2.1. Results on ${\mathsf{Var}} [g(X)]$

We recall (see, for instance, Section 2 of Wasserman [Reference Wasserman30]) that the delta method is often applied to approximate the moments of g(X) using Taylor expansions, provided that the function g is sufficiently differentiable and that the moments of X are finite. For instance, if g is differentiable and ${\mathbb{E}}(X^2)$ is finite, estimating the first moment and using a second-order approximation for the random variable X, one obtains

(22) \begin{equation} {\mathsf{Var}}[g(X)] \approx \left[g'({\mathbb{E}}(X))\right]^2 {\mathsf{Var}}(X).\end{equation}

Nevertheless, we can now provide an identity analogous to (22).

Remark 6. Under the assumption of Corollary 3, from Eq. (19) we have

(23) \begin{equation} {\mathsf{Var}}[g(X)] = \left[g'(\xi)\right]^2 {\mathsf{Var}}(X), \end{equation}

where $\xi\in \mathbb R^+$ is such that

(24) \begin{equation} [g'(\xi)]^2=\frac{ {\mathbb{E}} [g'(X_e)g(X_e)] - {\mathbb{E}}[g'(X_e)]\,{\mathbb{E}}[g(\tilde{X})]}{{\mathbb{E}} (X_e) - {\mathbb{E}} (\tilde{X})}. \end{equation}

In general, the computation of $\xi$ is not easy. It can be simplified when the inverse of $g'$ exists, and taking Remark 4 into account. Next we provide an example with two cases in which $\xi$ can be computed analytically.

As seen in (4), Theorem 1 allows us to express the variance of g(X) in terms of the product of the mean of X and a term depending on the random variables $X_e$ and $\tilde X$ . This result has been suitably extended in Theorem 3, where the difference $\mathsf{Var}[g(Y)] - \mathsf{Var}[g(X)]$ is expressed in a similar way. We aim to provide similar decompositions, where the variance of g(X) is given by the product of the variance of X and a suitable mean.

Remark 7. Let X be a non-negative random variable, and let $g\colon \mathbb{R}_0^+\times \mathbb{R}_0^+\to \mathbb{R}$ be a Riemann integrable function, such that $\mathsf{Var}\left[ \int_0^{+\infty} g (X, \theta) \,{\mathrm{d}} \theta \right]$ is finite. Then, from the properties of the covariance, we have

\begin{align*} \mathsf{Var} \left [ \int_0^{+\infty} g (X, \theta)\, {\mathrm{d}} \theta \right ] = 2 \int_0^{+\infty} {\mathrm{d}} \theta \int_{\theta}^{+\infty}\mathsf{Cov} \left[ g(X,\theta), g(X,\eta) \right]\, {\mathrm{d}} \eta.\end{align*}

Theorem 4. Let X be a non-negative random variable with CDF F and SF $\overline F$ , such that $\mathsf{Var}(X)$ is finite and non-zero. Let g be a differentiable function and let $g'$ be Riemann integrable such that ${\mathbb{E}} [ g'(X_1^*) g'(X_2^*)]$ is finite, where $(X_1^*,X_2^*)$ is a non-negative absolutely continuous random vector such that $X_1^* \leq X_2^*$ almost surely, having joint PDF

(25) \begin{equation} f^* (x_1,x_2)= \frac{2\,F(x_1) \overline{F} (x_2)}{\mathsf{Var}(X)}\, \mathbf{1}_{\{ 0 < x_1 < x_2 \}}. \end{equation}

Then we have

(26) \begin{equation} \mathsf{Var} [g(X)] = {\mathbb{E}} [ g'(X_1^*) g'(X_2^*)] \,\mathsf{Var} (X).\end{equation}

Proof. Following the probabilistic generalization of Taylor’s theorem (cf. [Reference Massey and Whitt21]), we have

\begin{align*} \mathsf{Var} [g(X)] = \mathsf{Var} [g(X) - g(0)]= \mathsf{Var} \left [ \int_0^X g' (x) \,{\mathrm{d}} x \right ] = \mathsf{Var} \left [ \int_0^{+\infty} \mathbf{1}_{\{X>x\}} \,g' (x)\, {\mathrm{d}} x \right ].\end{align*}

Hence, making use of Remark 7 and equation (25), it follows that

\begin{equation*}\begin{split} \mathsf{Var} [g(X)] = \ & 2 \int_0^{+\infty} {\mathrm{d}} x_1 \int_{x_1}^{+\infty} \mathsf{Cov}\left[ \mathbf{1}_{\{X>x_1\}}, \mathbf{1}_{\{X>x_2\}} \right] g'(x_1) g'(x_2)\, {\mathrm{d}} x_2 \\ = \ &2 \int_0^{+\infty} {\mathrm{d}} x_1 \int_{x_1}^{+\infty} \left[ \overline{F} (x_2) -\overline{F} (x_1) \overline{F} (x_2) \right] g'(x_1) g'(x_2) \,{\mathrm{d}} x_2 \\ = \ &2 \int_0^{+\infty} {\mathrm{d}} x_1 \int_{x_1}^{+\infty} F(x_1) \overline{F} (x_2) g'(x_1) g'(x_2)\, {\mathrm{d}} x_2 \\ = \ &\mathsf{Var} (X) \int_0^{+\infty} {\mathrm{d}} x_1 \int_{x_1}^{+\infty} f^* (x_1,x_2) g'(x_1) g'(x_2)\, {\mathrm{d}} x_2.\end{split}\end{equation*}

Then (26) immediately follows.

Remark 8. Under the assumptions of Theorem 4, by repeated use of Fubini’s theorem and straightforward calculations, one has

(27) \begin{align} \int\!\!\!\!\int_{\mathbb{R}^2} 2\,F(x_1) \overline{F} (x_2)\,\mathbf{1}_{\{ 0 < x_1 < x_2 \}} \,{\mathrm{d}} x_1{\mathrm{d}} x_2 & = \int_0^{+\infty} \mathrm{d}F (x_1) \int_0^{+\infty} (x_2-x_1)^2 \mathbf{1}_{\{x_2 > x_1\}} {\mathrm{d}} F (x_2) \nonumber \\ &= \frac{1}{2}\, {\mathbb{E}}\big[ \big(X-\hat X\big)^2 \big]=\mathsf{Var}(X), \end{align}

where $\hat X$ is an independent copy of X. Hence, from (25), we have that $(X_1^*,X_2^*)$ is an honest random vector.

Identity (27) recalls the analogous relation concerning the Gini’s mean difference of X that, under the same assumptions, is given by (cf. Yitzhaki and Schechtman [Reference Yitzhaki and Schechtman32], for instance)

(28) \begin{equation} \int_0^{+\infty} 2\,F(x) \overline{F} (x) \,{\mathrm{d}} x = {\mathbb{E}}\big[ \big|X-\hat X\big| \big]=:\mathsf{GMD}(X).\end{equation}

We remark that identity (26) is quite similar to an analogous result shown in Corollary 4.1 of Psarrakos [Reference Psarrakos24], namely $\mathsf{Var} [w(X)] = {\mathbb{E}} [ w'(X_{\tilde w})]\, {\mathbb{E}} [w'(X_{\star})] \,\mathsf{Var}(X)$ , where the distributions of $X_{\tilde w}$ and $X_{\star}$ are suitably expressed in terms of the distribution of a non-negative absolutely continuous random variable X, and where w is an increasing function.

