3.1. Violation to the usual stochastic order

Consider two random variables $X_1$ and $X_2$ defined on domain $\mathcal{X}$ . If $X_1 \leq_{st} X_2$ , then $\bar F_{X_1}(x)-\bar F_{X_2}(x) \leq 0$ for all $x\in \mathcal{X}$ . The violation to the usual stochastic order at $x \in \mathcal{X}$ is thus

\begin{align*} v(x;\; X_1 \leq_{st} X_2) = (\bar F_{X_1}(x) - \bar F_{X_2}(x))^+. \end{align*}

If the usual stochastic order is violated at x, the amount of violation $v(x;\ X_1 \leq_{st} X_2)$ is strictly positive. Then the overall, left-side, and right-side violations can be computed through (1)–(2).

The integral characterization of the usual stochastic order gives rise to the class of all increasing functions, i.e. $X_1\leq_{st} X_2$ , if and only if $\mathbb{E}[g(X_1)] \leq \mathbb{E}[g(X_2)]$ for all $g\in \textit{inc}$ . When the usual stochastic order is violated, the relationship $\mathbb{E}[g(X_1)] \leq \mathbb{E}[g(X_2)]$ fails to hold for some $g\in \textit{inc}$ . To identify the subclass of functions that maintains the usual stochastic order, define

\begin{align*} g^{(1)}_{\min} & = \inf_{x_1, x_2}\{ \delta_g(x_1,x_2)\colon \underline{x} \leq x_1 < x_2 \leq \overline{x} \},\\[5pt] g^{(1)}_{\max} & = \sup_{x_1, x_2}\{ \delta_g(x_1,x_2)\colon \underline{x} \leq x_1 < x_2 \leq \overline{x} \}, \\[5pt] g^{(1)}_{\min^-}(x) & = \inf_{x_1,x_2}\{ \delta_g(x_1,x_2)\colon \underline{x} \leq x_1 < x_2 \leq x\},\\[5pt] g^{(1)}_{\max^-}(x) & = \sup_{x_1,x_2}\{ \delta_g(x_1,x_2)\colon \underline{x} \leq x_1 < x_2 \leq x\},\\[5pt] g^{(1)}_{\min^+}(x) & = \inf_{x_1,x_2}\{ \delta_g(x_1,x_2)\colon x \leq x_1 < x_2 \leq \overline{x}\} ,\\[5pt] g^{(1)}_{\max^+}(x) & = \sup_{x_1,x_2}\{ \delta_g(x_1,x_2)\colon x \leq x_1 < x_2 \leq \overline{x}\} , \end{align*}

where

$\delta_g(x_1,x_2) =\frac{g(x_2) - g(x_1)}{x_2-x_1}.$

Thus $g^{(1)}_{\min}$ is the ‘smallest slope’ of function g, and

$g^{(1)}_{\min^-}(x) \bigl[g^{(1)}_{\min^+}(x)\bigr]$

is the smallest slope of g below (above) x. With these, we can define subclasses of increasing functions parameterized by some $\Delta\in[0,\infty)$ :

(3) \begin{align}\textit{inc}(\Delta) &= \Big\{ g \in \overline{\mathcal{G}}\colon 0 \leq g^{(1)}_{\min}\leq \delta_g(x_1,x_2) \leq (1+\Delta) g^{(1)}_{\min}, \, \underline{x}\leq x_1 \lt x_2 \leq \overline{x} \Big\},\end{align}

(4) \begin{align}\textit{inc}^{+}(\Delta) &= \Big\{ g\in\overline{\mathcal{G}}\colon 0 \leq g^{(1)}_{\min^+}(x_2) \leq \delta_g(x_1,x_2)\leq (1+\Delta) g^{(1)}_{\min^+}(x_2),\, \underline{x}\leq x_1\lt x_2\leq \overline{x}\Big\},\end{align}

(5) \begin{align}\textit{inc}^{-}(\Delta) &= \Big\{ g\in\overline{\mathcal{G}}\colon 0 \leq g^{(1)}_{\min^-}(x_1)\leq \delta_g(x_1,x_2) \leq (1+\Delta) g^{(1)}_{\min^-}(x_1), \, \underline{x}\leq x_1\lt x_2\leq \overline{x}\Big\}.\end{align}

These conditions are equivalent to

\begin{align*}\delta_g(x_1,x_2) \geq \frac{1}{1+\Delta} g^{(1)}_{\max},\! \quad\delta_g(x_1,x_2) \geq \frac{1}{1+\Delta}g^{(1)}_{\max^-}(x_1),\!\quad \text{and} \quad\delta_g(x_1,x_2) \geq \frac{1}{1+\Delta} g^{(1)}_{\max^+}(x_2),\end{align*}

respectively. It is clear that any function within these classes must be continuous. We further observe that the subclasses get larger with a larger $\Delta$ . In particular, for $\Delta \in (0,\infty)$ ,

\begin{align*} \textit{ilin}&= \textit{inc}(0) \subset \textit{inc}(\Delta) \subset \textit{inc}(\infty) = \textit{inc}, \\[5pt] \textit{icx}&=\textit{inc}^{+}(0) \subset \textit{inc}^{+}(\Delta) \subset \textit{inc}^{+}(\infty)=\textit{inc}, \\[5pt] \textit{icv} &= \textit{inc}^{-}(0)\subset \textit{inc}^{-}(\Delta) \subset \textit{inc}^{-}(\infty) = \textit{inc}, \end{align*}

where $\textit{ilin}$ is the class of linearly increasing functions. The parameter $\Delta$ specifies the extent to which the increasingness of the function is restricted. The functions in $inc^{+}(\Delta)$ ( $inc^{-}(\Delta)$ ) are close to, but can deviate from, increasing convex (concave) functions. Thus the associated integral stochastic orders are weaker than the usual stochastic order, but can be stronger than the increasing convex (concave) order. We can relax the usual stochastic orders by evaluating the overall, left-side, and right-side violations, as described in the next theorem.

Proof of Theorem 1 . Part (i) is proved by Leshno and Levy [Reference Leshno and Levy7] and Tsetlin et al. [Reference Tsetlin, Winkler, Huang and Tzeng23]. We first show part (iii). Take $g\in \textit{inc}^- (\Delta)$ . We derive

\begin{align*} & \mathbb{E}[g(X_2)] - \mathbb{E}[g(X_1)] \\[5pt] &\quad = \int_{\xi\in\mathcal{X}} \bigl(\bar F_{X_2}(\xi)-\bar F_{X_1}(\xi)\bigr)^+\,\mathrm{d} g(\xi) - \int_{\xi\in\mathcal{X}} \bigl(\bar F_{X_1}(\xi)-\bar F_{X_2}(\xi)\bigr)^+\,\mathrm{d} g(\xi) \\[5pt] &\quad \geq \int_{\xi\in\mathcal{X}} g^{(1)}_{\min^-}(\xi) \bigl(\bar F_{X_2}(\xi)-\bar F_{X_1}(\xi)\bigr)^+\,\mathrm{d} \xi - (1+\Delta)\int_{\xi\in\mathcal{X}} g^{(1)}_{\min^-}(\xi) \bigl(\bar F_{X_1}(\xi)-\bar F_{X_2}(\xi)\bigr)^+\,\mathrm{d} \xi \\[5pt] &\quad = \int_{\xi\in\mathcal{X}} g^{(1)}_{\min^-}(\xi) \bigl(\bigl(\bar F_{X_2}(\xi)-\bar F_{X_1}(\xi)\bigr) - \Delta \bigl(\bar F_{X_1}(\xi)-\bar F_{X_2}(\xi)\bigr)^+\bigr) \,\mathrm{d} \xi. \end{align*}

Because $g^{(1)}_{\min^-}$ is decreasing and bounded by definition, we further deduce

\begin{align*} & \mathbb{E}[g(X_2)] - \mathbb{E}[g(X_1)] \\[5pt] &\quad\geq \int_{\xi\in\mathcal{X}}\biggl(b_1 - \int_{\xi}^{\overline{x}} \,\mathrm{d} g^{(1)}_{\min^-}(x) \biggr) \bigl(\bigl(\bar F_{X_2}(\xi)-\bar F_{X_1}(\xi)\bigr) - \Delta \bigl(\bar F_{X_1}(\xi)-\bar F_{X_2}(\xi)\bigr)^+\bigr) \,\mathrm{d} \xi\\[5pt] &\quad= b_1 \int_{\xi \in \mathcal{X}} \bigl(\bigl(\bar F_{X_2}(\xi)-\bar F_{X_1}(\xi)\bigr) - \Delta \bigl(\bar F_{X_1}(\xi)-\bar F_{X_2}(\xi)\bigr)^+\bigr) \,\mathrm{d} \xi \\[5pt] & \quad\quad + \int_{x\in\mathcal{X}} \int_{\underline{x}}^x \bigl(\bigl(\bar F_{X_2}(\xi)-\bar F_{X_1}(\xi)\bigr) - \Delta \bigl(\bar F_{X_1}(\xi)-\bar F_{X_2}(\xi)\bigr)^+\bigr) \,\mathrm{d} \xi \,\mathrm{d} \bigl(-g^{(1)}_{\min^-}(x)\bigr), \end{align*}

where $b_1 = g^{(1)}_{\min}(\overline{x}) \geq 0$ . When (8) holds, the right-hand side is non-negative.

