2. Main results and examples

As mentioned in the introduction, Block and Savits (Reference Block and Savits2015) have proved that, for two independent random lifetimes X and Y, whenever X is IFR then

$$r_{X+Y}(t) \le r_X(t) \quad \text{for all }t\ge 0,$$

where $r_{X+Y}$ denotes the hazard rate of $X+Y$ .

Unfortunately, there are many situations where the components are dependent, which means that the previous theorem cannot be applied. Therefore, a natural question arises in this context. Is the thesis of the previous result still valid for dependent random lifetimes? The following example shows that the answer to this question is not always positive.

Example 2.1. Let us consider a bivariate random vector (X,Y) with Fairlie–Gumbel–Morgernstern copula given by

$$C(u,v)= uv[1 + \theta (1-u)(1-v)],\quad \text{for all }0\le u,v \le 1,$$

where $-1 \le \theta \le 1$ is a dependence parameter such that the dependence is positive for $0 < \theta \le 1$ , negative for $-1 \le \theta <0$ , and the components are independent for $\theta =0$ . The marginal distributions follow a gamma-distributed model, denoted by $G(r, \sigma)$ , with density function given by

$$f(x)=\Big(\frac{x}{\sigma}\Big)^{r-1} \frac{{\rm e}^{-\frac{x}{\sigma}}}{\Gamma(r)},\quad \text{for all }x\ge 0,$$

where r is the shape parameter and $\sigma$ is the scale parameter. Let us consider $X\sim G(1.5, 1)$ , $Y\sim G(1.25, 1)$ , and $\theta =0.5$ . In such a case, the hazard rate function of the convolution $X+Y$ crosses the hazard rate functions of both components (see Figure 1). Analogously, taking $\theta =-0.5$ (in other words, assuming negative association), the conclusion remains the same (see again Figure 1).

Figure 1: Plot of the hazard rate function of $X+Y$ (continuous line), X (dashed and dotted line), and Y (dashed line).

Therefore, in order to obtain a bound for the hazard rate of a convolution of dependent random lifetimes, we need to consider a different approach. Next, we provide a set of conditions which generalizes the result given for the case of independent components. Let us fix the following notation prior to stating the result. Given a bivariate random vector (X,Y), we denote the hazard rate function of the conditional random variable ${(X \mid Y=y)}$ , where y belongs to the support of Y, by $r_X(t \mid Y=y)$ .

Theorem 2.1. Let (X, Y) be a non-negative bivariate random vector with joint density function f. If

(2.1) \begin{equation} r_X(t \mid Y=y) \text{ is increasing in }y \text{ for all }t\geq 0 \end{equation}

and

(2.2) \begin{equation}r_X(t \mid Y=y) \text{ is increasing in }t,\text{ for all }y\geq 0,\label{eqn2}\end{equation}

then $r_{X+Y}(t)\leq r_X(t \mid Y=t),\text{ for all }t\geq 0$ .

Proof. Let us denote by h and $\skew2\overline H $ respectively the density and the survival function of the convolution $X+Y$ . Then

$$h(t)= \int^{t}_{0} f(t-y,y)\,{\rm d} y,\quad \text{for all } t\geq 0$$

and

\begin{align} \skew2\overline H(t) & = \int_{0}^{\infty}\int_{t-y}^{\infty} f(x,y) \,{\rm d}\kern0.5ptx\,{\rm d} y \\ & = \int_{0}^{t}\int_{t-y}^{\infty} f(x,y) \,{\rm d}\kern0.5ptx\,{\rm d} y + \int_{t}^{\infty}\int_{0}^{\infty} f(x,y) \,{\rm d}\kern0.5ptx\,{\rm d} y \\ & = \int_{0}^{t}\int_{t-y}^{\infty} f(x,y) \,{\rm d}\kern0.5ptx\,{\rm d} y + \skew3\overline G(t) , \end{align}

for all $t\geq 0$ , where $\skew3\overline G$ is the marginal survival function of Y. Consequently, the condition

