GC_{\widetilde{H}}^{2}\left( p_{i}\right) \right] \right\} +\varepsilon _{2}\left\{ \sum \limits _{i=1}^{N}\left[ GC_{H}^{2}\left( p_{i}\right) -GC_{\widetilde{H}}^{2}\left( p_{i}\right) \right] \right\} }{ \left( 1-2\varepsilon _{2}\right) \frac{\sum \limits _{i=1}^{N}x_{i}}{N} +\varepsilon _{2}}, \end{aligned}$$

(A.13)

where $x_{i}$ is assigned a value of 0 or 1. However, the resulting problem is a nonlinear integer programming problem. It is complex and hard to solve.

In order to overcome this difficulty, we show how the objective function of Eq. (A.13) can be rewritten as a linear two-stage optimization problem. A constrained linear integer programming problem can be described as follows:

$$\begin{aligned}&\underset{x_{1},x_{2},\ldots ,x_{N}}{\max }\mathcal {L}=\left( 1-2\varepsilon _{2}\right) \left\{ \sum \limits _{i=1}^{N}x_{i}\left[ GC_{H}^{2}\left( p_{i}\right) -GC_{\widetilde{H}}^{2}\left( p_{i}\right) \right] \right\} \nonumber \\&\qquad +\varepsilon _{2}\left\{ \sum \limits _{i=1}^{N}\left[ GC_{H}^{2}\left( p_{i}\right) -GC_{\widetilde{H}}^{2}\left( p_{i}\right) \right] \right\} \nonumber \\&\quad \text { s.t. }\left. \left( 1-2\varepsilon _{2}\right) \frac{ \sum \nolimits _{i=1}^{N}x_{i}}{N}+\varepsilon _{2}=\lambda \right. , \end{aligned}$$

(A.14)

where $\lambda \in \left[ \varepsilon _{2},1-\varepsilon _{2}\right]$. In the first stage, we maximize $\mathcal {L}$ by setting $\lambda$. Since the values of $\sum \limits _{i=1}^{N}x_{i}$range from 0 to N, the candidate of $\lambda$ can be represented as $\frac{j+\left( N-2j\right) \varepsilon _{2}}{N}$ for $j=0$, 1, ..., N. By repeating the procedure $N+1$ times at this stage, a sample of $N+1$ optimal objective values is generated. In the second stage, based on the optimal objective values ${\mathcal{L}}_{\lambda }$ determined in the previous stage for each $\lambda$, we maximize $\frac{{{\mathcal{L}}_{\lambda } }}{\lambda }$ over $\lambda$. In general, when considering all possible $\lambda$, this two-stage procedure is equivalent to solving the nonlinear problem in Eq. (A.13).

Note that, alternatively, there is a trick that makes the problem much easier to handle in the first stage. By sorting $GC_{H}^{2}\left( p_{i}\right) -GC_{\widetilde{H}}^{2}\left( p_{i}\right)$ in descending order, the summation of the first j sorted values denotes the optimal value of $\max _{x_{1},x_{2},\ldots ,x_{N}} \sum \nolimits _{i=1}^{N}x_{i} \left[ GC_{H}^{2}\left( p_{i}\right) -GC_{\widetilde{H}}^{2}\left( p_{i}\right) \right]$ subject to $\sum \nolimits _{i=1}^{N}x_{i}=j$, where $j=0$ , 1, ..., N. In this case we could directly identify the optimal objective values ${\mathcal{L}}_{\lambda }$ through the sorted values for each $\lambda$. Note that the second stage is the same as before. Thus we just need to determine which j is the best for optimization. This technique could require less computation time when N is large.