Table 1 shows some examples of densities $f^* (x_1,x_2)$ obtained for suitable choices of the CDF F(x), where $\gamma$ and $\Gamma$ denote respectively the lower and upper incomplete gamma functions. Note that the last example considered in Table 1 refers to a case in which X is discrete (i.e., Bernoulli), whereas in all other cases X is absolutely continuous.

Table 1. Examples of joint PDFs $f^* (x_1,x_2)$ for some choices of the distribution of X.

Remark 9. The identity given in (26) can be suitably specialized for suitable choices of g. For instance, if the assumptions of Theorem 4 are satisfied for

(i) $g(x)=x^{\alpha}$ , $\alpha\in [1,+\infty)$ , then

\begin{align*} \mathsf{Var} \big[ X^{\alpha}\big] = \alpha^2\, {\mathbb{E}} \big[ (X_1^* X_2^*)^{\alpha-1}\big] \,\mathsf{Var} (X);\end{align*}

(ii) $g(x)=e^{\beta x}$ , $\beta \in \mathbb{R}^+$ , then

\begin{align*} \mathsf{Var} \big[e^{\beta X}\big] = \beta^2 \, {\mathbb{E}} \big[e^{\beta (X_1^* + X_2^*)}\big] \,\mathsf{Var} (X).\end{align*}

2.2. Results on the marginal distributions

In this section we study the marginal distributions of $X^*_1$ and $X^*_2$ . In order to give a probabilistic representation of their PDFs, we recall that, given a non-negative random variable X with SF $\overline{F}(x)$ , its stop-loss function is defined as

(29) \begin{equation} \mathbb{E}\left[(X-x)_+\right]=\int_x^{+\infty} \overline{F}(t)\, {\mathrm{d}} t, \quad x\in\mathbb R^+_0.\end{equation}

The stop-loss function is useful in the context of actuarial risks (cf. Section 1.2.2 of Belzunce et al. [Reference Belzunce, Martnez-Riquelme and Mulero5], for instance). It is worth mentioning that in a similar way we can define the reversed stop-loss function of X as

(30) \begin{equation} \mathbb{E}\left[(x-X)_+\right]= \int_0^{x} {F}(t)\, {\mathrm{d}} t, \quad x\in\mathbb R^+_0,\end{equation}

where F is the CDF of X. Clearly, due to (29) and (30), these functions are related by

\begin{equation*} \mathbb{E}\left[(X-x)_+\right]-\mathbb{E}\left[(x-X)_+\right]=\mathbb{E}\left[X\right]-x, \quad x\in\mathbb R^+_0.\end{equation*}

Corollary 4. (i) Under the assumptions of Theorem 4, the PDFs of $X^*_1$ and $X^*_2$ are given respectively by

(31) \begin{equation} \begin{split} f_{1}^*(x) &= \frac{2\,F(x)\,{\mathbb{E}} [(X-x)_{+}]}{\mathsf{Var}(X)}\,\mathbf{1}_{\{x \in\mathbb R^+_0 \}}, \\ f_{2}^* (x) &= \frac{2\,\overline{F}(x)\, {\mathbb{E}} [(x-X)_{+}]}{\mathsf{Var}(X)}\,\mathbf{1}_{\{x \in\mathbb R^+_0 \}}.\end{split}\end{equation}

(ii) Moreover, for $x\in\mathbb R^+_0$ the SFs of $X_1^*$ and $X_2^*$ can be expressed as

\begin{equation*} \begin{split} \overline{F}_{1}^*(x) & = \frac{1}{\mathsf{Var}(X)} \left [ \int_x^{+\infty} \mathbb{E} \left [((X-z)_+)^2 \right ] f(z)\mathrm{d}z + F(x) \mathbb{E} \left [ ((X-x)_+)^2 \right ] \right ], \\ \overline{F}_{2}^* (x) & = \frac{1}{\mathsf{Var}(X)} \left [\int_x^{+\infty} \mathbb{E} \left [((z-X)_+)^2 \right ] f(z)\mathrm{d}z -\overline{F}(x) \mathbb{E} \left [ ((x-X)_+)^2 \right ] \right ]. \end{split}\end{equation*}

Table 2 shows some examples of the PDFs of $X^*_1$ and $X^*_2$ provided in (31), where $E_k(x)=(k! /\sqrt{\pi})\int_0^x e^{-t^k} {\rm d}t$ is the generalized error function, and ${\rm erf}(x)=E_2(x)$ denotes the Gauss error function. In this table, the PDF $f_1^* (x)$ of case (i) is a special instance of the weighted exponential PDF (see, for instance, Gupta and Kundu [Reference Gupta and Kundu17]).

Table 2. PDFs $f_1^* (x)$ and $f_2^* (x)$ for the examples of Table 1.

Recalling Proposition 3.9 of Asadi and Berred [Reference Asadi and Berred3], we can express $X_1^*$ and $X_2^*$ in terms of the equilibrium distribution, the second-order equilibrium distribution and the length-biased distribution of the original random variable X.

Remark 10. Under the assumptions of Theorem 4, if ${\mathbb{E}} (X_e) \in \mathbb R^+$ , then the random variable $X_1^*$ can be written as

\begin{align*}X_1^* \stackrel{d}{=} [X_{e_2}|X < X_{e_2}],\end{align*}

where $X_{e_2}$ is the second-order equilibrium distribution of X, having PDF $f_{e_2}(x) = f_{e}(x) / {\mathbb{E}} (X_e)$ , for all $x \in \mathbb R^+$ .

On the other hand, we can write $X_2^*$ as a generalized mixture, given by

\begin{align*}X_2^* \stackrel{d}{=} c \, X_e^L -[X_e|\widehat{X}_e < X_e],\quad c= \frac{{\mathbb{E}}(X^2)}{{\mathsf{Var}} (X)},\end{align*}

where $X_e^L$ is the length-biased distribution of $X_e$ , whose PDF is $f_e^L (x)=x f_e(x)/{\mathbb{E}} (X_e)$ , and $\widehat{X}_e$ is an independent copy of $X_e$ .

The random variables introduced in Theorem 4 satisfy $X_1^* \leq X_2^*$ almost surely, so that from Theorem 1.A.1 of Shaked and Shanthikumar [Reference Shaked and Shanthikumar28] one immediately has that $X_1^*$ and $X_2^*$ are ordered according to the usual stochastic order (see Chapter 1 of [Reference Shaked and Shanthikumar28] for details). Let us now investigate if the stronger relation based on the likelihood ratio order can be established between such random variables. To this end, we adopt the notation $S_G\;:\!=\;\{x\in \mathbb R^+_0\colon G(x)>0\}$ , for any function $G\colon\mathbb R^+_0\to \mathbb R$ .