Now take $X_1$ and $X_2$ such that $\mathbb{E}[g(X_1)]\leq E[g(X_2)]$ for any $g\in \textit{inc}^-(\Delta)$ . Let

$g_y(x) = \int_{\underline{x}}^x g^{(1)}_y(\xi) \,\mathrm{d} \xi,$

where

$g^{(1)}_y(x) =\bigl(1 + \Delta {\mathbb{I}}_{\left\{\bar F_{X_1}(x) > \bar F_{X_2}(x)\right\}}\bigr){\mathbb{I}}_{\{x \leq y\}}\quad\text{for $x \in \mathcal{X}$.}$

It is clear that $g^{(1)}_y(x) \geq0$ for $x\in\mathcal{X}$ , and thus $g_y$ is increasing. Moreover, for $x\leq y$ ,

$(1+\Delta)g^{(1)}_{y \min^-}(x) \geq 1 + \Delta \geq g^{(1)}_{y \max^+}(x) \geq g^{(1)}_y(x);$

for $x > y$ ,

$g^{(1)}_{y \min^-}(x) = g^{(1)}_{y\max^+}(x)=g^{(1)}_y(x)=0.$

Thus $g_y\in \textit{inc}^-(\Delta)$ . Moreover, for a random variable Z with distribution $F_Z$ over $\mathcal{X}$ ,

\begin{align*} \mathbb{E}[g_y(Z)] &= -\int_{\xi\in\mathcal{X}}\int_{\underline{x}}^{\xi} g^{(1)}_y(x) \,\mathrm{d} x \,\mathrm{d} \bar F_Z(\xi)\\[5pt] & = -\int_{\xi\in\mathcal{X}}\int_{\underline{x}}^{\xi} \bigl(1+\Delta {\mathbb{I}}_{\left\{\bar F_{X_1} (x) > \bar F_{X_2}(x) \right\}}\bigr) {\mathbb{I}}_{\{x \leq y\}} \,\mathrm{d} x \,\mathrm{d} \bar F_Z(\xi) \\[5pt] &= \int_{x\in\mathcal{X}}{\mathbb{I}}_{\{x \leq y\}} \bar F_Z(x)\,\mathrm{d} x + \Delta \int_{x\in\mathcal{X}} {\mathbb{I}}_{\left\{\bar F_{X_1}(x)> \bar F_{X_2}(x)\right\}} {\mathbb{I}}_{\{x \leq y\}} \bar F_{Z}(x) \,\mathrm{d} x\\[5pt] &= \int_{\underline{x}}^y\bar F_Z(x)\,\mathrm{d} x + \Delta\int_{\underline{x}}^y {\mathbb{I}}_{\left\{\bar F_{X_1}(x)> \bar F_{X_2}(x)\right\}} \bar F_{Z}(x) \,\mathrm{d} x. \end{align*}

Because $g_y \in \textit{inc}^-(\Delta)$ , we deduce

\begin{align*} 0&\leq \mathbb{E}[g_y(X_2)] - \mathbb{E}[g_y(X_1)] \\[5pt] &= \int_{\underline{x}}^y \bigl(\bar F_{X_2}(x) - \bar F_{X_1}(x)\bigr) \,\mathrm{d} x - \Delta \int_{\underline{x}}^y \bigl(\bar F_{X_1}(x) - \bar F_{X_2}(x)\bigr)^+ \,\mathrm{d} x, \end{align*}

yielding (8).

The proof of part (ii) can be carried out similarly. In particular, we can define

$g^{(1)}_y(x) =\bigl(1 + \Delta {\mathbb{I}}_{\{\bar F_{X_1}(x) > \bar F_{X_2}(x)\}}\bigr){\mathbb{I}}_{\{x \geq y\}},\quad x \in \mathcal{X}.$

For $x < y$ , $g^{(1)}_y(x) = g^{(1)}_{y\min^+}(x) = 0$ . For $x \geq y$ , $ g^{(1)}_{y\min^+}(x) \geq 1$ and thus

$(1+\Delta)g^{(1)}_{y\min^+}(x) \geq 1+\Delta \geq g^{(1)}_{y}(x),$

suggesting $g_y\in \textit{inc}^+(\Delta)$ . For any random variable Z with distribution $F_Z$ , we derive

\begin{align*} \mathbb{E}[g_y(Z)] &= \int_y^{\overline{x}} \bar F_Z(x) \,\mathrm{d} x + \Delta \int_y^{\overline{x}} {\mathbb{I}}_{\{\bar F_{X_1}(x) > \bar F_{X_2}(x)\}} \bar F_Z(x) \,\mathrm{d} x. \end{align*}

It follows that

\begin{align*} \mathbb{E}[g_y(X_2)] - \mathbb{E}[g_y(X_1)] = \int_y^{\overline{x}} (\bar F_{X_2}(x) - \bar F_{X_1}(x)) \,\mathrm{d} x - \Delta \int_y^{\overline{x}} (\bar F_{X_1}(x) - \bar F_{X_2}(x))^+ \,\mathrm{d} x. \end{align*}

Because $g_y\in \textit{inc}^+(\Delta)$ , we conclude that (7) holds.

The proof of Theorem 1 directly utilizes the upper and lower bounds for the slope of function g defined (3)–(4) to evaluate, respectively, the positive part and the negative part of $g(X_1)-g(X_2)$ , which gives rise to the distributional characterization of the relaxed usual stochastic orders.

The stochastic order associated with $inc(\Delta)$ is the $\epsilon$ -AFSD, first introduced in [Reference Leshno and Levy7] with $\Delta = 1/\epsilon -2 \in [0,\infty)$ . Leshno and Levy [Reference Leshno and Levy7] derive Theorem 1(i) for differentiable g and Tsetlin et al. [Reference Tsetlin, Winkler, Huang and Tzeng23] analyze this order for strictly increasing functions. Müller et al. [Reference Müller, Scarsini, Tsetlin and Winkler18] analyze differentiable functions within $inc^-(\Delta)$ and define the $(1+\gamma)$ -SD order with $1+\Delta=1/\gamma$ interpreted as the index of greediness in the utility theory. This order is sometimes referred to as the ‘fractional stochastic dominance’ [Reference Huang, Tzeng and Zhao5, Reference Mao, Wu and Hu16]. Müller et al. [Reference Müller, Scarsini, Tsetlin and Winkler18] recognize that this order is in between the usual stochastic order and the increasing concave order, and show that two discrete random variables are $(1+\gamma)$ -SD ordered if and only if one can be obtained through a sequence of mean-preserving contraction mass transports from the other one. Klar and Müller [Reference Klar and Müller6] connect the $(1+\gamma)$ -SD order to the $\Omega$ ratio, $\Omega(\xi) = \mathbb{E}[(X-\xi)^+]/\mathbb{E}[(X-\xi)^-]$ , used for fund performance. Yang et al. [Reference Yang, Zhou and Zhuang25], among others, explore useful properties of this parametric order.

The characterization in (6) for $\epsilon$ -AFSD associated with $inc(\Delta)$ functions suggests the possibility of violating the usual stochastic order with the amount of violation bounded by $1/\Delta$ of the mean difference. It is easy to show that (6) implies

\begin{align*} 1 + \Delta \leq \dfrac{nv_{O}(X_1 \leq_{st} X_2)}{v_{O}(X_1 \leq_{st} X_2)}, \end{align*}

where

$nv_{O}(X_1 \leq_{st} X_2)=v_{O}(X_1 \geq_{st} X_2) =\int_{x\in\mathcal{X}} (\bar F_{X_2}(x) - \bar F_{X_1}(x))^+ \,\mathrm{d} x$

is the overall measure of $X_1$ dominating $X_2$ in the usual stochastic order. Thus $(1+\Delta)$ is the minimum ratio of dominance violation. Similarly, (7) and (8), respectively, imply

\begin{align*} 1+\Delta \leq \dfrac{nv_{\geq}(x;\ X_1 \leq_{st} X_2)}{v_{\geq}(x;\ X_1 \leq_{st} X_2)} \quad \text{and}\quad 1+\Delta \leq \dfrac{nv_{\leq}(x;\ X_1 \leq_{st} X_2)}{v_{\leq}(x;\ X_1 \leq_{st} X_2)}. \end{align*}

3.2. Generalization with two-sided restrictions

We may generalize the discussion in the previous subsection by imposing asymmetric two-sided restrictions to linear functions and define the parametric order that allows for the asymmetric two-sided violation to the usual stochastic order. Specifically, define

(9) \begin{align}& \textit{inc}^\pm(\Delta_+,\Delta_-) \notag \\[5pt] &\quad = \Bigl\{ g \in \overline{\mathcal{G}}\colon 0 \leq \delta_g(x_1,x_2) \leq \min\{(1+\Delta_+) g^{(1)}_{\min^+}(x_2), (1+\Delta_-) g^{(1)}_{\min^-}(x_1)\}, \underline{x}\leq x_1 < x_2 \leq \overline{x} \Bigr\}, \notag \\[5pt] &\hspace{80mm} \Delta_+ \in [0,\infty), \Delta_-\in [0, \infty). \end{align}

It is clear that $inc^\pm(\Delta_+,\Delta_-)= \textit{inc}^+(\Delta_+)\cap \textit{inc}^-(\Delta_-)$ . The stochastic order associated with this class of functions is named as $(1+\gamma_{\textit{cx}},1+\gamma_{\textit{cv}})$ -SD by Müller et al. [Reference Müller, Scarsini, Tsetlin and Winkler18, Reference Müller, Scarsini, Tsetlin and Winkler19] with $\gamma_{\textit{cx}} = {{1}/{(1+\Delta_+)}}\in [0,1]$ and $\gamma_{\textit{cv}} = {{1}/{(1+\Delta_-)}} \in [0,1]$ . The characterization in Theorem 2 below, using the left- and right-side violation measures, facilitates an intuitive proof.

