2. Preliminaries

Before proceeding to give the main results of the paper, we overview some preliminary concepts of ageing and stochastic orders. (For more details of these concepts, see, e.g., [Reference Lai and Xie23,Reference Shaked and Shanthikumar33].)

Let $X$ and $Y$ be two nonnegative random variables with their density functions $f$ and $g$, distribution functions $F$ and $G$ and survival functions $\bar {F}=1-F$ and $\bar {G}=1-G$, respectively. Furthermore, $m_X(t)={\int _t^{\infty }\bar {F}(x)\,\mathrm {d}x}/{\bar {F}(t)}$ denotes the mean residual life function corresponding to random variable $X$ ($m_Y$ is defined analogously). Throughout this paper, we assume that these functions all exist and increasing (decreasing) means nondecreasing (nonincreasing).

The above stochastic orders are related as the following.

$$X\leq_{{\rm lr}}Y\Rightarrow X \leq_{{\rm st}} Y \Rightarrow X\leq_{{\rm icx}}Y, \quad X\leq_{{\rm dmrl}}Y\Rightarrow X\leq_{{\rm nbue}}Y \Rightarrow X\leq_{\rm L}Y$$

3. Some basic properties and characterizations

In this section, we investigate the main properties of the proposed order such as its characterization and its closure properties. First, note that

(3.1)\begin{align} \int_p^{1}W_X(u)\,\mathrm{d}u & = \int_p^{1}\int_{F^{{-}1}(u)}^{\infty}\bar{F}(x)\,\mathrm{d}x\,\mathrm{d}u =\int_{F^{{-}1}(p)}^{\infty}\int_y^{\infty}\bar{F}(x)\,\mathrm{d}x\,\mathrm{d}F(y)\nonumber\\ & =\int_{F^{{-}1}(p)}^{\infty}\int_{F^{{-}1}(p)}^{x}\bar{F}(x)\,\mathrm{d}F(y)\,\mathrm{d}x\nonumber\\ & =\int_{F^{{-}1}(p)}^{\infty}\bar{F}(x)[F(x)-p]\,\mathrm{d}x \end{align}

(3.2)\begin{align} & =\int_p^{1}(1-u)(u-p)\,\mathrm{d}F^{{-}1}(u) \end{align}

(3.3)\begin{align} & =\int_p^{1}(2u-p-1)F^{{-}1}(u)\,\mathrm{d}u, \end{align}

where the latter equation is obtained by using the integration by parts. Belzunce et al. [Reference Belzunce, Pinar, Ruiz and Sordo9] have provided their measure for some statistical parametric models. For comparison purposes, Table 1 gives the $\int _p^{1}W_X(u)\,\mathrm {d}u$ for the same statistical models.

TABLE 1.

The $\int _p^{1}W_X(u)\,\mathrm {d}u$ for some distributions.

It is clear that the excess wealth order implies the second-order excess wealth order. However, the following example shows that the reverse is not necessarily true.

Example 3.1. Let $X$ and $Y$ be distributed as Weibull and gamma with the density functions $f(x)=2xe^{-x^{2}}$ and $g(x)=\frac {1}{\sqrt {2}}x^{-0.5}e^{-x}$, $x>0$, respectively. Figure 2 shows that $X\leq _{{\rm sew}}Y$ but the excess wealth order dose not hold.

FIGURE 2.

Plots of $W(p)$ and $\int _p^{1}W(u)\,\mathrm {d}u$ for distributions given in Example 3.1.

It is easy to see that in similar with the excess wealth order, the order $\leq _{{\rm sew}}$ is also location independent. That is, if $X\leq _{{\rm sew}}Y$ then $X+c\leq _{{\rm sew}}Y$ and $X\leq _{{\rm sew}}Y+c$ for all $c\in (-\infty,\infty )$. The following theorem gives a characterization result for the second-order excess wealth order. For a proof, we will need the following lemma.

Lemma 3.1 (Barlow and Proschan [Reference Barlow and Proschan5, p. 120]).

Let $W(x)$ be a measure, not necessarily positive, for which $\int _t^{\infty } \mathrm {d}W(x)\geq 0$ for all $t$, and let $h(x)\geq 0$ be increasing. Then

$$\int_t^{\infty} h(x)\,\mathrm{d}W(x)\geq 0, \quad \textrm{for all} \ t.$$

Theorem 3.1. $X\leq _{{\rm sew}}Y$ if and only if,

(3.4)\begin{equation} \int_0^{1}(1-u)\phi(u)\,\mathrm{d}F^{{-}1}(u)\leq \int_0^{1}(1-u)\phi(u)\,\mathrm{d}G^{{-}1}(u),\end{equation}

for any increasing convex function $\phi :[0,1]\rightarrow \mathrm {R}$, such that $\phi (0)=0$.