Remark 11. Under the assumptions of Theorem 4, due to (31), the following identity holds:

(32) \begin{equation} \frac{f_{1}^*(x)}{f_{2}^* (x)} =\frac{\mathsf{mrl}(x)}{\mathsf{mit}(x)}, \quad x\in\mathbb R^+_0\cap S_{F},\end{equation}

where

(33) \begin{equation} \mathsf{mrl}(x) = \left\{ \begin{array}{l@{\quad}l} \mathbb{E}\left[X-x \, | \, X > x \right] = \displaystyle\frac{1}{\overline{F}(x)} \int_x^{+\infty} \overline{F}(t) \, {\rm d}t, & x\in\mathbb R^+_0\cap S_{\overline{F}}, \\0, & \hbox{otherwise},\end{array}\right. \end{equation}

and

(34) \begin{equation} \mathsf{mit}(x) = \left\{ \begin{array}{l@{\quad}l} \mathbb{E}\left[x-X \, | \, X \le x \right] = \displaystyle \frac{1}{F(x)} \int_0^{x} F(t) \, {\rm d}t, & x\in\mathbb R^+_0 \cap S_{F}, \\0, & \hbox{otherwise},\end{array}\right. \end{equation}

denote respectively the mean residual lifetime and the mean inactivity time of X (see, for instance, Nanda et al. [Reference Nanda, Bhattacharjee and Balakrishnan22] and Section 2 of Navarro [Reference Navarro23]).

If X is a random variable such that its mean residual lifetime and mean inactivity time are increasing, then $X_1^*$ and $X_2^*$ can be viewed as weighted versions of the equilibrium distribution $X_e$ of X, as we see in the following remark. We recall that, given a random variable X having PDF f, its weighted version $X^w$ is a random variable having PDF defined by

(35) \begin{equation} f_{X^w}(x)=\frac{w(x)}{{\mathbb{E}}[w(X)]}\, f(x), \quad x \in \mathbb{R}^+,\end{equation}

where $w:[0,+\infty)\to [0,+\infty)$ is a continuous function, such that ${\mathbb{E}}[w(X)] \in \mathbb{R}^+$ . This construction modifies the original distribution of X by assigning more weight to certain values according to the function w.

Remark 12. Under the assumptions of Theorem 4, if X has finite mean residual lifetime and mean inactivity time, then one has

\begin{align*} X_1^* \stackrel{d}{=} X_e^{w_1} \quad \text{and} \quad X_2^* \stackrel{d}{=} X_e^{w_2}, \end{align*}

where $w_1(x)= F(x) \,\mathsf{mrl}(x)$ and $w_2(x)= F(x)\,\mathsf{mit}(x)$ , for all $x\in \mathbb R_0^+ \cap S_{F \cdot \overline{F}}$ .

Usually, ratios of similar functions involving the mean residual lifetime and the mean inactivity time are encountered in reliability theory and survival analysis when dealing with stochastic orders, relative ageing and similar notions. See, for instance, Finkelstein [Reference Finkelstein16] for the ‘MRL ageing faster’ notion and relative ordering of mean residual lifetime functions, and Arriaza et al. [Reference Arriaza, Sordo and Suárez-Llorens2] for the quantile mean inactivity time order. Quite unexpectedly, the ratio of the marginal densities of $(X_1^*,X_2^*)$ is expressed in (32) as the ratio of $\mathsf{mrl}(x)$ and $\mathsf{mit}(x)$ , which appears to be new in the literature.

Identity (32) suggests that we investigate relations between $X_1^*$ and $X_2^*$ based on the likelihood ratio order and expressed in terms of properties of the functions introduced in (33) and (34). In order to establish a likelihood ratio ordering between $X_1^*$ and $X_2^*$ (see (11)), we recall that for a non-negative absolutely continuous random variable X with PDF f, CDF F and SF $\overline F$ , the hazard rate and the reversed hazard rate are given respectively by (cf. Navarro [Reference Navarro23] for the main properties and applications of these functions)

(36) \begin{equation} \lambda(x) = \frac{f(x)}{\overline{F}(x)}, \ x\in S_{\overline F}, \quad \overline{\lambda}(x) = \frac{f(x)}{F(x)}, \ x\in S_{F}\end{equation}

so that $\lambda(x) + \overline{\lambda}(x) =\frac{\lambda(x)}{F(x)}=\frac{\overline{\lambda}(x)}{\overline{F}(x)}$ .

Theorem 5. Under the assumptions of Theorem 4, if X is an absolutely continuous random variable then

\begin{align*} X_1^* \leq_{\mathrm{lr}} X_2^* \ \text{if and only if} \,\, \lambda(x) + \overline{\lambda}(x) \le \frac{1}{\mathsf{mrl}(x)} + \frac{1}{\mathsf{mit}(x)} \quad \forall\, x\in S_{F\cdot \overline F}.\end{align*}

Proof. Due to (33) and (34) we have

\begin{align*}\lim_{x \to \ell^+} \frac{\mathsf{mrl}(x)}{\mathsf{mit}(x)} = + \infty,\end{align*}

where $\ell=\inf S_{F\cdot \overline F}$ , so that the ratio $\frac{\mathsf{mrl}(x)}{\mathsf{mit}(x)}$ cannot be increasing in x. Moreover, recalling (36), we have

(37) \begin{equation} \frac{\rm d}{{\rm d}x}\mathsf{mrl}(x) = \lambda(x) \, \mathsf{mrl}(x) - 1, \ \frac{\rm d}{{\rm d}x}\mathsf{mit}(x) =1 - \overline{\lambda}(x) \, \mathsf{mit}(x), \quad x\in S_{F\cdot \overline F}.\end{equation}

and thus

\begin{align*}\begin{split} \frac{\rm d}{{\rm d}x} \left(\frac{\mathsf{mrl}(x)}{\mathsf{mit}(x)}\right) \le 0 \quad & \iff \quad \mathsf{mit}(x) \, \frac{\rm d}{{\rm d}x} \mathsf{mrl}(x) \le \mathsf{mrl} (x) \,\frac{\rm d}{{\rm d}x} \mathsf{mit}(x) \\ & \iff \quad \lambda(x) + \overline{\lambda}(x) \le \frac{1}{\mathsf{mrl}(x)} + \frac{1}{\mathsf{mit}(x)}.\end{split} \end{align*}

The desired result then follows from identity (32) and condition (11) for $X_1^*$ and $X_2^*$ .

Let us now discuss an immediate corollary of Theorem 5, based on suitable monotonicities of the functions $\mathsf{mrl}(x)$ and $\mathsf{mit}(x)$ .

Proof. The monotonicity assumptions of $\mathsf{mrl}(x)$ and $\mathsf{mit}(x)$ , due to (37), imply

\begin{align*}\lambda(x) \le \frac{1}{\mathsf{mrl}(x)} \quad \mbox{and} \quad \overline{\lambda}(x) \le \frac{1}{\mathsf{mit}(x)}.\end{align*}

Hence, the desired result follows from Theorem 5.