Theorem 2. (Two-sided relaxation of usual stochastic order.) For some $\Delta_+\in[0,\infty)$ and $\Delta_-\in [0,\infty)$ , two random variables $X_1$ and $X_2$ defined on support $\mathcal{X}$ satisfy $\mathbb{E}[g(X_1)] \leq \mathbb{E}[g(X_2)]$ for all $g\in \textit{inc}^\pm(\Delta_+,\Delta_-)$ , if and only if (where $a\wedge b = \min\{a,b\}$ ), for any $x \in \mathcal{X}$ ,

(10) \begin{align} & \dfrac{1}{1+\Delta_-}\biggl(\int_{\underline{x}}^x \bigl(\bar F_{X_2}(\xi) - \bar F_{X_1}(\xi)\bigr) \,\mathrm{d} \xi - (\Delta_+\wedge \Delta_-) v_{\leq}(x;\ X_1 \leq_{st} X_2) \biggr) \nonumber\\[5pt] &\quad + \dfrac{1}{1+\Delta_+}\biggl(\int_x^{\overline{x}} \bigl(\bar F_{X_2}(\xi) - \bar F_{X_1}(\xi)\bigr) \,\mathrm{d} \xi - (\Delta_+\wedge \Delta_-) v_{\geq}(x;\ X_1 \leq_{st} X_2) \biggr) \geq 0 . \end{align}

Proof of Theorem 2 . We first prove the ‘if’ part of the theorem by showing that (10) implies that $\mathbb{E}[g(X_1)] \leq \mathbb{E}[g(X_2)]$ for all $g\in \textit{inc}^\pm(\Delta_+,\Delta_-)$ . We derive the result for $\Delta_+ \leq \Delta_-$ , as that for $\Delta_+ \geq \Delta_-$ follows similarly. When $\Delta_+ \leq \Delta_-$ , the condition in (10) becomes

(11) \begin{align} \Xi(x)&\equiv \dfrac{1}{1+\Delta_-}\int_{\underline{x}}^x \bigl(\bar F_{X_2}(\xi) - \bar F_{X_1}(\xi)\bigr)^+ \,\mathrm{d} \xi - \dfrac{1+\Delta_+}{1+\Delta_-}\int_{\underline{x}}^x \bigl(\bar F_{X_1}(\xi) - \bar F_{X_2}(\xi)\bigr)^+ \,\mathrm{d} \xi \nonumber\\[5pt] &\quad + \dfrac{1}{1+\Delta_+}\int_x^{\overline{x}} \bigl(\bar F_{X_2}(\xi) - \bar F_{X_1}(\xi)\bigr)^+ \,\mathrm{d} \xi - \int_x^{\overline{x}} \bigl(\bar F_{X_1}(\xi) - \bar F_{X_2}(\xi)\bigr)^+ \,\mathrm{d} \xi \geq 0. \end{align}

Take $g \in \textit{inc}^\pm(\Delta_+,\Delta_-) = \textit{inc}^+(\Delta_+) \cap \textit{inc}^-(\Delta_-)$ . Then,

\begin{align*} \delta_g(x_1,x_2) \geq \max\biggl\{\dfrac{1}{1+\Delta_-} g^{(1)}_{\max^+}(x_2), \dfrac{1}{1+\Delta_+} g^{(1)}_{\max^-}(x_1)\biggr\}. \end{align*}

Because $g^{(1)}_{\max^+}(x)$ is decreasing and $g^{(1)}_{\max^-}(x)$ is increasing, and both attain $g^{(1)}_{\max}$ within $[\underline{x},\overline{x}]$ , we can define

$x_{\max} = \inf\bigl\{x\in\mathcal{X}\colon g^{(1)}_{\max^+}(x)=g^{(1)}_{\max^-}(x)\bigr\}$

and

$ x_0 = \sup\biggl\{x\in\mathcal{X}\colon \frac{1}{1+\Delta_-} g^{(1)}_{\max^+}(x) > \frac{1}{1+\Delta_+}g^{(1)}_{\max^-}(x)\biggr\}.$

We have $x_0 \leq x_{\max}$ as ${{1}/{(1+\Delta_-)}} \leq {{1}/{(1+\Delta_+)}}$ . We treat the situation where

$\frac{1}{1+\Delta_-} g^{(1)}_{\max^+}(x_0) = \frac{1}{1+\Delta_+}g^{(1)}_{\max^-}(x_0)\quad\text{for some $x_0\in[\underline{x},\overline{x}]$.}$

The situation for

$\frac{1}{1+\Delta_-} g^{(1)}_{\max^+}(x) < \frac{1}{1+\Delta_+}g^{(1)}_{\max^-}(x)\quad\text{for all $x\in[\underline{x},\overline{x}]$}$

can be easily deduced from the argument below.

Note that $g^{(1)}_{\max^-}$ must be strictly increasing at $x_0$ . Because $x_0 \leq x_{\max}$ , we have $g^{(1)}_{\max^+}(x_0) = g^{(1)}_{\max}$ . It follows that

\begin{align*} g^{(1)}_0 \equiv g^{(1)}_{\max^-}(x_0)= \dfrac{1+\Delta_+}{1+\Delta_-} g^{(1)}_{\max}. \end{align*}

Define

(12) \begin{align} \rho(x) = \begin{cases} g^{(1)}_0, & x \leq x_0, \nonumber\\[5pt] g^{(1)}_{\max^-}(x), & x_0 < x < x_{\max} ,\\[5pt] g^{(1)}_{\max}, & x \geq x_{\max}. \end{cases} \end{align}

We have two cases to consider. In the following derivation, we choose a sufficiently small $\epsilon>0$ to compute the left- or right-side deviation of g at given point x.

1. (i) For $x \leq x_0$ , $\delta_g$ is lower than

   $g^{(1)}_{\max^-}(x)\leq g^{(1)}_{\max^-}(x_0)= g^{(1)}_0$
   as $ g^{(1)}_{\max^-}$ is increasing. Moreover, $\delta_g$ is higher than
   $\frac{1}{1+\Delta_-} g_{\max^+}(x) =\frac{1}{1+\Delta_-}g^{(1)}_{\max} = \frac{1}{1+\Delta_+} g^{(1)}_0.$
   Therefore,
   \begin{align*} \dfrac{1}{1+\Delta_+} g^{(1)}_0 \leq \delta_{g} (x-\epsilon, x) \leq g^{(1)}_0. \end{align*}

2. (ii) For $x> x_0$ , we deduce, because $g\in \textit{inc}^+(\Delta_+)$ and $g^{(1)}_{\max^-}$ is strictly increasing at $x_0$ ,

   \begin{align*} \dfrac{1}{1+\Delta_+} g^{(1)}_{\max^-}(x) \leq \delta_{g} (x, x+\epsilon) \leq g^{(1)}_{\max^-}(x) . \end{align*}
   In particular, $g^{(1)}_{\max^-}(x) = g^{(1)}_{\max}$ for $x \geq x_{\max}$ .

Together with (12), we obtain

\begin{align*} \mathbb{E}[g(X_2)] - \mathbb{E}[g(X_1)] &= \int_{x\in\mathcal{X}} \bigl(\bar F_{X_2}(\xi) - \bar F_{X_1}(\xi)\bigr)^+ \,\mathrm{d} g(\xi) - \int_{x\in\mathcal{X}} \bigl(\bar F_{X_1}(\xi) - \bar F_{X_2}(\xi)\bigr)^+ \,\mathrm{d} g(\xi) \nonumber\\[5pt] &\geq \int_{x\in\mathcal{X}} \Gamma (\xi)\rho(\xi) \,\mathrm{d} \xi, \end{align*}

where

$\Gamma(\xi) = \frac{1}{1+\Delta_+} (\bar F_{X_2}(\xi) - \bar F_{X_1}(\xi))^+ - (\bar F_{X_1}(\xi) - \bar F_{X_2}(\xi))^+.$

Now define

\begin{align*} \mu(x) = \dfrac{g^{(1)}_{\max^-}(x) - g^{(1)}_0}{g^{(1)}_{\max}- g^{(1)}_0}\quad \text{for $ x_0 \leq x \leq x_{\max}$.} \end{align*}

It is clear that $\mu$ is a probability distribution that increases from zero to one. From (11), we have

\begin{align*} 0 &\leq g^{(1)}_{\max} \int_{x_0}^{x_{\max}} \Xi(x) \,\mathrm{d} \mu(x) \\[5pt] &= \int_{x_0}^{x_{\max}} \biggl(\dfrac{1}{1+\Delta_+} g^{(1)}_0\int_{\underline{x}}^x \bigl(\bar F_{X_2}(\xi) - \bar F_{X_1}(\xi)\bigr)^+ \,\mathrm{d} \xi - g^{(1)}_0\int_{\underline{x}}^x \bigl(\bar F_{X_1}(\xi) - \bar F_{X_2}(\xi)\bigr)^+ \,\mathrm{d} \xi \nonumber\\[5pt] &\quad + \dfrac{1}{1+\Delta_+}g^{(1)}_{\max}\int_x^{\overline{x}} \bigl(\bar F_{X_2}(\xi) - \bar F_{X_1}(\xi)\bigr)^+ \,\mathrm{d} \xi - g^{(1)}_{\max} \int_x^{\overline{x}} \bigl(\bar F_{X_1}(\xi) - \bar F_{X_2}(\xi)\bigr)^+ \,\mathrm{d} \xi \biggr) \,\mathrm{d}\mu(x) \\[5pt] &= \int_{x_0}^{x_{\max}} \biggl( g^{(1)}_0 \int_{\underline{x}}^x \Gamma(\xi)\,\mathrm{d}\xi + g^{(1)}_{\max} \int_x^{\overline{x}} \Gamma(\xi)\,\mathrm{d}\xi\biggr)\,\mathrm{d} \mu(x) \\[5pt] &= \int_{\underline{x}}^{\overline{x}} \Gamma(\xi) \rho (\xi) \,\mathrm{d}\xi \leq \mathbb{E}[g(X_2)]-\mathbb{E}[g(X_1)]. \end{align*}

Thus we obtain $\mathbb{E}[g(X_1)]\leq \mathbb{E}[g(X_2)]$ .