Proof. For the only if part, it is known that (cf. [Reference Belzunce, Franco-Pereira and Mulero12]), given an increasing and convex function $\phi$, then $\phi$ is continuous and there exists a positive and increasing function $h$ such that

(3.5)\begin{equation} \phi(b)-\phi(a)=\int_a^{b}h(v)\,\mathrm{d}v. \end{equation}

We have

(3.6)\begin{align} & \int_0^{1}(1-u)\phi(u)\,\mathrm{d}G^{{-}1}(u)-\int_0^{1}(1-u)\phi(u)\,\mathrm{d}F^{{-}1}(u)\nonumber\\ & \quad=\int_0^{1}(1-u)\phi(u)[\mathrm{d}G^{{-}1}(u)-\mathrm{d}F^{{-}1}(u)]\nonumber\\ & \quad=\int_0^{1}\int_0^{u}(1-u)h(v)\,\mathrm{d}v[\mathrm{d}G^{{-}1}(u)-\mathrm{d}F^{{-}1}(u)]\nonumber\\ & \quad=\int_0^{1}\int_v^{1}(1-u)[\mathrm{d}G^{{-}1}(u)-\mathrm{d}F^{{-}1}(u)]h(v)\,\mathrm{d}v \end{align}

where the second equality comes from (3.5) and the assumption $\phi (0)=0$. Under the hypothesis,

$$\int_p^{1}\int_v^{1}(1-u)[\mathrm{d}G^{{-}1}(u)-\mathrm{d}F^{{-}1}(u)]\,\mathrm{d}v =\int_p^{1}[W_Y(v)-W_X(v)]\,\mathrm{d}v\geq 0.$$

The result now follows from Lemma 3.1.

For the if part, note that, for a fixed $p\in [0,1]$, the function $\phi (u)=(u-p)^{+}=\max \{u-p,0\}$ is increasing and convex such that $\phi (0)=0$ and

$$\int_0^{1}(1-u)\phi(u)\,\mathrm{d}F^{{-}1}(u)=\int_p^{1}(1-u)(u-p)\,\mathrm{d}F^{{-}1}(u) =\int_p^{1}W_X(u)\,\mathrm{d}u,$$

where, the last equality is just Eq. (3.2). Thus, the result easily follows.

One may consider

$$\int_0^{1}(1-u)\phi(u)\,\mathrm{d}F^{{-}1}(u)=\int_0^{\infty}\bar{F}(x)\phi(F(x))\,\mathrm{d}x,$$

as a generalized version of the cumulative residual entropy (CRE, introduced by Rao et al. [Reference Rao, Chen, Vemuri and Wang31]) which gives the CRE by taking $\phi (u)=-\ln (1-u)$.

By using the integration by parts, one can also see that the inequality (3.4) is equivalent to that

(3.7)\begin{equation} \int_0^{1}W_X(u)\,\mathrm{d}\phi(u)\leq\int_0^{1}W_Y(u)\,\mathrm{d}\phi(u), \quad \textrm{or} \quad \int_0^{1}\phi(u)\,\mathrm{d}W_X(u)\geq\int_0^{1}\phi(u)\,\mathrm{d}W_Y(u). \end{equation}

Belzunce [Reference Belzunce7] gives a characterization of the excess wealth order in terms of the increasing convex order. In general, neither of the orders $\leq _{{\rm sew}}$ and $\leq _{{\rm icx}}$ implies the other (let, for example, $X$ be distributed as Weibull with the shape parameter equal to 4 and 0.9 and the scale parameter 1 and $Y$ be distributed as gamma with the shape parameter equal to 2.2 and 0.8 and scale parameter 1). However, the following theorem also gives a characterization of the second-order excess wealth order in terms of the increasing convex order.

Theorem 3.2. Let $X_1,X_2$ and $Y_1,Y_2$ be independent copies of continuous random variables $X$ and $Y$, respectively. Then, $X\leq _{{\rm sew}}Y$ if and only if

(3.8)\begin{equation} (X_2-X_1)_+I(X_1>F^{{-}1}(p))\leq_{{\rm icx}}(Y_2-Y_1)_+I(Y_1>G^{{-}1}(p)), \quad \textrm{for all}\ p\in[0,1]. \end{equation}

Proof. The only if part follows from Eq. (1.1) and the fact that the order $\leq _{{\rm icx}}$ implies $E(X)\leq E(Y)$.

To prove the if part, one can observe that the survival function of $\mathcal {X}_p=(X_2-X_1)_+I(X_1>F^{-1}(p))$ at $x\geq 0$ is given by

$$\bar{K}_{\mathcal{X}_p}(x)=\int_{F^{{-}1}(p)}^{\infty}\bar{F}(x+z)\,\mathrm{d}F(z) =\int_p^{1}\bar{F}(x+F^{{-}1}(u))\,\mathrm{d}u.$$

The $\bar {K}_{\mathcal {Y}_p}(\cdot )$, the survival function of $\mathcal {Y}_p=(Y_2-Y_1)_+I(Y_1>G^{-1}(p))$ is given analogously. We observe that for a fixed $p$, $\lim _{t\rightarrow \infty }\int _t^{\infty }[\bar {K}_{\mathcal {Y}_p}(x)-\bar {K}_{\mathcal {X}_p}(x)]\,{\rm d}x=0$ and under the hypothesis, $E[\mathcal {X}_p]\leq E[\mathcal {Y}_p]$. This implies that the number of sign changes of $\bar {K}_{\mathcal {X}}-\bar {K}_{\mathcal {Y}}$ is at most one. The result now follows from Theorem 4.A.22 in Shaked and Shanthikumar [Reference Shaked and Shanthikumar33, p. 194].

For any nonnegative random variable $X$ and for any $a>0$, we have

$$\int_p^{1}W_{aX}(u)\,\mathrm{d}u=a\int_p^{1}W_X(u)\,\mathrm{d}u,$$

from which, for any two nonnegative random variables $X$ and $Y$ we get that if $X\leq _{{\rm sew}}Y$, then $aX\leq _{{\rm sew}}aY$. The result for a general transform of $X$ is given in the following theorem.