$$r_{X+Y}(t)\leq r_X(t \mid Y=t),\quad \text{for all }t\geq 0$$

is equivalent to

$$h(t)-r_X(t \mid Y=t)\skew2\overline H(t)\leq0,\quad \text{for all }t\geq 0.$$

Let us prove the previous inequality. Under the assumptions, the following chain of equalities and inequalities holds:

\begin{align}h(t)-r_X(t \mid Y=t)\skew2\overline H(t) \\ & = \int_{0}^{t}f(t-y,y)\,{\rm d} y \\ & \quad - r_X(t \mid Y=t)\int_{0}^{t}\int_{t-y}^{\infty} f(x,y) \,{\rm d}\kern0.5ptx\,{\rm d} y + \skew3\overline G(t) \\ & = \int_{0}^{t}\Big[f(t-y,y) - r_X(t \mid Y=t)\int_{t-y}^{\infty} f(x,y) \,{\rm d}\kern0.5ptx\Big]\,{\rm d} y \\ & \quad - r_X(t \mid Y=t)\skew3\overline G(t) \\ & = \int_{0}^{t}\Big[r(t-y \mid Y=y)\int_{t-y}^{\infty}f(x,y)\,{\rm d}\kern0.5ptx \\ & \qquad \qquad - r_X(t \mid Y=t)\int_{t-y}^{\infty} f(x,y) \,{\rm d}\kern0.5ptx\Big]\,{\rm d} y \\ & \quad - r_X(t \mid Y=t)\skew3\overline G(t) \\ & = \int_{0}^{t}[r(t-y \mid Y=y) - r_X(t \mid Y=t)]\int_{t-y}^{\infty}f(x,y)\,{\rm d}\kern0.5ptx\,{\rm d} y \\ & \quad - r_X(t \mid Y=t)\skew3\overline G(t) \\ & \leq - r_X(t \mid Y=t)\skew3\overline G(t) \\ \le 0 \end{align}

for all $t\geq 0$ , where the first inequality follows by taking into account that conditions (2.1) and (2.2) imply that

$$r(t-y \mid Y=y) \leq r(t-y \mid Y=t) \leq r_X(t \mid Y=t), \quad \text{for all } t,y\geq 0.$$

Therefore, we conclude that $r_{X+Y}(t)\leq r_X(t \mid Y=t)$ for all $t\geq 0$ .

Let us make several remarks on the previous result.

Recently, Navarro and Sordo (Reference Navarro and Sordo2018) have characterized Condition (2.1) in terms of the survival copula. In particular, denoting by $\widehat C$ the survival copula of a bivariate random vector (X,Y), Navarro and Sordo (Reference Navarro and Sordo2018) proved that (2.1) is satisfied if, and only if,

$$\frac{\partial_1 \widehat C(u,v_2)}{\partial_1 \widehat C(u,v_1)}\text{ is increasing in }u, \text{ for all } 0\le v_1\le v_2\le 1.$$

Remark 2.3. Let us assume that (X,Y) is DRR(1,0) and DRR(0,1), and let us denote by $r_Y(t \mid X=x)$ the hazard rate function of $(Y \mid X=x)$ , for any x in the support of X. If $r_X(t \mid Y=y)$ satisfies Condition (2.2) and $r_Y(t \mid X=x)$ is increasing in t, for all x in the support of X, then, by applying Theorem 2.1, we obtain

$$r_{X+Y}(t) \le \min\{r_X(t \mid Y=t),r_Y(t \mid X=t)\},\quad \text{for all }t > 0.$$

Next, we apply the previous theorem to several examples of bivariate random vectors. First, we consider Gumbel’s bivariate exponential, Model I (see Kotz et al. (Reference Kotz, Balakrishnan and Johnson2000), p. 350).

Example 2.2. (Gumbel’s bivariate exponential (Model I).) Let (X,Y) be a bivariate random vector with joint density function given by

$$f(x,y)= \exp{(-x-y-\theta xy)}\{(1+\theta x)(1+\theta y)-\theta\},\quad \text{for }x,y>0,$$

where $0\le \theta \le 1$ .