Now we prove the ‘only if’ part of the theorem by showing that $\mathbb{E}[g(X_1)]\leq \mathbb{E}[g(X_2)]$ for any $g \in \textit{inc}^\pm(\Delta_+,\Delta_-)$ implies (10). Define

$g_y(x) = \int_{\underline{x}}^{x} g_y^{(1)}(\xi) \,\mathrm{d}\xi,$

where

\begin{align*} g_y^{(1)}(x) = \dfrac{\bigl(1+(\Delta_+\wedge \Delta_-){\mathbb{I}}_{\{\bar F_{X_1}(x) \geq \bar F_{X_2}(x)\}}\bigr){\mathbb{I}}_{\{x \leq y\}}}{1+\Delta_-} + \dfrac{\bigl(1+(\Delta_+\wedge \Delta_-){\mathbb{I}}_{\{\bar F_{X_1}(x) \geq \bar F_{X_2}(x)\}}\bigr){\mathbb{I}}_{\{x \geq y\}}}{1+\Delta_+}. \end{align*}

We can easily verify that $g_y(x) \in \textit{inc}^\pm(\Delta_+,\Delta_-)$ . We derive (following the proof of Theorem 1)

\begin{align*} 0 &\leq \mathbb{E}[g_y(X_2)] - \mathbb{E}[g_y(X_1)]\\[5pt] &= \dfrac{1}{1+\Delta_-} \biggl(\int_{\underline{x}}^y \bigl(\bar F_{X_2}(x) - \bar F_{X_1}(x)\bigr) \,\mathrm{d} x - (\Delta_+\wedge\Delta_-)\int_{\underline{x}}^y \bigl(\bar F_{X_1}(x) - \bar F_{X_2}(x)\bigr)^+ \,\mathrm{d} x \biggr)\\[5pt] &\quad +\dfrac{1}{1+\Delta_+} \biggl(\int_y^{\overline{x}} \bigl(\bar F_{X_2}(x) - \bar F_{X_1}(x)\bigr) \,\mathrm{d} x - (\Delta_+\wedge\Delta_-)\int_y^{\overline{x}} \bigl(\bar F_{X_1}(x) - \bar F_{X_2}(x)\bigr)^+ \,\mathrm{d} x \biggr), \end{align*}

which gives rise to (10).

It is clear that Theorem 2 generalizes Theorem 1. We recognize that the integral comparison $\mathbb{E}[g(X_1)]-\mathbb{E}[g(X_2)]$ is determined by both the slope $\delta_g$ and the distributional difference $\bar F_{X_1}-\bar F_{X_2}$ over the entire support $\mathcal{X}$ . The functional property described in (9) allows us to bound the positive and the negative part of $g(X_1)-g(X_2)$ with a monotone upper bound of $\delta_g$ . When we convert this upper bound into a probability measure for the distributional differences, the characterization in (10) can be easily obtained.

We name several special cases of the two-sided parametric functions.

• • When $\Delta_+ = \Delta_-=\Delta$ , we obtain $\epsilon$ -AFSD, which corresponds to the symmetric two-sided restriction of increasing functions:

  \begin{align*} \textit{inc}^{\pm}(\Delta,\Delta) = \textit{inc}(\Delta). \end{align*}

• • When $\Delta_- = \infty$ or $\Delta_+ =\infty$ , the restriction of increasing function is only one-sided:

  \begin{align*} \textit{inc}^{\pm}(\Delta,\infty) = \textit{inc}^+(\Delta) \quad \text{and} \quad \textit{inc}^{\pm}(\infty,\Delta) = \textit{inc}^-(\Delta). \end{align*}

• • When $\Delta_+ = 0$ or $\Delta_-=0$ , the functions are increasing convex or increasing concave with the two-sided restriction on the increasingness (see our later discussion in Section 4):

  \begin{align*} \textit{inc}^{\pm}(0,\Delta) = \textit{inc}(\Delta)\textit{-cx} \quad \text{and} \quad \textit{inc}^{\pm}(\Delta,0)= \textit{inc}(\Delta)\textit{-cv}. \end{align*}

4.1. Parametric increasing convex or concave functions

In Section 3 we discussed two subclasses of increasing functions, $inc^+(\Delta)$ and $inc^-(\Delta)$ , which are superclasses of increasing convex ( $\textit{icx}$ ) and increasing concave ( $\textit{icv}$ ) functions, respectively. In this subsection we analyze subclasses of $\textit{icx}$ and $\textit{icv}$ , and characterize the corresponding orders. For that, we can parameterize either the increasingness or the convexity (concavity) of the functions. Specifically, define

\begin{align*} \textit{inc-cx} (\Delta) &= \textit{inc} \cap \textit{cx}(\Delta)\\[5pt] &= \bigl\{ g\in\overline{\mathcal{G}}\colon \delta_g(x_1,x_2) \geq 0, 0 \leq \delta^2_g(x_1,x_2) \leq (1+\Delta) g^{(2)}_{\min}, \ \underline{x}\leq x_1\lt x_2\leq\overline{x} \bigr\},\\[5pt] \textit{inc-cv} (\Delta)&= \textit{inc} \cap \textit{cv}(\Delta)\\[5pt] &= \bigl\{ g\in \overline{\mathcal{G}}\colon \delta_g(x_1,x_2) \geq 0,\ 0 \geq \delta^2_g(x_1,x_2) \geq (1+\Delta) g^{(2)}_{\max}, \ \underline{x}\leq x_1\lt x_2\leq\overline{x} \bigr\}, \\[5pt] \textit{inc}(\Delta)\textit{-cx} &= \textit{inc}(\Delta) \cap \textit{cx}\\[5pt] &= \bigl\{ g\in \overline{\mathcal{G}}\colon 0\leq \delta_g(x_1,x_2) \leq (1+\Delta) g^{(1)}_{\min},\, \delta^2_g(x_1,x_2)\geq 0 ,\ \underline{x}\leq x_1\lt x_2\leq\overline{x} \bigr\},\\[5pt] \textit{inc}(\Delta)\textit{-cv} &= \textit{inc}(\Delta) \cap \textit{cv}\\[5pt] &= \bigl\{ g\in \overline{\mathcal{G}}\colon 0\leq \delta_g(x_1,x_2) \leq (1+\Delta) g^{(1)}_{\min},\, \delta^2_g(x_1,x_2) \leq 0, \ \underline{x}\leq x_1\lt x_2\leq\overline{x}\bigr\}. \end{align*}

Certainly $\textit{inc-cx}(\Delta)$ ( $\textit{inc-cv}(\Delta)$ ) and $\textit{inc}(\Delta)$ $\textit{-cx}$ ( $\textit{inc}(\Delta)$ $\textit{-cv}$ ) are different, though their intersection is non-empty. In particular, for $\Delta\in(0,\infty)$ ,

\begin{align*} \begin{array}{r} \textit{iquadcx} = \textit{inc-cx}(0) \subset \textit{inc-cx}(\Delta) \subset \textit{inc-cx}(\infty)\\[5pt] \textit{ilin} = \textit{inc(0)-cx} \subset \textit{inc}(\Delta)\textit{-cx} \subset \textit{inc}(\infty)\textit{-cx} \end{array} = \textit{icx},\\[5pt] \begin{array}{r} \textit{iquadcv} = \textit{inc-cv}(0) \subset \textit{inc-cv}(\Delta) \subset \textit{inc-cv}(\infty)\\[5pt] \textit{ilin}= \textit{inc(0)-cv} \subset \textit{inc}(\Delta)\textit{-cv} \subset \textit{inc}(\infty)\textit{-cv} \end{array} = \textit{icv}, \end{align*}

where $\textit{iquadcx}$ and $\textit{iquadcv}$ are the sets of increasing functions that are quadratically convex and concave, respectively. Theorem 4 below provides the distributional characterizations for the orders associated with the above function classes.

Proof of Theorem 4 . Part (i-a) follows from the proof of Theorem 3(i) by imposing the condition $\mathbb{E}[X_1]\leq \mathbb{E}[X_2]$ and using the fact that

$g(x)=\int_{\underline{x}}^x \int_{\underline{x}}^{\xi} g^{(2)}(\xi_0) \,\mathrm{d} \xi_0 \,\mathrm{d} \xi$

is increasing. Part (i-b) can be derived similarly, and it is shown for twice-differentiable g by Tzeng et al. [Reference Tzeng, Huang and Shih24]. We remark that the result can be generalized when the supports of $X_1$ and $X_2$ are different in view of the relations

$\int_{\underline{x}_i}^{\overline{x}_i}\int^{\overline{x}_i}_x \bar F_{X_i}(\xi)\,\mathrm{d}\xi \,\mathrm{d} x= \mathbb{E}[(X_i-\underline{x}_i)^2]$

and

$ \int_{\underline{x}_i}^{\overline{x}_i}\int_{\underline{x}_i}^x F_{X_i}(\xi)\,\mathrm{d}\xi\,\mathrm{d} x= \mathbb{E}[(\overline{x}_i-X_i)^2],$

where $\underline{x}_i$ and $\overline{x}_i$ , respectively, are the minimum and maximum realizations of $X_i,i=1,2$ .

To see part (ii-a), we note from (10) that $ \textit{inc}(\Delta)$ $\textit{-cx}=\textit{inc}(\Delta) \cap \textit{cx} = \textit{inc}^\pm(0,\Delta)$ . The relation in (16) can be derived from Theorem 2, which also implies $\mathbb{E}[X_2] \geq \mathbb{E}[X_1]$ . Part (ii-b) follows from an argument similar to that of part (ii-a).

The $\textit{inc-cv}(\Delta)$ functions lead to the ‘almost second-degree stochastic dominance’ ( $\epsilon$ -ASSD) introduced in [Reference Leshno and Levy7] and further discussed in [Reference Levy, Leshno and Leibovitch10] with the relation $\Delta=1/\epsilon-2$ .