Proof. First, note that, the survival and quantile functions of the random variable $\varphi (X)$ are $\bar {F}(\varphi ^{-1}(x))$ and $\varphi (F^{-1}(p))$, respectively. Using Eqs. (3.3) and (3.5), we have

(3.9)\begin{align} \int_p^{1}W_{\varphi(Y)}(u)\,\mathrm{d}u-\int_p^{1}W_{\varphi(X)}(u)\,\mathrm{d}u & =\int_p^{1}(2u-p-1)[\varphi(G^{{-}1}(u))-\varphi(F^{{-}1}(u))]\,\mathrm{d}u\nonumber\\ & =\int_p^{1}\int_{F^{{-}1}(u)}^{G^{{-}1}(u)}(2u-p-1)h(v)\,\mathrm{d}v\,\mathrm{d}u. \end{align}

Now, we obtain a lower bound for (3.9). First, assume that $G^{-1}(u)>F^{-1}(u)$. Then, we get

$$\int_{F^{{-}1}(u)}^{G^{{-}1}(u)}h(v)\,\mathrm{d}v\geq h(F^{{-}1}(u))[G^{{-}1}(u)-F^{{-}1}(u)].$$

On the other hand, if $G^{-1}(u)< F^{-1}(u)$, then

$$\int_{F^{{-}1}(u)}^{G^{{-}1}(u)}h(v)\,\mathrm{d}v={-}\int_{G^{{-}1}(u)}^{F^{{-}1}(u)}h(v)\,\mathrm{d}v\geq h(F^{{-}1}(u))[G^{{-}1}(u)-F^{{-}1}(u)].$$

Therefore, we obtain

$$\int_p^{1}\int_{F^{{-}1}(u)}^{G^{{-}1}(u)}(2u-p-1)h(v)\,\mathrm{d}v\,\mathrm{d}u \geq\int_p^{1}(2u-p-1)h(F^{{-}1}(u))[G^{{-}1}(u)-F^{{-}1}(u)]\,\mathrm{d}u.$$

By the hypothesis,

$$\int_p^{1}(2u-p-1)[G^{{-}1}(u)-F^{{-}1}(u)]\,\mathrm{d}u=\int_p^{1}[W_Y(u)-W_X(u)]\,\mathrm{d}u\geq 0.$$

Since $h(F^{-1}(u))$ is a positive and increasing function, applying Lemma 3.1 now gives that

$$\int_p^{1}(2u-p-1)h(F^{{-}1}(u))[G^{{-}1}(u)-F^{{-}1}(u)]\,\mathrm{d}u\geq 0,$$

which completes the proof.

As an application in reliability theory, let $X_1,X_2,\ldots,X_n$ be the independent random lifetimes of the components of a parallel system which are copies of $X$. Consider another parallel systems with $Y_1,Y_2,\ldots,Y_n$ being its components lifetime which are independent and are copies of $Y$. The following theorem shows that if $X\leq _{{\rm sew}}Y$, then the lifetime of the systems are also ordered in the sense of $\leq _{{\rm sew}}$.

Theorem 3.4. Let $X_1,X_2,\ldots,X_n$ and $Y_1,Y_2,\ldots,Y_n$ be independent copies of $X$ and $Y$, respectively. If $X\leq _{{\rm sew}}Y$, then

$$\max\{X_1,X_2,\ldots,X_n\}\leq_{{\rm sew}}\max\{Y_1,Y_2,\ldots,Y_n\}.$$

Proof. It is sufficient to consider only the case $n=2$. Using Eq. (3.2), $X\leq _{{\rm sew}}Y$ is equivalent to

(3.10)\begin{equation} \int_{\sqrt{p}}^{1}(1-u)(u-\sqrt{p})[\mathrm{d}G^{{-}1}(u)-\mathrm{d}F^{{-}1}(u)]\geq 0, \quad \textrm{for all} \ p\in[0,1]. \end{equation}

On the other hand, the quantile functions of $\max \{X_1,X_2\}$ and $\max \{Y_1,Y_2\}$ are $F^{-1}(\sqrt {u})$ and $G^{-1}(\sqrt {u})$, respectively. Hence, to prove $\max \{X_1,X_2\}\leq _{{\rm sew}}\max \{Y_1,Y_2\}$, we need to show that

$$\int_p^{1}(1-u)(u-p)[\mathrm{d}G^{{-}1}(\sqrt{u})-\mathrm{d}F^{{-}1}(\sqrt{u})]\geq 0, \quad \textrm{for all} \ p\in[0,1],$$

or, equivalently, that

$$\int_{\sqrt{p}}^{1}(1+u)(u+\sqrt{p})(1-u)(u-\sqrt{p})[\mathrm{d}G^{{-}1}(u)-\mathrm{d}F^{{-}1}(u)]\geq 0, \quad \textrm{for all} \ p\in[0,1],$$

which is obtained by using the inequality (3.10) and applying Lemma 3.1.

Consider also two series systems with $X_1,X_2,\ldots,X_n$ and $Y_1,Y_2,\ldots,Y_n$ being their components lifetime which are independent and are copies of $X$ and $Y$, respectively. For these series systems, the following result gives the reversed preservation property of the order $\leq _{{\rm sew}}$; that is, if $\min \{X_1,X_2,\ldots,X_n\}\leq _{{\rm sew}}\min \{Y_1,Y_2,\ldots,Y_n\}$, then the lifetimes of components in two systems are also ordered in the sense of $\leq _{{\rm sew}}$.

Theorem 3.5. For any integer $n>0$, if

$$\min\{X_1,X_2,\ldots,X_n\}\leq_{{\rm sew}}\min\{Y_1,Y_2,\ldots,Y_n\},$$

then $X\leq _{{\rm sew}}Y$.