We remark that the ratio $f_1^* (x)/f_2^* (x)$ is decreasing in x, that is, $X_1^* \leq_{\mathrm{lr}} X_2^*$ , for all the examples proposed in Table 2. With reference to the distribution of X pertaining to Theorem 4, we note that X is DMRL and IMIT, that is to say, it possesses the properties mentioned in Corollary 5, for the following cases treated in Table 1: (i), (ii) for $\alpha \geq 1$ , (iii), (iv), (vii) and (viii). In case (v), the Lomax random variable has increasing mean residual life and increasing mean inactivity time. Moreover, the mean residual life and the mean inactivity time are non-monotonous for (vi) the U-quadratic distribution and (ii) the power distribution for $0<\alpha <1$ .

Let us now discuss a case when the ratio $f_1^* (x)/f_2^* (x)$ is not monotone, whereas $\overline F_1^* (x)/\overline F_2^* (x)$ is decreasing. This suggests that we investigate the relations between $X_1^*$ and $X_2^*$ based on a stochastic order that is weaker than the likelihood ratio order. To this end, we recall that if X and Y have SFs $\overline F_X(x)$ and $\overline F_Y(x)$ , respectively, then X is smaller than Y in the hazard rate order, expressed as $X \le_{\mathrm{hr}} Y$ , if

(38) \begin{equation} \frac{\overline F_X(x)}{\overline F_Y(x)} \ \hbox{is decreasing in $x \in \mathbb R^+_0\cap S_{\overline{F}_Y}$}.\end{equation}

Example 2. Let X be a non-negative absolutely continuous random variable having CDF

\begin{align*}F(x)= e^{-1-\frac{1}{x}} \, \mathbf{1}_{\{0 < x \leq 1 \}} + e^{\frac{x^2-5}{2}} \, \mathbf{1}_{\{1 < x \leq 2\}} + \left [ 1- e^{-(x-2-\ln (1- e^{-1/2}))} \right ] \, \mathbf{1}_{\{ x > 2 \}}\end{align*}

(this is a modification of a CDF treated in Section 2 of Block et al. [Reference Block, Savits and Singh7]). Then it is not hard to show that the assumptions of Theorem 4 are satisfied. Moreover, due to Corollary 4, one has that the ratio $f_1^* (x)/f_2^* (x)$ is not decreasing in $x\in \mathbb R^+$ , whereas $\overline F_1^* (x)/\overline F_2^* (x)$ is decreasing in $x\in \mathbb R^+$ (see Figure 1). Hence, recalling (11) and (38), in this case we have $X_1^* \not \le_{\mathrm{lr}} X_2^*$ and $X_1^* \le_{\mathrm{hr}} X_2^*$ .

Figure 1. For the case treated in Example 2, a plot of $f_1^* (x)/f_2^* (x)$ (left) and $\overline F_1^* (x)/\overline F_2^* (x)$ (right).

3.1. Centred mean residual lifetime

Let us now recall some stochastic orders and introduce some useful notions. Given two non-negative absolutely continuous random variables X and Y having respectively mean residual lifetimes $\mathsf{mrl}_X(x)$ and $\mathsf{mrl}_Y(x)$ defined as in (33), it is well known that X is said to be smaller than Y in the mean residual life order (denoted by $X \leq_{\mathrm{mrl}} Y$ ) if (see, for instance, Shaked and Shanthikumar [Reference Shaked and Shanthikumar28])

(39) \begin{equation} \mathsf{mrl}_X(x)\leq \mathsf{mrl}_Y(x)\quad \hbox{for all }\textit{x}. \end{equation}

This stochastic order is often used in reliability theory since it provides an effective tool for comparing systems’ reliability. However, since we shall need a modified version of this notion, let us now provide the following definition.

Definition 1. Let X be a non-negative absolutely continuous random variable such that $\mathbb{E}(X)$ is finite. Then we define the centred mean residual lifetime of X as the function

(40) \begin{equation} \mathsf{cmrl}(x) = \mathsf{mrl}(x) -\mathsf{mrl}(0) = \mathbb{E}\left[X-x \, | \, X>x \right] - \mathbb{E}\left[X\right], \quad x\in\mathbb R^+_0\cap S_{\overline{F}}.\end{equation}

Clearly, the CMRL can be viewed as a measure of positive or negative ageing of X. With this in mind, we recall the following notions (cf. Deshpande et al. [Reference Deshpande, Kochar and Singh9], for instance), for a non-negative random variable X having SF $\overline F(x)$ .

$\bullet$ X is new better than used (NBU) if $[X-x\, | \, X>x] \leq_{\mathrm{st}} X$ for all $x\in\mathbb R^+_0\cap S_{\overline{F}}$ or, equivalently, if $\overline F(x+t)\leq \overline F(x) \overline F(t)$ for all $x,t\in\mathbb R^+_0\cap S_{\overline{F}}$ . Moreover, X is new worse than used (NWU) if the above inequalities are reversed.

We also have the following weaker notions of ageing.

$\bullet$ X is new better than used in expectation (NBUE) if $\mathbb{E}[X-x\, | \, X>x] \leq \mathbb{E}[X]$ for all $x\in\mathbb R^+_0\cap S_{\overline{F}}$ . Moreover, X is new worse than used in expectation (NWUE) if the above inequality is reversed.

We are now ready to define a new order. In the following, the subscripts refer to the proper random variables.

Definition 2. Let X and Y be non-negative absolutely continuous random variables such that $\mathbb{E}(X)$ and $\mathbb{E}(Y)$ are finite. We say that X is less than Y in the CMRL order, denoted by $X\leq_{\mathrm{cmrl}} Y$ , if

\begin{equation*} \mathsf{cmrl}_X(x) \leq \mathsf{cmrl}_Y(x) \quad \hbox{for all } x\in\mathbb R^+_0\cap S_{\overline{F}_X \cdot \overline{F}_Y}.\end{equation*}

We observe that the relation $X\leq_{\mathrm{cmrl}} Y$ defines a partial stochastic order for suitable equivalence classes. Indeed, the CMRL order given in Definition 2 is reflexive and transitive, but for the antisymmetric property we have the following remark.

Example 3. Let $X\sim GP(a,b)$ have generalized Pareto distribution, with SF given by

\begin{align*} \overline{F}_X(x)=\left(\frac{b}{a x + b}\right)^{\frac{1}{a}+1}_+, \quad x\in \mathbb{R}_0^+,\end{align*}

with $a>-1$ , $a\neq 0$ , $b>0$ , and where the subscript $+$ stands for the positive part of the expression in parentheses. This distribution, which is also known as the Hall–Wellner family, includes the Pareto distribution (for $a>0$ ), the exponential distribution (for $a\to 0$ ), and the rescaled beta distribution with support $[0, -b/a]$ , also known as the power distribution (for $-1<a<0$ ). Moreover, let $Y\sim GP(c,d)$ , with $c>-1$ , $c\neq 0$ , $d>0$ . It is well known that the mean residual lifetimes of X and Y are linear, that is,

\begin{align*} \mathsf{mrl}_X(x) = (a\,x+b)_+, \ \mathsf{mrl}_Y(x) = (c\,x+d)_+, \quad x\in \mathbb{R}_0^+.\end{align*}

Hence, from Definitions 1 and 2, we have

$\bullet$ $X\leq_{\mathrm{mrl}} Y$ if $a\leq c$ and $b\leq d$ ,

$\bullet$ $X\leq_{\mathrm{cmrl}} Y$ if $a\leq c$ .