4.2. Other integral orders

There are other ways of defining integral stochastic orders and parameterizing the orders, depending on the applications. For example, Tsetlin et al. [Reference Tsetlin, Winkler, Huang and Tzeng23] consider restrictions on both the first-order and the second-order properties by considering the following function class:

\begin{align*} \textit{inc}(\Delta_1)\textit{-cv}(\Delta_2) =\textit{inc}(\Delta_1) \cap \textit{cx}(\Delta_2). \end{align*}

They name the corresponding order as the ‘generalized almost second-degree stochastic dominance’ (GASSD). They show that $X_1 \leq_{\rm GASSD} X_2$ if and only if $\mathbb{E}[(\overline{x} -X_1)^2]\geq \mathbb{E}[(\overline{x}-X_2)^2]$ and

\begin{align*} \max_{h(x)} \int_{x\in\mathcal{X}} h(x) \bigl(F_{X_1}^{(2)}(x) - F_{X_2}^{(2)}(x)\bigr) \,\mathrm{d} x \leq \dfrac{1}{\Delta_1}(\mathbb{E}[(\overline{x} -X_1)^2]-\mathbb{E}[(\overline{x}-X_2)^2] ), \end{align*}

where

\begin{align*} h(x) \geq 0, \ \int_{x\in\mathcal{X}} h(x)\,\mathrm{d} x = 1, \ \sup_{x\in\mathcal{X}}\{ h(x)\} \leq (1+\Delta_2) \inf_{x\in\mathcal{X}}\{ h(x)\}. \end{align*}

This characterization can be derived by modifying the proof of Theorem 3(ii). Specifically, we recognize that the function class is closed under scaling, i.e. $g\in \textit{inc}(\Delta_1)$ $\textit{-cv}(\Delta_2)$ implies $\alpha g \in \textit{inc}(\Delta_1)$ $\textit{-cv}(\Delta_2)$ . Moreover, any increasing concave function with bounded first- and second-order deviations can be approximated by twice-differentiable functions. For twice-differentiable $g\in \textit{inc}(\Delta_1)$ $\textit{-cv}(\Delta_2)$ with appropriate scaling such that $g^{(1)}_{\max}-g^{(1)}_{\min}=1$ , we have

$g^{(1)}_{\min^+}(\underline{x}) =g^{(1)}_{\min} = g^{(1)}_{\max} -1 \leq (1+\Delta_1) g^{(1)}_{\min},$

implying

$g^{(1)}_{\min^+}(\underline{x}) =g^{(1)}_{\min} \geq \frac{1}{\Delta_1}.$

Moreover,

$g^{(1)}_{\max}-g^{(1)}_{\min}=\int_{\xi\in\mathcal{X}} g^{(2)}(\xi) \,\mathrm{d} \xi.$

One may similarly define the $\textit{inc}(\Delta_1)$ $\textit{-cx}(\Delta_2)$ class, and obtain, by modifying the proof of Theorem 3(i), the characterization that $\mathbb{E}[(X_1-\underline{x})^2]\leq \mathbb{E}[(X_2-\underline{x})^2]$ and

\begin{align*} \max_{h(x)} \int_{x\in\mathcal{X}} h(x) \bigl(\bar F_{X_2}^{(2)}(x) - \bar F_{X_1}^{(2)}(x)\bigr) \,\mathrm{d} x \leq \dfrac{1}{\Delta_1}(\mathbb{E}[(X_2-\underline{x})^2] - \mathbb{E}[(X_1-\underline{x})^2]), \end{align*}

where

\begin{align*} h(x) \geq 0, \ \int_{x\in\mathcal{X}} h(x)\,\mathrm{d} x = 1, \ \sup_{x\in\mathcal{X}}\{ h(x)\} \leq (1+\Delta_2) \inf_{x\in\mathcal{X}}\{ h(x)\}. \end{align*}

As we can see, the distributional characterizations for $\textit{inc}(\Delta_1)$ $\textit{-cv}(\Delta_2)$ and $\textit{inc}(\Delta_1)$ $\textit{-cx}(\Delta_2)$ , involving maximization over a functional space, may not be easy to verify in general.

A similar analysis can be used to characterize the ‘almost risk-averse stochastic dominance’ (ARSD) defined by Luo and Tan [Reference Luo and Tan14]. They consider the following subclass of increasing concave (twice-differentiable) functions parameterized by $\kappa \geq 0$ :

\begin{align*} \biggl\{ 0 \leq g^{(1)}(x) \leq (1+\Delta) g^{(1)}_{\min}, \ 0 \leq \dfrac{-g^{(2)}(x)}{g^{(1)}(x)} \leq \dfrac{\kappa}{1+\Delta},\ \Delta \in [0,\infty) \biggr\}, \end{align*}

and obtain the characterization of $\mathbb{E}[X_1] \leq \mathbb{E}[X_2]$ and

\begin{align*} v_{O}(X_1 \leq_{\textit{icv}} X_2) = \int_{x\in\mathcal{X}} \biggl(\int_{\underline{x}}^x F_{X_2}(\xi) \,\mathrm{d} \xi - \int_{\underline{x}}^x F_{X_1}(\xi) \,\mathrm{d} \xi\biggr)^+ \,\mathrm{d} x \leq \dfrac{1}{\kappa} (\mathbb{E}[X_2]-\mathbb{E}[X_1]). \end{align*}

In a related discussion, Huang et al. [Reference Huang, Tzeng and Zhao5] define the ‘fractional degree stochastic dominance’, denoted by $(1+c)$ -SD, for the following class of functions:

\begin{align*} \biggl\{g\in\overline{\mathcal{G}}\colon g^{(1)}(x)\geq 0,\ \dfrac{- g^{(2)}(x)}{g^{(1)}(x)} \geq -\biggl(\dfrac{1}{c}-1\biggr),\ x\in\mathcal{X}\biggr\} . \end{align*}

They reach the characterization

\begin{align*} \int_{\underline{x}}^x (F_{X_2}(\xi) - F_{X_1}(\xi) )\,\mathrm{d} h_c(\xi) \geq 0 , \end{align*}

where

\begin{align*} h_c(\xi) = \begin{cases} {\mathrm{e}}^{(1/c-1)\xi}, & 0 \lt c\lt 1,\\[5pt] x, & c = 1. \end{cases} \end{align*}

They show that $\epsilon$ -AFSD implies $(1+c)$ -SD order for appropriately chosen c (as a function of $\epsilon$ ). Thus this order is more general than $\epsilon$ -AFSD.

5.1. Parametric m-convex and m-concave functions Parametric m-convex and m-concave functions

The m-convex and m-concave functions, respectively, are defined by (e.g. [Reference Shaked and Shanthikumar21])

\begin{align*} \textit{m-cx} &= \bigl\{g\in\overline{\mathcal{G}}^{m} \colon g^{(m)}(x) \geq 0,\, x\in\mathcal{X}\bigr\} , \\[5pt] \textit{m-cv} &= \bigl\{g\in\overline{\mathcal{G}}^{m} \colon (\!-\!1)^{m-1} g^{(m)}(x) \geq 0, \, x\in\mathcal{X}\bigr\}. \end{align*}

A random variable $X_1$ is smaller than another random variable $X_2$ in the m-convex (m-concave) order, i.e. $X_1 \leq_{\textit{m-cx}}\ (\leq_{\textit{m-cv}})\ X_2$ , if and only if $\mathbb{E}[g(X_1)] \leq \mathbb{E}[g(X_2)]$ for all $g \in \textit{m-cx}$ ( $\textit{m-cv}$ ). An equivalent condition is that $\mathbb{E}[X_1^n] = \mathbb{E}[X_2^n],n=1,2,\ldots, m-1$ , and

\begin{align*} \bar F_{X_1}^{(m-1)} (x) \leq \bar F_{X_2}^{(m-1)} (x) \ \bigl[ F_{X_1}^{(m-1)} (x) \geq F_{X_2}^{(m-1)} (x)\bigr], \quad x \in \mathcal{X}, \end{align*}

where

$\bar F^{(n)}_X (x) = \int_x^{\overline{x}} \bar F^{(n-1)}_X(\xi) \,\mathrm{d} \xi ,$

with

$\bar F^{(0)}_X = \bar F_X \biggl[ F^{(n)}_X (x) = \int_{\underline{x}}^x F^{(n-1)}_X(\xi) \,\mathrm{d} \xi\ \text{with}\ F^{(0)}_X = F_X\biggr].$

It is straightforward to verify that

\begin{align*} \bar F^{(n)}_X(x) = \dfrac{1}{n!} \mathbb{E}[((X-x)^+)^n] \quad \text{and} \quad F^{(n)}_X(x) = \dfrac{1}{n!} \mathbb{E}[((x-X)^+)^n], \quad x \in \mathcal{X}, \quad n = 1,2, \ldots . \end{align*}

We can define the restricted function classes as

\begin{align*} \textit{m-cx}(\Delta) &= \bigl\{g \in \overline{\mathcal{G}}^m\colon 0 \leq g^{(m)}(x) \leq (1+\Delta) g_{\min}^{(m)}, x\in\mathcal{X}\bigr\} , \\[5pt] \textit{m-cv}(\Delta) &= \bigl\{g \in \overline{\mathcal{G}}^m\colon g= (\!-\!1)^{m-1}\hat g,\, 0 \leq \hat g^{(m)}(x) \leq (1+\Delta) \hat g_{\min}^{(m)},\, x\in\mathcal{X}\bigr\}. \end{align*}

For an odd m, $\textit{m-cx}(\Delta)=\textit{m-cv}(\Delta)$ , e.g. 1 $\textit{-cx}(\Delta)=1$ $\textit{-cv}(\Delta)=\textit{inc}(\Delta)$ . For an even m, $g \in \textit{m-cx}(\Delta)$ is equivalent to $-g \in \textit{m-cv}(\Delta)$ , e.g. $g \in \textit{2-cx}(\Delta)=\textit{cx}(\Delta)$ , if and only if $-g \in \textit{2-cv}(\Delta)=\textit{cv}(\Delta)$ .

The violations to the $\textit{m-cx}$ and $\textit{m-cv}$ orders, respectively, are defined as

\begin{align*} v(x;\ X_1 \leq_{\textit{m-cx}} X_2)&= \bigl(\bar F^{(m-1)}_{X_1}(x) - \bar F^{(m-1)}_{X_2}(x)\bigr)^+ = \bigl((\!-\!1)^m \bigl( F^{(m-1)}_{X_1}(x) - F^{(m-1)}_{X_2}(x)\bigr)\bigr)^+ , \\[5pt] v(x;\ X_1 \leq_{\textit{m-cv}} X_2) &= \bigl(F^{(m-1)}_{X_2}(x) - F^{(m-1)}_{X_1}(x)\bigr)^+ . \end{align*}

With these measures, we characterize the stochastic orders associated with $\textit{m-cx}(\Delta)$ and $\textit{m-cv}(\Delta)$ functions in the next theorem.