It is easy to see that the hazard rate function of $(X \mid Y=y)$ is given by

$$r_X(t \mid Y=y)=\frac{(1+\theta y )(1+ \theta t) - \theta}{1+\theta t},\quad \text{for }t\ge 0.$$

Analogously, ${(Y \mid X=x)}$ has a hazard rate function given by

$$r_Y(t \mid X=x)=\frac{(1+\theta x )(1+ \theta t) - \theta}{1+\theta t},\quad \text{for }t\ge 0.$$

On the one hand, it is obvious that $r_X(t \mid Y=y)$ and $r_Y(t \mid X=x)$ are increasing in y and x, respectively, for all $t\geq0$ . On the other hand, it is not difficult to see that $r_X(t \mid Y=y)$ and $r_Y(t \mid X=x)$ are increasing in $t\ge 0$ for all $y\ge 0$ and $x\ge 0$ , respectively. Therefore, the sufficient conditions in Theorem 2.1 are satisfied and, consequently, it is ensured that

$$r_{X+Y}(t) \leq \frac{(1+\theta t )^2- \theta}{1+\theta t},\quad \text{for all }t\geq 0.$$

Figure 2 shows the particular case for $\theta=0.2$ .

Figure 2: Left: The hazard rate of the convolution $X+Y$ (continuous line) and the bound $((1+\theta t )^2- \theta)/{(1+\theta t)}$ (dashed line). Right: The joint density function of (X,Y).

Next, we consider a parametric family where the conditional distributions are gamma distributed (see Arnold et al. (Reference Arnold, Castillo and Sarabia1999)). Let us apply Theorem 2.1 to this model.

Example 2.3. (Gamma conditionals (Model II).) Let (X,Y) be a bivariate random vector with joint density function given by

$$f(x,y)= \frac{k_{r,s}(\theta)}{\sigma_1^r \sigma_2^s \Gamma(r) \Gamma(s)} x^{r-1} y^{s-1}\exp\Big\{-\frac{x}{\sigma_1}-\frac{y}{\sigma_2}-\frac{\theta xy}{\sigma_1 \sigma_2}\Big\} , \quad \text{for }x,y>0,$$

where $\sigma_1, \sigma_2,r,s>0$ are non-negative, $\theta\ge0$ , and $k_{r,s}(\theta)$ is a normalizing constant. Observe that $\sigma_1$ and $\sigma_2$ are scale parameters, r and s are shape parameters, and $\theta$ is a dependence parameter.

In this case it is known that $(X \mid Y=y)$ follows a gamma distribution with shape parameter r and scale parameter ${(1+cy/{\sigma_2})/{\sigma_1}}$ . Analogously, ${(Y \mid X=x)}$ follows a gamma distribution with shape parameter s and scale parameter ${(1+cx/\sigma_1)/\sigma_2}$ .

It is not difficult to see that $r_X(t \mid Y=y)$ and $r_Y(t \mid X=x)$ are increasing in y and x, respectively, for all $t \geq 0$ (see Table 1 in Belzunc e et al. (2016)). In addition, it is well known

that gamma-distributed random variables have increasing failure rates if the shape parameter is greater than or equal to 1. Therefore, $r_X(t \mid Y=t)$ [ $r_Y(t \mid X=t)$ ] is increasing in $t\ge 0$ if the parameter $r \geq 1$ [ $s\geq 1$ ]. To summarize, if $r\ge 1$ [ $s\ge 1$ ], then

$$r_{X+Y}(t) \leq r_X(t \mid Y=t)\text{ }[r_Y(t \mid X=t)] ,\quad \text{for all }t\geq 0 %,$$

by applying Theorem 2.1.

Figure 3 shows the particular case for $\theta=0.5$ , $r=1.2$ , $s=1.5$ , $\sigma_1=4$ , and $ \sigma_2=3$ .

Figure 3: Left: The hazard rate of the convolution $X+Y$ (continuous line), the function $r_X(t \mid Y=t)$ (dashed line), and the function $r_Y(t \mid X=t)$ (dashed and dotted line). Right: The joint density function of (X,Y).

Next, let us apply the result to two semiparametric models. The first one was introduced by Navarro and Sarabia (Reference Navarro and Sarabia2013).