It thus follows that $X\leq_{\mathrm{cmrl}} Y$ and $Y\leq_{\mathrm{cmrl}} X$ hold simultaneously if $a=c$ , irrespective of the values of b and d (cf. Remark 14).

Even if the previous example shows a case in which the conditions $X\leq_{\mathrm{mrl}} Y$ and $X\leq_{\mathrm{cmrl}} Y$ may hold simultaneously, in general there are no implications between them, as shown in the following examples.

Example 4. Let X and Y respectively have exponential distribution with parameter $\lambda>0$ and Lomax distribution with parameters $\alpha>1$ and $\beta>0$ such that

(41) \begin{equation} {\mathbb{E}}(X) = \frac{1}{\lambda} > \frac{\beta}{\alpha-1}= {\mathbb{E}}(Y). \end{equation}

Their survival functions are

\begin{align*} \overline{F}_X (x)= e^{-\lambda x} , \ \overline{F}_Y (x) = \left(\frac{\beta}{x+\beta}\right)^{\alpha}, \quad x\in \mathbb{R}_0^+,\end{align*}

so that the respective hazard rates are

\begin{align*} \lambda_X (x)= \lambda, \ \lambda_Y (x) = \frac{\alpha}{x+\beta}, \quad x\in \mathbb{R}_0^+.\end{align*}

Under assumption (41) we have $ \lambda_X (0) = \lambda < \frac{\alpha}{\beta}= \lambda_Y (0) $ , so that $\lim_{x \rightarrow +\infty} \lambda_X (x) > \lim_{x \rightarrow +\infty} \lambda_Y (x)$ , and thus $X \nleq _{\mathrm{hr}} Y$ . On the other hand, recalling (33) and (40), we have, respectively,

\begin{align*}\begin{split} \mathsf{mrl}_X (x) = \frac{1}{\lambda}, \, \mathsf{mrl}_Y (x) = \frac{ x+\beta}{\alpha-1}, \quad x \in \mathbb{R}_0^+,\end{split}\end{align*}

\begin{align*}\begin{split} \mathsf{cmrl}_X (x) = 0, \, \mathsf{cmrl}_Y (x) = \frac{x}{\alpha-1}, \quad x \in \mathbb{R}_0^+.\end{split}\end{align*}

Hence, from Definition 2, one has $X \leq _{\mathrm{cmrl}} Y$ . Moreover, from (39) and (41) it follows that $X \not\leq _{\mathrm{mrl}} Y$ . In conclusion, this example shows that, in general,

\begin{align*} X \leq _{\mathrm{cmrl}} Y \, \nRightarrow \, X \leq _{\mathrm{hr}} Y \quad \hbox{and} \quad X \leq _{\mathrm{cmrl}} Y \, \nRightarrow \, X \leq _{\mathrm{mrl}} Y.\end{align*}

Example 5. Assume that X is exponentially distributed with parameter 1, and Y is Erlang distributed with scale 1 and shape 2, so that $X \leq _{\mathrm{st}} Y$ . Since $\mathsf{mrl}_X (x) = 1$ and $\mathsf{mrl}_Y (x) = \frac{ x+2}{x+1}$ , $x \in \mathbb{R}_0^+$ , we have $X \leq _{\mathrm{mrl}} Y$ . Furthermore, one has

\begin{align*} \mathsf{cmrl}_X (x) = 0 \;\geq \; \mathsf{cmrl}_Y (x) = - \frac{ x}{x+1}, \quad x \in \mathbb{R}_0^+,\end{align*}

and thus $Y \leq _{\mathrm{cmrl}} X$ . This shows that, in general,

\begin{align*} X \leq _{\mathrm{st}} Y \, \nRightarrow \, X \leq _{\mathrm{cmrl}} Y \quad \hbox{and} \quad X \leq _{\mathrm{mrl}} Y \, \nRightarrow \, X \leq _{\mathrm{cmrl}} Y.\end{align*}

Based on the CMRL order introduced in Definition 2, we are now able to compare the random variables Z and V considered in Theorem 3. For this purpose we also recall the following variability order (cf. Shaked and Shanthikumar [Reference Shaked and Shanthikumar27]).

As noted in [Reference Shaked and Shanthikumar27], this order allows us to compare variances of functions rather than just expected values, so that it is useful in applications concerning the variability of stochastic systems.

Proof. The assumption $X \leq _{\mathrm{cmrl}} Y$ , due to (40), is equivalent to

\begin{align*} \mathsf{mrl}_X(x) + \frac{1}{2} \, [\mathbb{E}(Y)-\mathbb{E}(X)] \le \mathsf{mrl}_Y(x) - \frac{1}{2} \, [\mathbb{E}(Y)-\mathbb{E}(X)]\end{align*}

for all $x\in\mathbb R^+_0\cap S_{\overline{F}_X \cdot \overline{F}_Y}$ . Hence, since $\overline{F}_X(x) \leq \overline{F}_Y(x)$ for all x and $ \mathbb{E}(X) < \mathbb{E}(Y) < +\infty$ by assumptions, for any x we have

\begin{align*} \overline{F}_X(x) \, \{\mathsf{mrl}_{X}(x) + \frac{1}{2} \, [\mathbb{E}(Y)-\mathbb{E}(X)] \} \le \overline{F}_Y(x) \, \{\mathsf{mrl}_{Y}(x) - \frac{1}{2} \, [\mathbb{E}(Y)-\mathbb{E}(X)] \}\end{align*}

or, equivalently,

\begin{align*} \frac{1}{2} \, \overline{F}_X(x) + \frac{1}{2} \, \overline{F}_Y(x) \le \frac{\int_x^{+\infty} \overline{F}_Y(t) \, {\rm d}t - \int_x^{+\infty} \overline{F}_X(t) \, {\rm d}t}{\mathbb{E}(Y)-\mathbb{E}(X)}.\end{align*}

Recalling (15) and (8), we get $\overline{F}_V(x) \le \overline{F}_Z(x)$ for any x, which completes the proof of (i).

Let g be an increasing convex function. Since g and $g'$ are increasing functions, we have

(42) \begin{equation}\mathsf{Cov}(g'(Z), g(Z)) =\mathbb{E} [g'(Z)g(Z)] - \mathbb{E}[g'(Z)] \, \mathbb{E}[g(Z)]\geq 0.\end{equation}

Moreover, from (i) one has $\mathbb{E}[g(Z)] \ge \mathbb{E}[g(V)]$ . Hence, we obtain

\begin{align*} \mathbb{E} [g'(Z)g(Z)] \geq \mathbb{E}[g'(Z)] \, \mathbb{E}[g(Z)] \geq \mathbb{E} [g'(Z)] \, \mathbb{E}[g(V)].\end{align*}

This implies that $\mathsf{Var}[g(X)] \leq \mathsf{Var}[g(Y)]$ due to (12) and assumption $ \mathbb{E}(X) < \mathbb{E}(Y) < +\infty$ . Statement (ii) thus follows.