Proof. The survival functions of $\min \{X_1,X_2,\ldots,X_n\}$ and $\min \{Y_1,Y_2,\ldots,Y_n\}$ are given by $\bar {F}_{(n)}(x)=\bar {F}^{n}(x)$ and $\bar {G}_{(n)}(x)=\bar {G}^{n}(x)$, respectively. Notice also that $G^{-1}_{(n)}F_{(n)}(x)=G^{-1}F(x)$. It is not difficult to see from Eq. (3.1) that $X\leq _{{\rm sew}}Y$ is equivalent to

(3.11)\begin{equation} \int_t^{\infty}\bar{F}(x)[\bar{F}(t)-\bar{F}(x)]\,\mathrm{d}(G^{{-}1}F(x)-x)\geq 0, \quad \textrm{for all} \ t\geq 0. \end{equation}

Thus, the assumption is equivalent to, for all $t\geq 0$,

$$\int_t^{\infty}\bar{F}^{n}(x)[\bar{F}^{n}(t)-\bar{F}^{n}(x)]\,\mathrm{d}(G^{{-}1}F(x)-x)\geq 0.$$

Using the fact that

$$\bar{F}^{n}(t)-\bar{F}^{n}(x)=[\bar{F}(t)-\bar{F}(x)]\sum_{i=1}^{n}\bar{F}^{n-i}(t)\bar{F}^{i-1}(x),$$

we obtain

\begin{align*} & \int_t^{\infty}\bar{F}(x)[\bar{F}(t)-\bar{F}(x)]\,\mathrm{d}(G^{{-}1}F(x)-x)\\ & \quad = \int_t^{\infty}[\bar{F}^{n-1}(x)\sum_{i=1}^{n}{\bar{F}}^{n-i}(t)\bar{F}^{i-1}(x)]^{{-}1} \bar{F}^{n}(x)[\bar{F}^{n}(t)-\bar{F}^{n}(x)]\,\mathrm{d}(G^{{-}1}F(x)-x)\geq 0, \end{align*}

where, the inequality follows from Lemma 3.1 and the fact that

$$\left[\bar{F}^{n-1}(x)\sum_{i=1}^{n}{\bar{F}}^{n-i}(t)\bar{F}^{i-1}(x)\right]^{{-}1},$$

is increasing in $x$. Thus, the proof is complete.

Some other properties of the second excess wealth order in reliability theory can also be tracked. For the residual random variable $X_t=X-t|X>t$ with survival function $\bar {F}_{X_t}(x)={\bar {F}(x+t)}/{\bar {F}(t)}$ and corresponding inverse function $F_{X_t}^{-1}(p)=F^{-1}(p\bar {F}(t)+F(t))-t$, we get that

$$W_{X_t}(p)=(1-p)m_X(F^{{-}1}(p\bar{F}(t)+F(t)))=\frac{W_X(p\bar{F}(t)+F(t))}{\bar{F}(t)}$$

and

(3.12)\begin{align} \int_p^{1}W_{X_t}(u)\,\mathrm{d}u & =\int_p^{1}(1-u)m_X(F^{{-}1}(u\bar{F}(t)+F(t)))\,\mathrm{d}u\nonumber\\ & =\frac{1}{\bar{F}^{2}(t)}\int_{p\bar{F}(t)+F(t)}^{1}W_X(u)\,\mathrm{d}u. \end{align}

Now, we have the following result which is a characterization of the DMRL(IMRL) and NBUE(NWUE) distributions in terms of the second-order excess wealth order.

Proof. We give proof of parts (b) and (c). For part (b), regarding Eq. (3.12), one can see that $X_t\geq _{{\rm sew}}(\leq _{{\rm sew}})X$ if and only if

\begin{align*} \int_p^{1}W_{X_t}(u)\,\mathrm{d}u & = \int_p^{1}(1-u)m_X(F^{{-}1}(u\bar{F}(t)+F(t)))\,\mathrm{d}u \\ & \leq({\geq}) \int_p^{1}(1-u)m_X(F^{{-}1}(u))\,\mathrm{d}u=\int_p^{1}W_X(u)\,\mathrm{d}u, \quad \textrm{for all} \ p\in[0,1], \end{align*}

which holds if and only (the only if parts holds using the integral comparison theorem) $m_X(F^{-1}(u\bar {F}(t)+F(t)))\leq (\geq )m_X(F^{-1}(u))$ for all $u\in [0,1]$ or equivalently $m_X(F^{-1}(u_2))\leq (\geq )m_X(F^{-1}(u_1))$ for all $u_1< u_2$. This implies that $m_X(F^{-1}(u))$ or $m_X(u)$ is decreasing (increasing). That is $X\in {\rm DMRL}({\rm IMRL})$.

To prove part (c), we see from Table 1 that if $Y$ is distributed as exponential with mean $E(X)$, then $\int _p^{1}W_Y(u)\,{\rm d}u={(E(X)(1-p)^{2})}/{2}$. On the other hand, if

$$\int_p^{1}W_X(u)\,\mathrm{d}u=\int_p^{1}(1-u)m_X(F^{{-}1}(u))\,\mathrm{d}u\leq({\geq})\frac{E(X)(1-p)^{2}}{2},\quad \textrm{for all}\ p\in[0,1],$$

then

$$\int_p^{1}(1-u)[m_X(F^{{-}1}(u))-E(X)]\,\mathrm{d}u\leq({\geq}) 0,$$

which follows that $m_X(x)\leq (\geq ) E(X)$ for all $x\geq 0$. This means that $X\in NBUE(NWUE)$ if and only if $X\leq _{{\rm sew}}(\geq _{{\rm sew}})Y$, where $Y$ is an exponentially distributed random variable with mean $E(X)$.