Example 2.4. (Bivariate conditional proportional hazard rate model.) Let (X,Y) be a bivariate random vector with joint density function given by

\begin{multline} f(x,y)= k(\phi)\sigma_1 \sigma_2 \lambda_1(x) \lambda_2(y)\exp\{-\sigma_1 \Lambda_1(x) -\sigma_2 \Lambda_2(y) -\phi \sigma_1 \sigma_2 \Lambda_1(x)\Lambda_2(y)\} \\ , \text{for }x,y>0, \end{multline}

where $\sigma_1,\sigma_2>0$ are scale parameters, $\phi\ge 0 $ is a dependence parameter, and $k(\phi)$ is a normalizing constant. The functions $\Lambda_i$ , $i=1,2$ , are cumulative hazard rate functions for non-negative random variables with hazard rates $\lambda_i(x)=\Lambda_i'(x)$ , $i=1,2$ , respectively. It is known that the hazard rate function of ${(X \mid Y=y)}$ is given by

$$r_X(t \mid Y=y)=\sigma_1 [1 + \phi \sigma_2 \Lambda_2(y)]\lambda_1(t).$$

Analogously, the hazard rate function of ${(Y \mid X=x)}$ is given by

$$r_Y(t \mid X=x)=\sigma_2 [1 + \phi \sigma_1 \Lambda_1(x)]\lambda_2(t).$$

Since $\Lambda_1$ and $\Lambda_2$ are cumulative hazard functions, it is obvious that $r_X(t \mid Y=y)$ and $r_Y(t \mid X=x)$ are increasing in y and x, respectively, for all $t\geq0$ . Moreover, if the hazard rates $\lambda_1$ and/or $\lambda_2$ are increasing (IFR), then $r_X(t \mid Y=t)$ and/or $r_Y(t \mid X=t)$ are also increasing in $t\ge 0$ . To sum up, if $\lambda_1$ [ $\lambda_2$ ] is increasing, then

$$r_{X+Y}(t) \leq \sigma_1 (1 + \phi \sigma_2 \Lambda_2(t))\lambda_1(t)\text{ }[\sigma_2 (1 + \phi \sigma_1 \Lambda_1(t))\lambda_2(t)] %, \quad \text{for all }t\geq 0.$$

Next, let us consider the semiparametric family given by Navarro et al. (Reference Navarro, Esna-Ashari, Asadi and Sarabia2015).

Example 2.5. (Bivariate conditional proportional generalized odds rate model) Let (X,Y) be a bivariate random vector with joint density function given by

$$f(x,y)= \frac{K\sigma_1 \sigma_2 \lambda_1(x) \lambda_2(y)}{ [\sigma_0 + \theta \sigma_1 \Lambda_1(x) + \theta \sigma_2 \Lambda_2(y) + \theta \phi \sigma_1 \sigma_2 \Lambda_1(x) \Lambda_2(x) ]^{1+\frac1c}} , \quad \text{for }x,y>0,$$

where $\sigma_1,\sigma_2,\theta,K>0$ and $\sigma_0,\phi\ge 0 $ . The functions $\Lambda_i$ , $i=1,2$ , are univariate generalized odds functions such that $\lambda_i(x)=\Lambda_i'(x)$ , $i=1,2$ .

Let us define the functions

\begin{equation} \theta_1(y)=\frac{\sigma_0 + \theta \sigma_2 \Lambda_2(y)}{\sigma_1 + \phi \sigma_1 \sigma_2 \Lambda_2(y)} ,\qquad \theta_2(x)=\frac{\sigma_0 + \theta \sigma_1 \Lambda_1(x)}{\sigma_2 + \phi \sigma_1 \sigma_2 \Lambda_1(x)}. \end{equation}