Example 6. Suppose that X and Y have Lomax distributions (cf. (iv) of Table 1) with parameters $\lambda_1>0,\alpha_1>2$ and $\lambda_2>0,\alpha_2>2$ , respectively. Let $\alpha_1 > \alpha_2$ and $\lambda_1 < \lambda_2$ , so that $X \le_{\mathrm{st}} Y$ and $\mathbb{E}(X) = \frac{\lambda_1}{\alpha_1-1} < \frac{\lambda_2}{\alpha_2-1} = \mathbb{E}(Y)<+\infty$ . Moreover, for all $x\in\mathbb R^+_0$ we have

\begin{align*}\mathsf{cmrl}_{Y}(x)- \mathsf{cmrl}_{X}(x)= \frac{x}{\alpha_2-1} - \frac{x}{\alpha_1-1} \; \geq \; 0,\end{align*}

so that the assumptions of Theorem 6 are satisfied. See Figure Figure 2 for an instance of the relevant SFs in this case.

We can now obtain a lower bound for the ratio of the increment of variances of transformed random variables over the increment of variances. The following result involves the condition $V \leq_{\mathrm{st}} Z$ , which has been exploited in point (i) of Theorem 6. See also case (ii) of Remark 3.

Figure 2. For $\alpha_1=6$ , $\alpha_2=5$ , $\lambda_1=1$ and $\lambda_2=2$ , (left) the survival functions $\overline{F}_X(x)$ (solid line) and $\overline{F}_Y(x)$ (dashed line); (right) the survival functions $\overline{F}_V(x)$ (solid line) and $\overline{F}_Z(x)$ (dashed line).

Theorem 7. Under the assumptions of Theorem 3, if $\mathsf{Var} (X)< \mathsf{Var} (Y)$ , if $V \leq_{\mathrm{st}} Z$ and if g is an increasing convex (or a decreasing concave) function, then

\begin{equation*} \frac{{\mathsf{Var}} [g(Y)] - {\mathsf{Var}} [g(X)]}{{\mathsf{Var}} (Y)-{\mathsf{Var}} (X)} \geq {\mathbb{E}} [g'(Z)] \, {\mathbb{E}} [g'(T)],\end{equation*}

where the random variable $T \in \mathrm{PMVT}(V,Z)$ has PDF

(43) \begin{equation} f_T(x)=\frac{\overline{F}_Z(x)-\overline{F}_V(x)}{{\mathbb{E}} (Z) - {\mathbb{E}} (V)},\quad x \in \mathbb R_0^+.\end{equation}

Proof. The proof follows from Corollary 2 and Lemma 1, and making use of (42), so that

\begin{align*}\begin{split} \frac{{\mathsf{Var}} [g(Y)] - {\mathsf{Var}} [g(X)]}{{\mathsf{Var}} (Y)-{\mathsf{Var}} (X)} & = \frac{ {\mathbb{E}} [g'(Z)g(Z)] - {\mathbb{E}}[g'(Z)]\,{\mathbb{E}}[g(V)]}{{\mathbb{E}} (Z) - {\mathbb{E}} (V)} \\ & \geq {\mathbb{E}}[g'(Z)]\,\frac{ {\mathbb{E}} [g(Z)] - {\mathbb{E}}[g(V)]}{{\mathbb{E}} (Z) - {\mathbb{E}} (V)} \\ & = {\mathbb{E}}[g'(Z)]\, {\mathbb{E}} [g'(T)].\end{split} \end{align*}

The proof is thus complete.

Remark 15. Due to Theorem 7, the PDF given in (43) can be expressed as

\begin{align*}f_T (x)=2\frac{{\mathbb{E}}[(Y-x)_+]-{\mathbb{E}}[(X-x)_+]}{{\mathsf{Var}} (Y)-{\mathsf{Var}} (X)}-\frac{{\mathbb{E}}(Y) -{\mathbb{E}}(X)}{{\mathsf{Var}} (Y)-{\mathsf{Var}} (X)} \left(\overline{F}_X(x)+\overline{F}_Y(x)\right)\!.\end{align*}

3.2. Additive hazards model

In this section we refer to the additive hazards model (see, for example, Bebbington et al. [Reference Bebbington, Lai and Zitikis4], and Section 6 of Di Crescenzo and Psarrakos [Reference Di Crescenzo and Psarrakos12]). This model, denoted by $(X,Y,\delta)$ -AHM, involves (i) a baseline non-negative absolutely continuous random variable Y having finite mean, survival function $\overline{F}_Y(x)$ and hazard rate $\lambda_Y(x)$ , and (ii) another non-negative absolutely continuous random variable X, whose survival function and hazard rate are given respectively by

(44) \begin{equation} \overline{F}_{X}(x;\;\delta) = e^{-\delta x} \, \overline{F}_Y(x), \ \lambda_{X}(x;\;\delta) = \lambda_Y(x) + \delta, \quad x\in \mathbb{R}_0^+,\end{equation}

where $\delta > 0$ . From (44) it is clear that $X \le_{\mathrm{hr}} Y$ , that is, $\lambda_{X}(x;\;\delta)\geq \lambda_Y(x)$ for all $x\in \mathbb{R}_0^+$ (cf. Shaked and Shanthikumar [Reference Shaked and Shanthikumar28]). Moreover, recalling (40), we have

(45) \begin{equation} \mathsf{cmrl}_{Y}(x) = \mathsf{mrl}_{Y}(x) - \mathbb{E}(Y) = \int_0^{+\infty} \left[\frac{\overline{F}_Y(x+t)}{\overline{F}_Y(x)}- \overline{F}_Y(t) \right] {\rm d}t, \quad x\in \mathbb{R}_0^+\end{equation}

and

(46) \begin{equation} \mathsf{cmrl}_{X}(x) = \mathsf{mrl}_{X}(x)- \mathbb{E}(X) = \int_0^{+\infty} e^{-\delta t} \left[\frac{\overline{F}_Y(x+t)}{\overline{F}_Y(x)}- \overline{F}_Y(t) \right] {\rm d}t.\end{equation}

For more details, see Section 3 of Bebbington et al. [Reference Bebbington, Lai and Zitikis4]. The given assumptions ensure that the Theorem 3 can be applied; in particular, the survival functions of V and Z are given respectively by

\begin{equation*} \overline{F}_V(x) = \frac{1}{2} \left(1+ e^{-\delta x} \right) \overline{F}_Y(x), \ \overline{F}_Z(x) = \frac{\int_x^{+\infty}\left(1- e^{-\delta t} \right)\overline{F}_Y(t) \, {\rm d}t } {\int_0^{+\infty} \left(1-e^{-\delta t}\right) \overline{F}_Y(t) \, {\rm d}t},\quad x\in \mathbb{R}_0^+.\end{equation*}

Proof. Under the $(X,Y,\delta)$ -AHM we have $X\le_{\mathrm{hr}}Y$ and thus $X\le_{\mathrm{st}}Y$ . Furthermore, we have $\mathbb{E}(X) < \mathbb{E}(Y)<+\infty$ . Hence, from (45) and (46), we obtain

\begin{align*} \mathsf{cmrl}_{Y}(x) -\mathsf{cmrl}_{X}(x) =\int_0^{+\infty} \left(1-e^{-\delta t}\right) \left[\frac{\overline{F}_Y(x+t)}{\overline{F}_Y(x)}- \overline{F}_Y(t) \right] {\rm d}t \geq 0,\end{align*}

where the inequality is due to the assumption that Y is NWU. The results thus follow by Theorem 6.