We end this section by the following result which shows that the second-order excess wealth order is preserved under convergence.

4. Relationship with other stochastic orders

It is well-known that most of the classical variability orders agree with the comparison of variances and Gini's mean differences. This means that if, for example, $X\leq _{{\rm ew}}Y$, then ${\rm Var}(X)\leq {\rm Var}(Y)$ and ${\rm GMD}(X)\leq {\rm GMD}(Y)$ (cf. [Reference Shaked and Shanthikumar33, p. 166]), where ${\rm GMD}(X)=E|X_1-X_2|$ (and analogously for ${\rm GMD}(Y)$) is the Gini mean difference corresponding to $X$ with $X_1$ and $X_2$ being two independent copies of $X$. The Gini mean difference is a dispersion measure that shares many properties with the variance (cf. [Reference Yitzhaki37]). The order $\leq _{{\rm sew}}$ is also in agreement with comparison among the Gini mean differences. To see this, let $\phi (x)=x$ which is increasing and convex with $\phi (0)=0$. Therefore, it follows from (3.4) that if $X\leq _{{\rm sew}}Y$ then

$${\rm GMD}(X)=2\int_0^{\infty}\bar{F}(x)F(x)\,\mathrm{d}x\leq 2\int_0^{\infty}\bar{G}(x)G(x)\,\mathrm{d}x={\rm GMD}(Y).$$

The following theorem shows that the ordered Gini mean differences along with the order $\leq _{{\rm dmrl}}$ implies the second-order excess wealth order.

Proof. First, we observed that

(4.1)\begin{equation} \int_0^{1}W_X(u)\,{\rm d}u=\frac{1}{2}{\rm GMD}(X), \end{equation}

In addition, according to the definition (cf. [Reference Shaked and Shanthikumar33, p. 221]), $X\leq _{{\rm dmrl}}Y$ is equivalent to that ${m_X(F^{-1}(u))}/{m_Y(G^{-1}(u))}={W_Y(u)}/{W_X(u)}$ is increasing in $u\in [0,1]$ which is the same as the condition

$$\frac{\frac{1}{{\rm GMD}(Y)}W_Y(u)}{\frac{1}{{\rm GMD}(X)}W_X(u)} \quad \textrm{is incrasing in}\ u\in[0,1].$$

Note also that $({W_X(u)}/{{\rm GMD}(X)})({W_Y(u)}/{{\rm GMD}(Y)})$ is a density function corresponding to a random variable, say, $Z_X(Z_Y)$. Then, the above condition is equivalent to that $Z_X\leq _{{\rm lr}}Z_Y$ which implies that $Z_X\leq _{{\rm st}}Z_Y$. Now, this along with the hypothesis ${\rm GMD}(X)\leq {\rm GMD}(Y)$ gives that

$$\frac{\int_p^{1}W_X(u)\,\mathrm{d}u}{{\rm GMD}(X)}\leq\frac{\int_p^{1}W_Y(u)\,\mathrm{d}u}{{\rm GMD}(Y)}\leq\frac{\int_p^{1}W_Y(u)\,\mathrm{d}u}{{\rm GMD}(X)},$$

which implies that $X\leq _{{\rm sew}}Y$.

One can also easily show that if $X\leq _{{\rm nbue}}Y$ and $E(X)\leq E(Y)$, then $X\leq _{{\rm sew}}Y$.

It is not difficult to see that the excess wealth function can be expressed in terms of the function

$$A_X(p)=\int_0^{p}[F^{{-}1}(u)-E(X)]\,\mathrm{d}u, \quad 0\leq p\leq 1,$$

as

(4.2)\begin{equation} W_X(p)={-}A_X(p)-(1-p)A'_X(p), \end{equation}

where $A'_X(p)=({\mathrm {d}}/{\mathrm {d}p})A_X(p)$. $A_X(p)$ is called the absolute Lorenz curve (cf. [Reference Ramos and Sordo30]) and takes the values $A_X(0)=A_X(1)=0$ which follow that

$$A_X(p)={-}\int_p^{1}[F^{{-}1}(u)-E(X)]\,\mathrm{d}u, \quad 0\leq p\leq 1.$$

Now, we have the following theorem which shows that the dilation order implies the second-order excess wealth order.

Proof. Using Eq. (4.2) and the integration by parts, we get that

(4.3)\begin{align} \int_p^{1}W_X(u)\,\mathrm{d}u & ={-}\int_p^{1}A_X(u)\,\mathrm{d}u-\int_p^{1}(1-u)\,\mathrm{d}A_X(u)\nonumber\\ & =pA_X(p)+\int_p^{1}u\,\mathrm{d}A_X(u)-\int_p^{1}(1-u)\,\mathrm{d}A_X(u)\nonumber\\ & ={-}p\int_p^{1} \mathrm{d}A_X(u)+\int_p^{1}u\,\mathrm{d}A_X(u)-\int_p^{1}(1-u)\,\mathrm{d}A_X(u)\nonumber\\ & =\int_p^{1}[2u-1-p]\,\mathrm{d}A_X(u)\nonumber\\ & =\int_0^{1}[2u-1-p]I(u\geq p)\,\mathrm{d}A_X(u). \end{align}

This along with Theorem 2.2 in Ramos and Sordo [Reference Ramos and Sordo30] now follow the result.

The following example shows that the order $\leq _{{\rm sew}}$ dose not necessarily imply $X\leq _{{\rm dil}}Y$.