It is known that the hazard rate function of ${(X \mid Y=y)}$ is given by

$$r_X(t \mid Y=y)=\frac{\lambda_1(t)}{\theta_1(y) + \theta \Lambda_1(t)}$$

and, analogously, the hazard rate function of $(Y \mid X=x)$ is given by

$$r_Y(t \mid X=x)=\frac{\lambda_2(t)}{\theta_2(x) + \theta \Lambda_2(t)}.$$

It is easy to see that if $\theta< [>]\,\phi\sigma_0$ then $r_X(t \mid Y=y)$ and $r_Y(t \mid X=x)$ increase [decrease] in y and x, respectively. Furthermore, if $\theta_1(y) + \theta \Lambda_1(t)$ is logconvex [logconcave] in t then $r_X(t \mid Y=y)$ increases [decreases] in t, and analogously for $r_Y(t \mid X=x)$ . To sum up, if $\theta<\phi\sigma_0$ and $\theta_1(y) + \theta \Lambda_1(t)$ [ $\theta_2(y) + \theta \Lambda_2(t)$ ] is logconvex in t, then

$$r_{X+Y}(t) \leq \frac{\lambda_1(t)}{\theta_1(t) + \theta \Lambda_1(t)}\text{ }\Big[ \frac{\lambda_2(t)}{\theta_2(t) + \theta \Lambda_2(t)}\Big] , \quad \text{for all }t\geq 0.$$

3. Results for the reversed hazard rate function

In this section similar results to the previous ones given in Section 2 are provided for the reversed hazard rate function. First, we provide a lower bound for the reversed hazard rate function of a convolution of not necessarily independent components. Let us fix some notation prior to stating the result. Given a bivariate random vector (X,Y), we denote by $\skew1\overline r_X(t \mid Y=y)$ the reversed hazard rate function of $(X \mid Y=y)$ , where y is a value in the support of Y.

Theorem 3.1. Let (X, Y) be a non-negative bivariate random vector with joint density function f. If

(3.1) \begin{equation} \skew1\overline r_X(t \mid Y=y) \text{ is decreasing in }y\text{ for all }t\geq 0 \end{equation}

and

(3.2) \begin{equation} \skew1\overline r_X(t \mid Y=y) \text{ is decreasing in }t \text{ for all }y\geq 0, \end{equation}

then $\skew1\overline r_{X+Y}(t)\geq \skew1\overline r_X(t \mid Y=t) \text{ for all }t\geq 0$ .

Proof. According to the proof of Theorem 2.1, h and H denote the density and the distribution function of the convolution $X+Y$ and we will equivalently show that $h(t)-\skew1\overline r_X(t \mid Y=t) H(t)\geq0$ , where

$$H(t)= \int_{0}^{t}\int_0^{t-y}f(x,y) \,{\rm d}\kern0.5ptx\,{\rm d} y,$$

for all $t\geq 0$ .

Since conditions (3.1) and (3.2) imply that

$$\skew1\overline r(t-y \mid Y=y) \geq \skew1\overline r_X(t \mid Y=y) \geq \skew1\overline r_X(t \mid Y=t) \quad \text{for all } t\ge y\geq 0,$$

we have that

$$h(t)-\skew1\overline r_X(t \mid Y=t) H(t)= \int_{0}^{t}\Big(f(t-y,y)\,{\rm d} y - \skew1\overline r_X(t \mid Y=t)\int_{0}^{t-y} f(x,y) \,{\rm d}\kern0.5ptx\Big)\,{\rm d} y \ge 0 %,$$

for all $t\geq 0$ . Therefore, we conclude that $h(t)-\skew1\overline r_X(t \mid Y=t) H(t)\geq0$ or, equivalently, $\skew1\overline r_{X+Y}(t)\geq \skew1\overline r_X(t \mid Y=t)$ , for all $t\geq 0$ .

Despite the fact that Condition (3.1) can be considered as a negative dependence property, as far as we know this condition has never been seen from this point of view. Recently, however, Navarro and Sordo (Reference Navarro and Sordo2018) characterized this property in terms of the copula. In particular, they have proved that, given a bivariate random vector (X,Y) with copula C, Condition (3.1) is satisfied if, and only if,

$$\frac{\partial_1 C(u,v_2)}{\partial_1 C(u,v_1)}\text{ is decreasing in }u \text{ for all }0<v_1 \le v_2 < 1.$$

We also want to observe that, for independent components, Condition (3.1) is always satisfied and Condition (3.2) is equivalent to the DRHR property of the random variable X. Therefore, we can state the following corollary.