Proof of Theorem 5 . We prove part (i), and the proof of part (ii) follows similarly. Take an integer n and an nth differentiable function g. We have

\begin{align*} \mathbb{E}[g(X)] &= a_0 - \int_{x\in \mathcal{X}} \int_{\underline{x}}^x g^{(1)} (\xi) \,\mathrm{d} \xi \,\mathrm{d} \bar F_X(x)\\[5pt] & = a_0 + \int_{\xi\in\mathcal{X}} g^{(1)}(\xi) \bar F_X(\xi) \,\mathrm{d} \xi \\[5pt] &= a_0 + a_1\bigl(\mathbb{E}[X] - \underline{x}\bigr) + \int_{\xi \in\mathcal{X}} \bar F_X^{(1)}(\xi) g^{(2)}(\xi) \,\mathrm{d} \xi \\[5pt] &= a_0 + a_1\bigl(\mathbb{E}[X] - \underline{x}\bigr) + \int_{x \in\mathcal{X}} \bar F_X^{(1)}(x)\biggl(a_2+\int_{\underline{x}}^x g^{(3)}(\xi) \,\mathrm{d} \xi \biggr) \,\mathrm{d} x\\[5pt] &= a_0 + a_1\bigl(\mathbb{E}[X] - \underline{x}\bigr) + \dfrac{a_2}{2}\mathbb{E}[(X-\underline{x})^2] + \int_{x\in\mathcal{X}} \bar F_X^{(2)}(x) g^{(3)}(x) \,\mathrm{d} x\\[5pt] &= \cdots \\[5pt] &= \sum_{k=0}^{n} \dfrac{a_k}{k!} \mathbb{E}[(X-\underline{x})^k] + \int_{x \in \mathcal{X}} \bar F_X^{(n)}(x) g^{(n+1)}(x) \,\mathrm{d} x, \end{align*}

where $a_k = g^{(k)}(\underline{x})$ for $k = 1,2, \ldots, n$ . We deduce

(19) \begin{align} & \mathbb{E}[g(X_2)] - \mathbb{E}[g(X_1)] \notag \\[5pt] &\quad = \sum_{k=0}^n \dfrac{a_k}{k!} \mathbb{E}[(X_2-\underline{x})^k - (X_1-\underline{x})^k ] + \int_{x \in \mathcal{X}} \bigl(\bar F_{X_2}^{(n)}(x) - \bar F_{X_1}^{(n)}(x) \bigr) g^{(n+1)}(x) \,\mathrm{d} x. \end{align}

For any $g \in \textit{m-cx}(\Delta)$ , and two random variables $X_1$ and $X_2$ with

$\mathbb{E}[X_1^k]=\mathbb{E}[X_2^k], \quad k=1,2,\ldots, m-1,$

we can rewrite (19) as

\begin{align*} & \mathbb{E}[g(X_2)]-\mathbb{E}[g(X_1)]\\[5pt] &\quad = \int_{x \in \mathcal{X}}\bigl(\bar F_{X_2}^{(m-1)}(x) - \bar F_{X_1}^{(m-1)}(x) \bigr)^+ g^{(m)}(x) \,\mathrm{d} x - \int_{x \in \mathcal{X}}\bigl(\bar F_{X_1}^{(m-1)}(x) - \bar F_{X_2}^{(m-1)}(x) \bigr)^+ g^{(m)}(x) \,\mathrm{d} x \\[5pt] &\quad \geq g^{(m)}_{\min} \int_{x \in \mathcal{X}}\bigl(\bar F_{X_2}^{(m-1)}(x) - \bar F_{X_1}^{(m-1)}(x) \bigr)^+ \,\mathrm{d} x - (1+\Delta) g^{(m)}_{\min} \int_{x \in \mathcal{X}}\bigl(\bar F_{X_1}^{(m-1)}(x) - \bar F_{X_2}^{(m-1)}(x) \bigr)^+ \\[5pt] &\quad = \dfrac{g^{(m)}_{\min}}{m!}(\mathbb{E}[(X_2-\underline{x})^{m}] - \mathbb{E}[(X_1-\underline{x})^{m}]) - g^{(m)}_{\min} \Delta v_O(x;\ X_1 \leq_{\textit{m-cx}} X_2). \end{align*}

Thus, when $X_1$ and $X_2$ satisfy (17), we must have $\mathbb{E}[g(X_2)] \geq \mathbb{E}[g(X_1)]$ .

Now take $X_1$ and $X_2$ such that $\mathbb{E}[g(X_1)] \leq \mathbb{E}[g(X_2)]$ for all $g\in \textit{m-cx}(\Delta)$ . We first note that $\mathbb{E}[X_1^k] = \mathbb{E}[X_2^k]$ for $k=1,2,\ldots,m-1$ because $g(x) = x^k$ and $g(x) = - x^k$ both have $g^{(m)}(x) =0$ and thus both belong to $ \textit{m-cx}(\Delta)$ . Now take

$g^{(m)}(x) = 1 + \Delta {\mathbb{I}}_{\bigl\{\bar F_{X_1}^{(m-1)}(x) > \bar F_{X_2}^{(m-1)}(x)\bigr\}}$

and

$g^{(k)}(x) = \int_{\underline{x}}^x g^{(k+1)}(\xi) \,\mathrm{d} \xi\quad\text{for $k = 0,1,\ldots, m-1$.}$

It is easy to verify $g \in \textit{m-cx}(\Delta)$ . We deduce from (19) that

\begin{align*} 0 &\leq \mathbb{E}[g(X_2)] - \mathbb{E}[g(X_1)]\\[5pt] & = \int_{x \in \mathcal{X}}\bigl(\bar F_{X_2}^{(m-1)}(x) - \bar F_{X_1}^{(m-1)}(x) \bigr) g^{(m)}(x) \,\mathrm{d} x \\[5pt] &= \int_{x \in \mathcal{X}}\bigl(\bar F_{X_2}^{(m-1)}(x) - \bar F_{X_1}^{(m-1)}(x) \bigr) \Bigl(1 + \Delta {\mathbb{I}}_{\{\bar F_{X_1}^{(m-1)}(x) > \bar F_{X_2}^{(m-1)}(x)\}}\Bigr) \,\mathrm{d} x \\[5pt] &= \dfrac{1}{m!}(\mathbb{E}[X_2^m] - \mathbb{E}[X_1^m] ) - \Delta \int_{x \in \mathcal{X}} \bigl(\bar F_{X_1}^{(m-1)}(x) - \bar F_{X_2}^{(m-1)}(x)\bigr)^+ \,\mathrm{d} x, \end{align*}

which gives (17).

Tsetlin et al. [Reference Tsetlin, Winkler, Huang and Tzeng23] name the ‘almost nth-degree risk’ ( $\epsilon$ -AnR) order corresponding to $\textit{m-cv}(\Delta)$ functions. They derive (18) and $F_{X_1}^{(k)}(\overline{x}) = F_{X_2}^{(k)}(\overline{x})$ for $k=1,2,\ldots, m-1$ . The latter condition leads to equal kth moments as the random variables have the same upper support. Mao et al. [Reference Mao, Wang and Zhao15] identify several invariant properties the order associated with $\textit{m-cv}$ functions and view the $\textit{m-cv}(\Delta)$ functions as an interpolation between the non-parametric function classes.

To extend the analysis to functions with asymmetric two-sided restrictions, we define

\begin{align*} & \textit{m-cx}^\pm(\Delta_+, \Delta_-) \\[5pt] &\quad = \bigl\{ g \in \overline{\mathcal{G}}^m\colon 0 \leq g^{(m)}(x) \leq \min\bigl\{(1+\Delta_+) g^{(m)}_{\min^+}(x), (1+\Delta_-) g^{(m)}_{\min^-}(x)\bigr\}, \, x \in \mathcal{X} \bigr\}, \nonumber\\[5pt] & \textit{m-cv}^\pm(\Delta_+, \Delta_-) \\[5pt] &\quad = \bigl\{ g \in \overline{\mathcal{G}}^m\colon \hat g = (\!-\!1)^m g, 0 \leq \hat g^{(m)}(x) \leq \min\bigl\{(1+\Delta_+) \hat g^{(m)}_{\min^+}(x), (1+\Delta_-) \hat g^{(m)}_{\min^-}(x)\bigr\}, \, x \in \mathcal{X} \bigr\}. \nonumber \end{align*}

We can characterize the distributional properties of orders associated with the $(\Delta+,\Delta-)$ m-convex (concave) functions using an argument similar to that of Theorem 2.

With Theorem 6, we can easily deduce the violation conditions for functions with one-sided restrictions that lead to the $\textit{m-cx}^+(\Delta)$ , $\textit{m-cx}^-(\Delta)$ , $\textit{m-cv}^+(\Delta)$ , and $\textit{m-cv}^-(\Delta)$ functions. We note two other special cases (see Table 3).

• • When $\Delta_+ = 0$ for m-convex functions, we obtain the function class $\textit{m-cx}^\pm(0,\Delta) = \textit{m-cx} (\Delta)\ \cap\ $ (m+1) $\textit{-cx}$ . The characterization in (20) becomes

  \begin{align*} \int_x^{\overline{x}} \bigl(\bar F^{(m-1)}_{X_1}(\xi) - \bar F^{(m-1)}_{X_2}(\xi) \bigr)\,\mathrm{d} \xi \leq \dfrac{1}{m! \Delta}\bigl( \mathbb{E}[X_2^m] - \mathbb{E}[X_1^m] \bigr), \quad x \in \mathcal{X}. \end{align*}
  For $m=1$ , we obtain the $\textit{inc}(\Delta)$ $\textit{-cx}$ order discussed in Theorem 4(ii-a).

• • When $\Delta_- = 0$ for m-concave functions, we obtain the function class $\textit{m-cv}^\pm(\Delta,0) = \textit{m-cv} (\Delta)\ \cap\ $ (m+1) $\textit{-cv}$ . The characterization in (21) becomes

  \begin{align*} \int_{\underline{x}}^x \bigl( F^{(m-1)}_{X_2}(\xi) - F^{(m-1)}_{X_1}(\xi)\bigr) \,\mathrm{d} \xi \leq \dfrac{(\!-\!1)^{m-1}}{m! \Delta} \bigl(\mathbb{E}[X_2^m] - \mathbb{E}[X_1^m]\bigr), \quad x \in \mathcal{X}. \end{align*}
  For $m=1$ , we obtain the $\textit{inc}(\Delta)$ $\textit{-cv}$ order discussed in Theorem 4(ii-b).