4.1. Per-payment

We now focus on the per-payment defined as

(47) \begin{equation}Y_p(d)=[X\,|\,X>d].\end{equation}

This represents the amount actually paid by the policy, since no benefits are paid to the policyholder if the loss is less than d. Clearly, the PDF and the SF of $Y_p(d)$ are given respectively by

\begin{equation*}f_{Y_p(d)}(x)= \frac{f_X(x)}{\overline{F}_X(d)} \mathbf{1}_{\{x > d\}},\qquad\overline{F}_{Y_p(d)}(x)= \begin{cases} 1, & x < d,\\ \frac{\overline{F}_X(x)}{\overline{F}_X(d)} , & x \geq d,\end{cases}\end{equation*}

so that $X \leq_{\mathrm{st}} Y_p(d)$ . Moreover, recalling (33), one has

\begin{align*} {\mathbb{E}}[Y_p(d)] = d+\frac{1}{\overline{F}_X (d)} \int_d^{+\infty} \overline{F}_X (x) \,\mathrm{d}x =d+\mathsf{mrl}(d).\end{align*}

Further, due to (40), we have

(48) \begin{equation}{\mathbb{E}} [Y_p(d)]-{\mathbb{E}} (X)=d+\mathsf{mrl} (d)-{\mathbb{E}}(X)=d+\mathsf{cmrl} (d).\end{equation}

This quantity plays a role in the analysis of the effect of the transformation g on the per-payment $Y_p(d)$ with respect to the loss X in terms of the mean and the variance. Note that ${\mathbb{E}} [Y_p(d)]$ is increasing in d, and thus also the quantity given in (48) is increasing in d. Applying Lemma 1 and Theorem 3, we obtain

\begin{align*} {\mathbb{E}} [g(Y_p(d))]-{\mathbb{E}}[g(X)]= {\mathbb{E}} [g'(Z)] (d+\mathsf{cmrl}(d))\end{align*}

and

(49) \begin{equation}{\mathsf{Var}} [g(Y_p(d))]-{\mathsf{Var}}[g(X)]=2 \, \{{\mathbb{E}} [g'(Z) g(Z)] - {\mathbb{E}} [g'(Z)] {\mathbb{E}} [g(V)]\} (d+\mathsf{cmrl}(d)),\end{equation}

respectively, where the PDFs of Z and V are given by

\begin{align*} f_Z(x) =\frac{F_X (x) \mathbf{1}_{\{0\leq x < d\}}+\overline{F}_X(x) O_X(d) \mathbf{1}_{\{x> d\}}}{d+\mathsf{cmrl}(d)},\quad f_V(x)= \frac{1}{2} f_X(x)+ \frac{1}{2} \frac{f_X(x)}{\overline{F}_X(d)} \mathbf{1}_{\{x>d\}},\end{align*}

with $O_X(d)\;:\!=\;\frac{F_X (d)}{\overline{F}_X (d)}$ denoting the odds function (or odds ratio) of X.

An useful interpretation of the above equations can be found, for instance, when a special choice of g is taken into account. See, for instance, Remark 1 of Di Crescenzo and Psarrakos [Reference Di Crescenzo and Psarrakos12] for some comments on the usefulness and the meaning of g in some applied contexts. In the case when $g(x)\equiv x$ , (49) yields

(50) \begin{equation} {\mathsf{Var}} [Y_p(d)] - {\mathsf{Var}}[X] = 2 \, \{{\mathbb{E}} (Z) - {\mathbb{E}} (V)\} (d+\mathsf{cmrl}(d)). \end{equation}

This identity and the analogous relation for the means given in (48) allow us to study the difference of dangerousness by using the deductible d and without the deductible (i.e., as $d \rightarrow 0^+$ ), by taking into account that both Z and V depend on d. We remark that $d+\mathsf{cmrl} (d)$ and ${\mathbb{E}} (V)$ are increasing in d, but ${\mathbb{E}} (Z)$ can be non-monotonic in d. This, due to (50), suggests that ${\mathsf{Var}} (Y_p(d))$ and ${\mathsf{Var}}(X)$ cannot be comparable for any d. However, due to the relevance of the random variables on the left-hand side of (50), we investigate their stochastic comparison. We mention that the likelihood ratio order has been recalled in (11). Moreover, X is said to be smaller than Y in the dispersive order (denoted by $X\leq_{\mathrm{disp}} Y$ ) if $F_X^{-1}(\beta) - F_X^{-1}(\alpha) \leq F_Y^{-1}(\beta) - F_Y^{-1}(\alpha)$ whenever $0<\alpha\leq \beta<1$ , where $F_X^{-1}$ and $F_Y^{-1}$ denote the right-continuous inverses of the CDFs $F_X$ and $F_Y$ , respectively. Clearly, the dispersive order is suitable for comparing the variability of two random variables (see Section 3.B of Shaked and Shanthikumar [Reference Shaked and Shanthikumar28] for related properties and results). We also recall the notion of decreasing hazard rate, or decreasing failure rate (DFR), which holds for a non-negative absolutely continuous random variable X when its hazard rate (defined in (36) is monotonic decreasing. This notion is customary in reliability theory or risk theory for random occurrences related to improvement with age.

Theorem 9 is particularly useful since relation $X \leq_{\mathrm{disp}} Y_p(d)$ straightforwardly implies that ${\mathsf{Var}}[X] \leq {\mathsf{Var}} [Y_p(d)]$ .

4.2. Per-payment residual loss

In the same framework as Section 4.1, analogous results can be obtained for the per-payment residual loss, defined as

(51) \begin{equation} Z_{p}(d)=[X-d\,|\,X>d].\end{equation}

This is a non-negative absolutely continuous random variable useful for describing instances in which the insurer covers the portion of the loss that exceeds d, provided the loss is greater than d. The PDF and SF of $Z_{p}(d)$ are given by

(52) \begin{equation} f_{Z_{p}(d)}(x) = \frac{f_X(x+d)}{\overline{F}_X(d)}, \, \overline{F}_{Z_{p}(d)}(x) = \frac{\overline{F}_X(x+d)}{\overline{F}_X(d)}, \quad x\in \mathbb R^+_0,\end{equation}

where $\overline{F}_X(d)>0$ by assumption. Hence, recalling (33), we have ${\mathbb{E}}(Z_{p}(d)) = \mathsf{mrl}_X(d)$ . It follows that

1. (i) $X \le_{\mathrm{st}} Z_{p}(d)$ if and only if X is NWU,

2. (ii) $Z_{p}(d) \le_{\mathrm{st}} X$ if and only if X is NBU.