Example 4.1. Let $X$ and $Y$ be random variables with distribution functions $F(x)={x}/{3}$ and $G(x)=\sqrt {{x}/{3}}$, $0< x<3$, respectively. Then, we have $A_X(p)=\frac {3}{2}(p^{2}-p)$, $A_Y(p)=p^{3}-p,$ $0\leq p\leq 1$. Also,

$$\int_p^{1}W_X(u)\,\mathrm{d}u=0.5(1-3p-3p^{2}-p^{3}), \quad \int_p^{1}W_Y(u)\,\mathrm{d}u=0.5-p+p^{3}-0.5p^{4}.$$

It is not difficult to verify that $A_X(p)< A_Y(p)$ for $p<0.5$ and $A_X(p)>A_Y(p)$ for $p>0.5$ which by Theorem 2.2 in Ramos and Sordo [Reference Ramos and Sordo30] means that the dilation order dose not hold. Though,

$$\int_p^{1}W_Y(u)\,\mathrm{d}u-\int_p^{1}W_X(u)\,\mathrm{d}u=0.5(p(1-p^{3})+3p^{3}+3p^{2})\geq 0, \quad \textrm{for all} \ p\in[0,1].$$

One can see that Eq. (4.2) is a first-order differential equation in $A_X(p)$ whose solution can be found as

(4.4)\begin{equation} A_X(p)={-}(1-p)\int_0^{p}\frac{W_X(u)}{(1-u)^{2}}\,\mathrm{d}u. \end{equation}

At the case where the right endpoint of the support of $X$, $u_X=F^{-1}(1)=\inf \{x: F(x)=1\}$ is finite, it can also be found that

(4.5)\begin{equation} A_X(p)=(1-p)\int_p^{1}\frac{W_X(u)}{(1-u)^{2}}\,\mathrm{d}u+(1-p)[E(X)-u_X]. \end{equation}

As we already mentioned, the orders $\leq _{{\rm sew}}$ and $\leq _{{\rm icx}}$ do not imply each other. However, using the above equation, we are able to show that the order $\leq _{{\rm sew}}$ implies the $\leq _{{\rm icx}}$ for distributions with finite supports.

Proof. Under the hypothesise, by Eq. (4.5) and applying Lemma 3.1, we get that for all $p\in [0,1]$

\begin{align*} A_X(p)-A_Y(p) & = (1-p)\int_p^{1}\frac{[W_X(u)-W_Y(u)]}{(1-u)^{2}}\,\mathrm{d}u \\ & \quad +(1-p)[E(X)-E(Y)+u_Y-u_X]\\ & \geq (1-p)[E(X)-E(Y)], \end{align*}

or equivalently

$$-\int_p^{1}[F^{{-}1}(u)-E(X)]\,\mathrm{d}u+\int_p^{1}[G^{{-}1}(u)-E(Y)]\,\mathrm{d}u\geq (1-p)[E(X)-E(Y)].$$

That is,

$$\int_p^{1}F^{{-}1}(u)\,\mathrm{d}u\leq\int_p^{1}G^{{-}1}(u)\,\mathrm{d}u,\quad \textrm{for all}\ p\in[0,1],$$

which is equivalent to $X\leq _{{\rm icx}}Y$ (see Theorem 4.A.3 in [Reference Shaked and Shanthikumar33, p. 183]).

Example 4.2. Let $X$ and $Y$ be distributed as ${\rm Beta}(1,3)$ and ${\rm Beta}(3,1)$ with distribution functions $F(x)=1-(1-x)^{3}$ and $G(x)=x^{3}$, $0< x<1$, respectively. Then, we have

\begin{align*} & \int_t^{1}\bar{F}(x)\,\mathrm{d}x=\frac{(1-t)^{4}}{4}, \quad \int_p^{1}W_X(u)\,\mathrm{d}u=\frac{3}{4}(1+p)(1-p)^{{4}/{3}}-2\int_p^{1}u(1-u)^{{1}/{3}}\,\mathrm{d}u,\\ & \int_t^{1}\bar{G}(x)\,\mathrm{d}x=\frac{1-t^{4}}{4}, \quad \int_p^{1}W_Y(u)\,\mathrm{d}u=\frac{6}{7}(1-p^{{7}/{3}})-\frac{3}{4}(1+p)(1-p^{{4}/{3}}). \end{align*}

Figure 3 depicts that $X\geq _{{\rm sew}}Y$ and $X\leq _{{\rm icx}}Y$.

FIGURE 3.

Plots of $\int _p^{1}W(u)\,\mathrm {d}u$ and $\int _t^{1}\bar {F}(x)\,\mathrm {d}x$ for distributions given in Example 4.2.

For $p\in [0,1]$, denote recursively $F_1^{-1}(p)=F^{-1}(p)$, and $F_n^{-1}(p)=\int _p^{1}F_{n-1}^{-1}(u)\,\mathrm {d}u,$ for $n=2,3,\ldots$ (and analogously for $G_n^{-1}$). For any positive integer $m$ and all $p\in [0,1]$, if $F_m^{-1}(p)\leq G_m^{-1}(p)$, then it is denoted $X\leq ^{-1}_mY$ (see [Reference Shaked and Shanthikumar33, p. 213]). The result of the next theorem is similar with that in the above theorem between the orders $\leq _{{\rm sew}}$ and $\leq ^{-1}_3$.