5.2. Parametric monotone higher orders

The monotone higher-order function classes are defined as

$\textit{m-icx}= \{g\in\mathcal{G}^{m}\colon g^{(k)}(\underline{x})=a_k \geq 0, k=1,2,\ldots, m-1;\ g^{(m)}(x)\geq 0, x\in\mathcal{X}\}$

and

$\textit{m-icv}= \{g\colon g=\hat g, (\!-\!1)^{m-1} \hat g\in \textit{m-icx}\}.$

A random variable $X_1$ is smaller than another random variable $X_2$ in the m-increasing convex (concave) order, i.e. $X_1 \leq_{\textit{m-icx}} [\!\leq_{\textit{m-icv}}] X_2$ , if and only if

$\mathbb{E}[g(X_1)] \leq \mathbb{E}[g(X_2)]\quad\text{for all $g \in \textit{m-icx}$ ($\textit{m-icv}$).}$

We define the subclasses of functions

\begin{align*} \textit{m-icx}(\Delta) &= \left\{g\in\mathcal{G}^{m}\colon \begin{array}{l} g^{(k)}(\underline{x})=a_k \geq 0, k=1,2,\ldots, m-1; \\[5pt] 0\leq g^{(m)}(x)\leq (1+\Delta) g^{(m)}_{\min}, \ x \in\mathcal{X} \end{array}\right\} , \\[5pt] \textit{m-icv}(\Delta) &= \left\{ g\in\mathcal{G}^{m}\colon \begin{array}{l} (\!-\!1)^{k-1} g^{(k)}(\overline{x})=(\!-\!1)^{k-1} b_k \geq 0, k=1,2,\ldots, m-1; \\[5pt] \hat g (x) = (\!-\!1)^{m-1} g(x),\\[5pt] 0 \leq \hat g^{(m)}(x_2) \leq (1+\Delta) \hat g^{(m)}_{\min}, x\in\mathcal{X} \end{array}\right\}. \end{align*}

The $\textit{m-icv}(\Delta)$ function leads to the ‘almost Nth-degree stochastic dominance’ (ANSD) named by Tzeng et al. [Reference Tzeng, Huang and Shih24] and the ‘generalized almost nth-degree stochastic dominance’ ( $\epsilon$ -GAnSD) named by Tsetlin et al. [Reference Tsetlin, Winkler, Huang and Tzeng23]. Lu et al. [Reference Lu, Zhang, Xiao and Dhesi13] show the correspondence of this order to the higher-order $\Omega$ ratio.

It is clear that the monotone orders have the same violation measure as the non-monotone orders. The main difference lies in the additional conditions on the ordered moments of the random variables. The moment conditions may or may not be needed for functions with one-sided restrictions, depending on whether the restriction is on the left or the right side. Specifically, for m-convex functions, we define the subclasses with one-sided restrictions as

\begin{align*} \textit{m-icx}^+(\Delta) &= \left\{g\in\overline{\mathcal{G}}^{m}\colon \begin{array}{l} g^{(k)}(\underline{x})=a_k \geq 0, k=1,2,\ldots, m-1;\\[5pt] 0 \leq g^{(m)}(x)\leq (1+\Delta) g^{(m)}_{\min^+}(x), x \in\mathcal{X} \end{array}\right\} ,\\[5pt] \textit{m-icx}^-(\Delta) &= \left\{g\in\overline{\mathcal{G}}^{m}\colon \begin{array}{l} g^{(k)}(\underline{x})=a_k \geq 0, k=1,2,\ldots, m-1;\\[5pt] 0 \leq g^{(m)}(x)\leq (1+\Delta) g^{(m)}_{\min^-}(x), x \in\mathcal{X} \end{array}\right\}. \end{align*}

We also have $ \textit{m-icx}^\pm(\Delta_+,\Delta_-) = \textit{m-icx}^+(\Delta_+) \cap \textit{m-icx}^-(\Delta_-)$ .

Proof of Theorem 8 . To show part (i), we first note that the left-hand side of (22) is non-negative, implying $\bar F^{(m)}_{X_1}(x) \leq \bar F^{(m)}_{X_2}(x)$ for all $x \in \mathcal{X}$ . We deduce $X_1 \leq_{(m+1)\textit{-icx}} X_2$ . It follows that $\mathbb{E}[g(X_1)] \leq \mathbb{E}[g(X_2)]$ for any $g\in $ (m+1) $\textit{-icx}$ , and $g(x)=(x -\underline{x})^k \in $ (m+1) $\textit{-icx}$ for $k=1,2,\ldots, m$ , we must have $\mathbb{E}[(X_1-\underline{x})^k] \leq \mathbb{E}[(X_2-\underline{x})^k]$ .

Since $g^{(m)}$ has bounded variation, we derive, using (19) in the proof of Theorem 5,

\begin{align*} & \mathbb{E}[g(X_2)]-\mathbb{E}[g(X_1)]\\[6pt] &\quad = \sum_{k=0}^{m-1} \dfrac{a_k}{k!} \mathbb{E}[(X_2-\underline{x})^k - (X_1-\underline{x})^k] + \int_{x \in \mathcal{X}}\bigl(\bar F_{X_2}^{(m-1)}(x) - \bar F_{X_1}^{(m-1)}(x) \bigr) g^{(m)}(x) \,\mathrm{d} x \\[6pt] &\quad \geq \int_{x \in \mathcal{X}} g^{(m)}_{\min^+} (x) \Bigl(\bigl(\bar F_{X_2}^{(m-1)}(x) - \bar F_{X_1}^{(m-1)}(x) \bigr) -\Delta \bigl(\bar F_{X_1}^{(m-1)}(x) - \bar F_{X_2}^{(m-1)}(x) \bigr)^+\Bigr)\,\mathrm{d} x \\[6pt] &\quad = \int_{x \in \mathcal{X}} \biggl(g^{(m)}_{\min^+}(\underline{x}) + \int_{\underline{x}}^x \,\mathrm{d} g^{(m)}_{\min^+}(\xi) \biggr) \\[6pt] &\quad \quad \times \Bigl(\bigl(\bar F_{X_2}^{(m-1)}(x) - \bar F_{X_1}^{(m-1)}(x) \bigr) -\Delta \bigl(\bar F_{X_1}^{(m-1)}(x) - \bar F_{X_2}^{(m-1)}(x) \bigr)^+\Bigr)\,\mathrm{d} x \\[6pt] &\quad = g^{(m)}_{\min^+}(\underline{x}) \int_{\xi \in \mathcal{X}} \Bigl(\bigl(\bar F_{X_2}^{(m-1)}(x) - \bar F_{X_1}^{(m-1)}(x) \bigr) - \Delta \bigl(\bar F_{X_1}^{(m-1)}(x) - \bar F_{X_2}^{(m-1)}(x) \bigr)^+\Bigr) \,\mathrm{d} x \\[6pt] &\quad \quad + \int_{\xi \in \mathcal{X}} \int_{\xi}^{\overline{x}} \Bigl(\bigl(\bar F_{X_2}^{(m-1)}(x) - \bar F_{X_1}^{(m-1)}(x) \bigr) -\Delta \bigl(\bar F_{X_1}^{(m-1)}(x) - \bar F_{X_2}^{(m-1)}(x) \bigr)^+\Bigr) \,\mathrm{d} x \,\mathrm{d} g^{(m)}_{\min^+} (\xi) . \end{align*}

Because $g^{(m)}_{\min^+}$ is increasing and $g_{\min^+}^{(m)}(\underline{x}) \geq 0$ , the right-hand side is non-negative when (22) holds.

Next we show that $\mathbb{E}[g(X_1)] \leq \mathbb{E}[g(X_2)]$ for all $g \in \textit{m-icx}^+(\Delta)$ implies (22). Take $g_y$ such that

$g^{(m)}_y(x) = \Bigl(1+\Delta {\mathbb{I}}_{\left\{\bar F_{X_1}^{(m-1)}(x) > \bar F_{X_2}^{(m-1)}(x) \right\}}\Bigr){\mathbb{I}}_{\{x \geq y\}},$

and

$g^{(k)}_y(x) = \int_{\underline{x}}^x g^{(k+1)}_y(\xi) \,\mathrm{d} \xi\quad\text{for $k = 0,1,\ldots, m-1$.}$

It is clear that for $x \geq y$ , $g^{(m)}_{y\min^+} (x) \geq 1$ and thus

$ g^{(m)}_y(x) \leq (1+\Delta) \leq (1+\Delta)g^{(m)}_{y\min^+} (x).$

For $x < y$ , $g^{(m)}_{y\min^+} (x)=g_y(x) = 0$ . Thus $g_y \in \textit{m-icx}^+(\Delta)$ . Using (19) in the proof of Theorem 5 with $a_k = g^{(k)}_y(\underline{x}) =0$ for $k=0,1,\ldots, m-1$ , we derive

\begin{align*}& \mathbb{E}[g_y(X_2)] - \mathbb{E}[g_y(X_1)]\\[5pt] &\quad = \int_{x\in \mathcal{X}}\bigl(\bar F_{X_2}^{(m-1)}(x) - \bar F_{X_1}^{(m-1)}(x)\bigr) g_y^{(m)}(x) \,\mathrm{d} x \\[5pt] &\quad = \int_y^{\overline{x}} \bigl(\bar F_{X_2}^{(m-1)}(x) - \bar F_{X_1}^{(m-1)}(x)\bigr)\,\mathrm{d} x + \Delta \int_y^{\overline{x}}\bigl(\bar F^{(m-1)}_{X_2}(x) - \bar F^{(m-1)}_{X_1}(x)\bigr) {\mathbb{I}}_{\bigl\{\bar F_{X_1}^{(m-1)}(x)> \bar F_{X_2}^{(m-1)}(x)\bigr\}} \,\mathrm{d} x\\[5pt] &\quad = \int_y^{\overline{x}}\bigl(\bar F_{X_2}^{(m-1)}(x) - \bar F^{(m-1)}_{X_1}(x)\bigr) \,\mathrm{d} x - \Delta\int_y^{\overline{x}} \bigl(\bar F^{(m-1)}_{X_1}(x) - \bar F^{(m-1)}_{X_2}(x)\bigr)^+ \,\mathrm{d} x. \end{align*}

We deduce that $\mathbb{E}[g_y(X_1)] \leq \mathbb{E}[g_y(X_2)]$ implies (22). We obtain part (i).