For case (i), recalling Eq. (33), we obtain

\begin{align*}{\mathbb{E}}(Z_{p}(d)) - {\mathbb{E}} (X) = \mathsf{cmrl}_X(d),\end{align*}

so that, making use of Lemma 1 and Theorem 3, we have respectively

\begin{align*}{\mathbb{E}} [g(Z_p(d))]-{\mathbb{E}}[g(X)]= {\mathbb{E}} [g'(Z)] \, \mathsf{cmrl}_X(d)\end{align*}

and

\begin{align*}{\mathsf{Var}} [g(Z_p(d))]-{\mathsf{Var}}[g(X)]=2 \, \{{\mathbb{E}} [g'(Z) g(Z)] - {\mathbb{E}} [g'(Z)] {\mathbb{E}} [g(V)]\} \, \mathsf{cmrl}_X(d),\end{align*}

provided that $\mathsf{cmrl}(d)\neq 0$ . Here, the PDFs of Z and V, for $x\in \mathbb R_0^+$ , are given respectively by

\begin{align*}f_Z(x) = \frac{1}{\mathsf{cmrl}_X(d)} \, \left(\frac{\overline{F}_X(x+d)}{\overline{F}_X(d)} - \overline{F}_X(x) \right),\quad f_V(x) = \frac{1}{2} \, \frac{\overline{F}_X(x+d)}{\overline{F}_X(d)} + \frac{1}{2} \, \overline{F}_X(x).\end{align*}

We are now able to provide a result analogous to Theorem 9 under a different assumption.

Proof. Recalling the definition of hazard rate given in the first part of Eq. (36) and the PDF of $Z_p(d)$ expressed in the first part of Eq. (52), we have

\begin{align*}\lambda_{Z_p(d)} (x)=\frac{f_{Z_p(d)}(x)}{\overline{F}_{Z_p(d)} (x)}=\frac{f_X(x+d)}{\overline{F}_X(x+d)}=\lambda_X (x+d),\quad x \in \mathbb R^+.\end{align*}

Hence, from the assumption that X has DFR it follows that $X \leq_{\mathrm{hr}} Z_p(d)$ for all $d\in \mathbb R^+$ . Moreover, making use of Theorem 3.B.20, case (a), of [Reference Shaked and Shanthikumar28] one has $X \leq_{\mathrm{disp}} Z_p(d)$ . The proof is thus complete.

The analysis of case (ii) proceeds in a similar way.

4.3. Per-loss residual loss

A situation in which the insurer pays zero for losses smaller than d can be described by using the per-loss residual loss $Z_l(d)$ . This is defined as

(53) \begin{equation} Z_l(d)=\max\{X-d,0\}=\begin{cases} X-d, \ &\ X > d,\\ 0, \ &\ X \leq d.\end{cases}\end{equation}

Clearly, $Z_l(d)$ is a non-negative mixed random variable, composed of an absolutely continuous component over $\mathbb R^+$ and a discrete one at 0. The SF of $Z_l(d)$ can be expressed as

\begin{align*} \overline{F}_{Z_l(d)}(x) =\begin{cases} 1, \ &\ x < 0,\\ \overline{F}_X(x+d), \ &\ x \geq 0. \end{cases}\end{align*}

It thus follows that $Z_l(d) \leq_{\mathrm{st}} X$ . Consequently, for $x\in \mathbb R^+_0$ , we have

\begin{align*}\overline{F}_V (x) =\frac{1}{2} \overline{F}_{X}(x+d) + \frac{1}{2} \overline{F}_X (x),\quad f_Z (x) =\frac{\overline{F}_X (x)-\overline{F}_{X}(x+d)}{{\mathbb{E}}(X) - \int_{d}^{+\infty} \overline{F}(x) \, \mathrm{d}x}.\end{align*}

In the last term, ${\mathbb{E}}(X) - \int_{d}^{+\infty} \overline{F}(x) \, \mathrm{d}x = \int_{0}^{d} \overline{F}(x) \, \mathrm{d}x$ is of interest because it is associated with the loss elimination ratio (LER), an index with some interpretation in this topic (see, for example, Dimitriyadis and Oney [Reference Dimitriyadis and Öney14]). In particular, the LER of X evaluated at d is the equilibrium distribution given by

\begin{equation*} LER_{X}(d) = \frac{{\mathbb{E}}(X) - \left[{\mathbb{E}}(X) - {\mathbb{E}}(X \wedge d)\right]}{{\mathbb{E}}(X)} = \frac{{\mathbb{E}}(X \wedge d)}{{\mathbb{E}}(X)} = \frac{\int_{0}^d \overline{F}(x) \, \mathrm{d}x}{{\mathbb{E}}(X)},\end{equation*}

(where $\wedge$ means minimum) so that $f_Z (x)$ can be rewritten as

\begin{align*}f_Z (x) =\frac{\overline{F}_X (x)-\overline{F}_{X}(x+d)}{{\mathbb{E}}(X) \left(1-LER_X(d)\right)}, \quad x\in \mathbb R^+_0.\end{align*}

From Lemma 1 and Theorem 3, we have respectively

\begin{align*}{\mathbb{E}} [g(X)]-{\mathbb{E}}[g(Z_l(d))]= {\mathbb{E}} [g'(Z)] \, {\mathbb{E}}(X) \left(1-LER_X(d)\right)\end{align*}

and

\begin{align*}{\mathsf{Var}}[g(X)] - {\mathsf{Var}} [g(Z_l(d))]=2 \, \{{\mathbb{E}} [g'(Z) g(Z)] - {\mathbb{E}} [g'(Z)] {\mathbb{E}} [g(V)]\} \, {\mathbb{E}}(X) \left(1-LER_X(d)\right).\end{align*}

Our aim is now to compare the variance of two per-loss residual loss functions. Let X and Y be two non-negative absolutely continuous random variables such that $X \le_{\mathrm{st}} Y$ . By Eq. (53), their corresponding per-loss residual loss functions are given respectively by

\begin{align*}Z_{X,l}(d)=\max\{X-d,0\}\equiv g(X),\quad Z_{Y,l}(d)=\max\{Y-d,0\}\equiv g(Y),\end{align*}

for $g(x) = (x - d)_+$ . We remark that $Z_{X,l} (d)$ and $Z_{Y,l}(d)$ are mixed random variables with an atom at 0. Hence, using Theorem 3 and recalling Remark 5, we can obtain an expression for the difference between the variances of $Z_{X,l} (d)$ and $Z_{Y,l}(d)$ in terms of the difference between the expectations of X and Y, and in terms of the stop-loss functions of Z and V. Indeed, after few calculations we have

\begin{align*}{\mathsf{Var}}(Z_{Y,l}(d)) - {\mathsf{Var}}(Z_{X,l}(d))= 2 \, \left\{{\mathbb{E}}[(Z-d)_+] - \overline{F}_Z(d) \, {\mathbb{E}}[(V-d)_+] \right\}\, [{\mathbb{E}}(Y) - {\mathbb{E}}(X)],\end{align*}

where Z and V are the random variables given in (8) and (15), respectively.