Proof. From Eqs. (4.4) and (4.5), we have

(4.6)\begin{align} \int_p^{1}A_X(u)\,\mathrm{d}u & ={-}\int_p^{1}(1-u)\int_0^{u}\frac{W_X(v)}{(1-v)^{2}}\,\mathrm{d}v\,\mathrm{d}u\nonumber\\ & ={-}\frac{(1-p)^{2}}{2}\int_0^{p}\frac{W_X(v)}{(1-v)^{2}}\,\mathrm{d}v-\frac{1}{2}\int_p^{1}W_X(v)\,\mathrm{d}v\nonumber\\ & =\frac{(1-p)}{2}A_X(p)-\frac{1}{2}\int_p^{1}W_X(v)\,\mathrm{d}v\nonumber\\ & =\frac{(1-p)^{2}}{2}\int_p^{1}\frac{W_X(v)}{(1-v)^{2}}\,\mathrm{d}v+\frac{(1-p)^{2}}{2}[E(X)-u_X] -\frac{1}{2}\int_p^{1}W_X(v)\,\mathrm{d}v,\nonumber\\ & =\int_p^{1}\left[\frac{(1-p)^{2}}{2(1-v)^{2}}-\frac{1}{2}\right]W_X(v)\,\mathrm{d}v +\frac{(1-p)^{2}}{2}[E(X)-u_X]. \end{align}

where the second equality follows from integration by parts. Now, using the hypothesis and applying Lemma 3.1, we see that for all $p\in [0,1]$

\begin{align*} \int_p^{1}A_X(u)\,\mathrm{d}u-\int_p^{1}A_Y(u)\,\mathrm{d}u & = \int_p^{1}\left[\frac{(1-p)^{2}}{2(1-v)^{2}}-\frac{1}{2}\right][W_X(v)-W_Y(v)]\,\mathrm{d}v \\ & \quad +\frac{(1-p)^{2}}{2}[E(X)-E(Y)+u_Y-u_X]\\ & \geq \frac{(1-p)^{2}}{2}[E(X)-E(Y)], \end{align*}

or equivalently

$$\int_p^{1}\int_u^{1}F^{{-}1}(v)\,\mathrm{d}v\,\mathrm{d}u\leq\int_p^{1}\int_u^{1}G^{{-}1}(v)\,\mathrm{d}v\,\mathrm{d}u,$$

which completes the proof.

As a corollary, the above theorem along with Theorem 4.A.72 in Shaked and Shanthikumar [Reference Shaked and Shanthikumar33, p. 213] follow that if $X\geq _{{\rm sew}}Y$ and $u_Y\geq u_X$, then

$$E[\max\{X_1,X_2,\ldots,X_k\}]\leq E[\max\{Y_1,Y_2,\ldots,Y_k\}], \quad k\geq 2,$$

where $X_i's[Y_i's]$ are independent copies of $X[Y]$.

It is known that [Reference Shaked and Shanthikumar33, p. 165] $X\leq _{{\rm ew}}Y$ if and only if,

(4.7)\begin{equation} \frac{G_2^{{-}1}(p)-F_2^{{-}1}(p)}{(1-p)}=\frac{1}{(1-p)}\int_p^{1}[G^{{-}1}(u)-F^{{-}1}(u)]\,\mathrm{d}u \quad \textrm{is increasing in} \ p\in(0,1). \end{equation}

In the next theorem, we give a similar equivalence for the second-order excess wealth order.

Theorem 4.5. $X\leq _{{\rm ew}}Y$ if and only if,

(4.8)\begin{equation} \frac{G_3^{{-}1}(p)-F_3^{{-}1}(p)}{(1-p)^{2}}=\frac{1}{(1-p)^{2}}\int_p^{1}\int_u^{1}[G^{{-}1}(v)-F^{{-}1}(v)]\,\mathrm{d}v\mathrm{d}u \quad \textrm{is increasing in} \ p\in[0,1]. \end{equation}

Proof. First, we have from Eq. (4.2) that

$$\int_p^{1}W_X(u)\,\mathrm{d}u=(1-p)A_X(p)-2\int_p^{1}A_X(u)\,\mathrm{d}u,$$

which implies that $X\leq _{{\rm sew}}Y$ if and only if, for all $p\in [0,1]$

(4.9)\begin{equation} 2\int_p^{1}[A_X(u)-A_Y(u)]\,\mathrm{d}u\geq(1-p)[A_X(p)-A_Y(p)], \end{equation}

or, equivalently,

(4.10)\begin{equation} \frac{1}{(1-p)^{2}}\int_p^{1}[A_X(u)-A_Y(u)]\,\mathrm{d}u \quad \textrm{is increasing in} \ p\in[0,1]. \end{equation}

On the other hand, one can write

\begin{align*} \int_p^{1}[A_X(u)-A_Y(u)]\,\mathrm{d}u& =\int_p^{1}\int_u^{1}[G^{{-}1}(v)-F^{{-}1}(v)]\,\mathrm{d}v\,\mathrm{d}u\\ & \quad + \frac{(1-p)^{2}}{2}[E(X)-E(Y)]. \end{align*}

Therefore, the condition (4.10) is equivalent to that

$$\frac{1}{(1-p)^{2}}\int_p^{1}\int_u^{1}[G^{{-}1}(v)-F^{{-}1}(v)]\,\mathrm{d}v\,\mathrm{d}u \quad \textrm{is increasing in} \ p\in[0,1].$$

This completes the proof.

Under Definition 3.1 in Ramos and Sordo [Reference Ramos and Sordo30], $X$ is said to be smaller than $Y$ in the second-order absolute Lorenz order if $\int _p^{1}A_X(u)\,\mathrm {d}u\geq \int _p^{1}A_Y(u)\,\mathrm {d}u,$ for all $p\in [0,1]$. The following theorem shows that the second-order absolute Lorenz order implies the second excess wealth order.