To see part (ii), we note that the left-hand side of (23) is non-negative, which implies (by setting $x=\overline{x}$ ) $\mathbb{E}[(X_1-\underline{x})^m] \leq \mathbb{E}[(X_2-\underline{x})^m]$ . Using an argument similar to that for part (i), we can show that (23), together with the condition $\mathbb{E}[(X_1-\underline{x})^k] \leq \mathbb{E}[(X_2-\underline{x})^k]$ for $k=1,2, \ldots, m-1$ , implies $\mathbb{E}[g(X_1)]\leq\mathbb{E}[g(X_2)]$ for all $g \in \textit{m-icx}^-(\Delta)$ . To show the reverse relation, we note that $g(x)=(x-\underline{x})^k \in \textit{m-icx}^-(\Delta)$ for any $k =1,2,\ldots, m-1$ , leading to $\mathbb{E}[(X_1-\underline{x})^k] \leq \mathbb{E}[(X_2-\underline{x})^k]$ for $k=1,2,\ldots, m-1$ . Then we take

$g^{(m)}_y(x) = \Bigl(1+\Delta {\mathbb{I}}_{\bigl\{\bar F_{X_1}^{(m-1)}(x) > \bar F_{X_2}^{(m-1)}(x) \bigr\}}\Bigr){\mathbb{I}}_{\{x \leq y\}}$

and apply an argument similar to that for part (i).

Next we prove part (iii). For the ‘if’ part, we show that (24) and the condition $\mathbb{E}[(X_1-\underline{x})^k] \leq \mathbb{E}[(X_2-\underline{x})^k]$ for $k=1,2, \ldots, m-1,$ imply $\mathbb{E}[g(X_1)] \leq \mathbb{E}[g(X_2)]$ for all $g \in \textit{m-icx}^\pm(\Delta_+,\Delta_-)$ . We focus on the case where $\Delta_+ \leq \Delta_-$ . Then (24) becomes

\begin{align*} \Xi(x)&\equiv \dfrac{1}{1+\Delta_-}\int_{\underline{x}}^x \bigl(\bar F^{(m-1)}_{X_2}(\xi) - \bar F^{(m-1)}_{X_1}(\xi)\bigr)^+ \,\mathrm{d} \xi - \dfrac{1+\Delta_+}{1+\Delta_-}\int_{\underline{x}}^x \bigl(\bar F^{(m-1)}_{X_1}(\xi) - \bar F^{(m-1)}_{X_2}(\xi)\bigr)^+ \,\mathrm{d} \xi \nonumber\\[5pt] & \quad + \dfrac{1}{1+\Delta_+}\int_x^{\overline{x}} \bigl(\bar F^{(m-1)}_{X_2}(\xi) - \bar F^{(m-1)}_{X_1}(\xi)\bigr)^+ \,\mathrm{d} \xi - \int_x^{\overline{x}} \bigl(\bar F^{(m-1)}_{X_1}(\xi) - \bar F^{(m-1)}_{X_2}(\xi)\bigr)^+ \,\mathrm{d} \xi \geq 0. \end{align*}

We derive

\begin{align*} & \mathbb{E}[g(X_2)] - \mathbb{E}[g(X_1)]\\[5pt] &\quad = \sum_{k=0}^m \dfrac{a_k}{k!}\bigl(\mathbb{E}[(X_2-\underline{x})^k] - \mathbb{E}[(X_1-\underline{x})^k]\bigr) + \int_{\underline{x}}^{\overline{x}}\bigl(\bar F^{(m-1)}_{X_2}(\xi) - \bar F^{(m-1)}_{X_1}(\xi)\bigr) g^{(m)}(\xi)\,\mathrm{d} \xi \\[5pt] &\quad \geq \int_{\underline{x}}^{\overline{x}}\bigl(\bar F^{(m-1)}_{X_2}(\xi) - \bar F^{(m-1)}_{X_1}(x)\bigr) g^{(m)}(\xi)\,\mathrm{d} \xi \\[5pt] &\quad \geq \int_{x\in\mathcal{X}} \Gamma (\xi)\rho(\xi) \,\mathrm{d} \xi\\[5pt] &\quad = g^{(m)}_{\max} \int_{x_0}^{x_{\max}} \Xi(x) \,\mathrm{d} \mu(x)\\[5pt] &\quad \geq 0, \end{align*}

where

$\Gamma(\xi) = \frac{1}{1+\Delta_+} \bigl(\bar F^{(m-1)}_{X_2}(\xi) - \bar F^{(m-1)}_{X_1}(\xi)\bigr)^+ - \bigl(\bar F^{(m-1)}_{X_1}(\xi) - \bar F^{(m-1)}_{X_2}(\xi)\bigr)^+$

and

$\mu(x) = \frac{g^{(m)}_{\max^-}(x) - g^{(m)}_0}{g^{(m)}_{\max}- g^{(m)}_0}\quad\text{for $ x_0 \leq x \leq x_{\max}$,}$

with $x_{\max}$ and $x_0$ defined similarly to those in the proof of Theorem 6.

To prove the ‘only if’ part, we first note that $g(x)=(x-\underline{x})^k \in \textit{m-icx}^\pm(\Delta_+,\Delta_-)$ and thus $\mathbb{E}[(X_1-\underline{x})^k] \leq \mathbb{E}[(X_2-\underline{x})^k]$ for $k=1,2,\ldots, m-1$ . Define

\begin{align*} g_y^{(m)}(x) &= \dfrac{\bigg(1+(\Delta_+\wedge\Delta_-){\mathbb{I}}_{\big\{\bar F^{(m-1)}_{X_1}(x) \geq \bar F^{(m-1)}_{X_2}(x)\big\}}\bigg){\mathbb{I}}_{\{x \leq y\}}}{1+\Delta_-}\\[5pt] & \quad + \dfrac{\bigg(1+(\Delta_+\wedge\Delta_-){\mathbb{I}}_{\big\{\bar F^{(m-1)}_{X_1}(x) \geq \bar F^{(m-1)}_{X_2}(x)\big\}}\bigg){\mathbb{I}}_{\{x \geq y\}}}{1+\Delta_+}, \end{align*}

and

$g^{(k)}_y(x) = \int_{\underline{x}}^x g^{(k+1)}_y(\xi) \,\mathrm{d} \xi\quad\text{for $k = 0,1,\ldots, m-1$.}$

We can easily verify that $g_y \in \textit{m-icx}^\pm(\Delta_+,\Delta_-)$ . We can derive (25) with the relation $\mathbb{E}[g_y(X_2)] \geq \mathbb{E}[g_y(X_1)]$ .

By Theorem 8, the condition of ordered moments, that is,

$\mathbb{E}[(X_1-\underline{x})^k] \leq \mathbb{E}[(X_2-\underline{x})^k]\quad\text{for $k=1,2,\ldots, m-1$,}$

is not needed for the parametric order associated with the right-side violation. This is because the violation measure in (22) is non-negative, and thus $X_1\leq_{(m+1)\textit{-icx}} X_2$ , which implies the order of the moments.

In the special case of $\Delta_- = 0$ , we obtain the function class $\textit{m-icx}^\pm(\Delta,0)=\textit{m-icx}(\Delta) \cap (m+1)\textit{-icx}$ . By (24), the distributional characterization of this order is

\begin{align*} \int_{\underline{x}}^x \bigl(\bar F^{(m-1)}_{X_1}(\xi) - \bar F^{(m-1)}_{X_2}(\xi) \bigr)\,\mathrm{d} \xi \leq \dfrac{1}{m! \Delta}\bigl( \mathbb{E}[(X_2-\underline{x})^m] - \mathbb{E}[(X_1-\underline{x})^m] \bigr), \quad x \in \mathcal{X}. \end{align*}

For $m=1$ , $1\textit{-icx}^\pm(\Delta,0) =1$ $\textit{-cx}^\pm(\Delta,0)$ , and we recover the $\textit{inc}(\Delta)$ $\textit{-cx}$ function discussed in Theorem 4(i-a). For higher orders, however, $\textit{m-icx}^\pm(\Delta,0) \neq \textit{m-cx}^\pm(\Delta,0)$ in general due to the different conditions on the moments.

For m-concave functions, we define the subclasses with one-sided restrictions as

\begin{align*} \textit{m-icv}^-(\Delta) = \left\{ g\in\overline{\mathcal{G}}^{m}\colon \begin{array}{l} (\!-\!1)^{k-1} g^{(k)}(\overline{x})=(\!-\!1)^{k-1} b_k \geq 0, k=1,2,\ldots, m-1, \\[5pt] \hat g (x) = (\!-\!1)^{m-1} g(x),\\[5pt] 0 \leq \hat g^{(m)}_{\min^-}(x_1) \leq \hat g^{(m)}(x_2) \leq (1+\Delta) \hat g^{(m)}_{\min^-}(x_1), \underline{x}\leq x_1\lt x_2\leq \overline{x} \end{array}\right\},\\[5pt] \textit{m-icv}^+(\Delta) = \left\{ g\in\overline{\mathcal{G}}^{m}\colon \begin{array}{l} (\!-\!1)^{k-1} g^{(k)}(\overline{x})=(\!-\!1)^{k-1} b_k \geq 0, k=1,2,\ldots, m-1, \\[5pt] \hat g (x) = (\!-\!1)^{m-1} g(x),\\[5pt] 0 \leq \hat g^{(m)}_{\min^+}(x_2) \leq \hat g^{(m)}(x_1) \leq (1+\Delta) \hat g^{(m)}_{\min^+}(x_2), \underline{x}\leq x_1\lt x_2\leq \overline{x} \end{array}\right\}. \end{align*}

We also define $ \textit{m-icv}^\pm(\Delta_+,\Delta_-) = \textit{m-icv}^+(\Delta_+) \cap \textit{m-icv}^-(\Delta_-)$ . We can obtain the following characterization using an argument similar to Theorem 8.