Proof. First, note that the inequality (4.9) gives that

$$\int_0^{1}[A_X(u)-A_Y(u)]\,\mathrm{d}u\geq 0.$$

On the other hand, it follows from (4.10) that

$$\int_p^{1}[A_X(u)-A_Y(u)]\,\mathrm{d}u\geq(1-p)^{2}\int_0^{1}[A_X(u)-A_Y(u)]\,\mathrm{d}u.$$

These two latter inequalities now give the result.

Example 4.3. For the random variables in Example 4.1, we have

$$\int_p^{1}A_X(u)\,\mathrm{d}u-\int_p^{1}A_Y(u)\,\mathrm{d}u=0.25p^{2}[1+2p(1-p)]\geq 0.$$

We also had $\int _p^{1}W_X(u)\,\mathrm {d}u\leq \int _p^{1}W_Y(u)\,\mathrm {d}u$.

Using Eq. (3.3), one can write

(4.11)\begin{equation} \int_p^{1}W_X(u)\,\mathrm{d}u = E(X)\int_p^{1}(2u-p-1)\,\mathrm{d}L_X(u)=E(X)\int_p^{1}[1-2L_X(u)]\,\mathrm{d}u,\end{equation}

where $L_X(u)=({1}/{E(X)})\int _0^{u}F^{-1}(v)\,\mathrm {d}v$ is the Lorenz curve corresponding to $X$. The following result gives the relationship between the Lorenz and second-order excess wealth orders.

Proof. Under the hypothesises and by using Eq. (4.11), we obtain

$$\frac{\int_p^{1}W_X(u)\,\mathrm{d}u}{E(X)}=\int_p^{1}[1-2L_X(u)]\,\mathrm{d}u \leq\int_p^{1}[1-2L_Y(u)]\,\mathrm{d}u=\frac{\int_p^{1}W_Y(u)\,\mathrm{d}u}{E(Y)} \leq\frac{\int_p^{1}W_Y(u)\,\mathrm{d}u}{E(X)},$$

which completes the proof.

Belzunce et al. [Reference Belzunce, Pinar, Ruiz and Sordo9] have developed a new criterion to compare risks based on the notion of expected proportional shortfall. They have said that $X$ is smaller than $Y$ in the expected proportional shortfall order (in short, the PS order), denoted by $X\leq _{\rm PS}Y$, if

$$\frac{W_X(p)}{F^{{-}1}(p)}\leq\frac{W_Y(p)}{G^{{-}1}(p)}, \quad \textrm{for all} \ p\in D_X\cap D_Y,$$

where $D_X=\{p\in (0,1): F^{-1}(p)>0\}$ (and analogously for $D_Y$). The following theorem connects the second-order excess wealth order with the PS order.

An application of the excess wealth order in comparing the epoch times of two nonhomogeneous poisson processes (NHPP) has been given in Belzunce et al. [Reference Belzunce, Hu and Khaledi8]. We end this section by considering the same comparison problem with respect to the second-order excess wealth order.

Let $T_{1,n}$ and $T_{2,n}$, $n\geq 1$ be the epoch points of two NHPPs with intensity functions $\lambda _1$ and $\lambda _2$, respectively, such that $\int _t^{\infty }\lambda _i(u)\,{\rm d}u=\infty,$ $i=1,2$, for all $t\geq 0$. Let also $X$ and $Y$ be two nonnegative random variables with hazard rates $\lambda _1$ and $\lambda _2$ and with distribution functions $F$ and $G$, respectively. The distribution functions of $T_{1,n}$ and $T_{2,n}$ can be expressed as (see [Reference Belzunce, Hu and Khaledi8])

$$F_n(t)=\Phi_n(F(t)), \quad \textrm{and} \quad G_n(t)= \Phi_n(G(t)), \quad t\geq 0,$$

respectively, where $\Phi _n(p)=\Gamma _n(-\ln (1-p))$ for $p\in (0,1)$, and $\Gamma _n$ is the distribution function of a gamma distribution with scale parameter $1$ and shape parameter $n$.

Proof. Regarding Theorem 3.1, the assertion follows if we show that, for any increasing and convex function $\phi$

$$\int_0^{1}(1-u)\phi(u)\,\mathrm{d}F^{{-}1}_n(u)\leq \int_0^{1}(1-u)\phi(u)\,\mathrm{d}G^{{-}1}_n(u),$$

or equivalently,

$$\int_0^{1}(1-u)\phi(u)\,\mathrm{d}F^{{-}1}(\Phi^{{-}1}_n(u))\leq \int_0^{1}(1-u)\phi(u)\,\mathrm{d}G^{{-}1}_n(\Phi^{{-}1}_n(u)).$$

Note that

\begin{align*} \int_0^{1}(1-u)\phi(u)\,\mathrm{d}F^{{-}1}(\Phi^{{-}1}_n(u))& = \int_0^{1}\bar{\Phi}_n(u)\phi(\Phi_n(u))\,\mathrm{d}F^{{-}1}(u) \\ & =\int_0^{1}(1-u)\phi^{*}(u)\,\mathrm{d}F^{{-}1}(u), \end{align*}

where $\phi ^{*}(u)={(\bar {\Phi }_n(u)\phi (\Phi _n(u)))}/{(1-u)}$. It is not difficult to see that $\phi ^{*}(u)$ is an increasing and convex function. Now, the assumption along with Theorem 3.1 implies that

$$\int_0^{1}(1-u)\phi^{*}(u)\,\mathrm{d}F^{{-}1}(u)\leq\int_0^{1}(1-u)\phi^{*}(u)\,\mathrm{d}G^{{-}1}(u).$$

This completes the proof.