Corollary 3.1. Let X and Y be two independent non-negative random variables such that X is DRHR with reversed hazard rate function denoted by $\skew1\overline r$ . Then

$$\skew1\overline r_{X+Y}(t)\geq \skew1\overline r_X(t) \quad \text{for all }t\geq 0.$$

Next, we apply Theorem 3.1 to a parametric family such that the conditional distributions are exponentially distributed (see Arnold and Strauss (Reference Arnold and Strauss1988) and Arnold et al. (Reference Arnold, Castillo and Sarabia1999), p. 80).

Example 3.1. Let (X,Y) be a bivariate random vector with joint density function given by

$$f(x,y)= \frac{k(\theta)}{\sigma_1 \sigma_2} \exp\Bigg\{-\frac{x}{\sigma_1}-\frac{y}{\sigma_2}-\frac{\theta xy}{\sigma_1 \sigma_2}\Bigg\} \quad \text{for }x,y>0,$$

where $\sigma_1,\sigma_2>0$ , $\theta\ge0$ , and

$$k(\theta)=\frac{1}{\int_0^{+\infty} {\rm e}^{-u} (1+ \theta u)^{-1}\,{\rm d} u}.$$

Observe that $\sigma_1$ and $\sigma_2$ are scale parameters and $\theta$ is a dependence parameter. It is known that ${(X \mid Y=y)}$ follows an exponential distribution with parameter ${(1+\theta y/\sigma_2)/\sigma_1}$ ; therefore, the reversed hazard rate of ${(X \mid Y=y)}$ is given by

$$\skew1\overline r_X(t \mid Y=y)=\frac{\frac{1}{\sigma_1}\Big(1+\frac{\theta y}{\sigma_2}\Big)}{\exp\Big(\frac{t}{\sigma_1}\Big(1+\frac{\theta y}{\sigma_2}\Big)\Big) - 1}$$

and, analogously, the reversed hazard rate of ${(Y \mid X=x)}$ is given by

$$\skew1\overline r_Y(t \mid X=x)=\frac{\frac{1}{\sigma_2}\Big(1+\frac{\theta x}{\sigma_1}\Big)}{\exp\Big(\frac{t}{\sigma_2}\Big(1+\frac{\theta x}{\sigma_1}\Big)\Big) - 1},$$

for all $t>0$ .

It is not difficult to see that $\skew1\overline r_X(t \mid Y=y)$ and $\skew1\overline r_Y(t \mid X=x)$ are decreasing in y and x, respectively, for all $t\geq0$ , and $\skew1\overline r_X(t \mid Y=y)$ and $\skew1\overline r_Y(t \mid X=x)$ are increasing in $t\ge 0$ , for all $x,y>0$ . Therefore, the sufficient conditions in Theorem (3.1) are satisfied and, consequently, it is ensured that

$$\skew1\overline r_{X+Y}(t) \geq \min\left\{\frac{\frac{1}{\sigma_1}\Big(1+\frac{\theta t}{\sigma_2}\Big)}{\exp\Big(\frac{t}{\sigma_1}\Big(1+\frac{\theta t}{\sigma_2}\Big)\Big) - 1}, \frac{\frac{1}{\sigma_2}\Big(1+\frac{\theta t}{\sigma_1}\Big)}{\exp\Big(\frac{t}{\sigma_2}\Big(1+\frac{\theta t}{\sigma_1}\Big)\Big) - 1}\right\} \quad \text{for all }t\geq 0.$$

4. Discussion and remarks

In this paper, an upper bound for the hazard rate of a convolution of not necessarily independent random lifetimes is provided. This result is an extension of a recent result by Block and Savits (Reference Block and Savits2015) where the hazard rate of a convolution is upper-bounded by the hazard rate of an IFR component in the case of independent components. As long as the result of Block and Savits (Reference Block and Savits2015) provides an upper bound in terms of the hazard rate function of one of the marginals, this is not possible in the case of dependent random lifetimes. In particular, negative dependence among the components has to be assumed, as well as monotonicity of the hazard rate function of the conditional distribution ${(X \mid Y=y)}$ [or ${(Y \mid X=x)}$ ]. Moreover, a similar result for the reversed hazard rate function is also provided, where a lower bound for the reversed hazard rate function of the convolution $X+Y$ is given. Applications of these results to several parametric and semiparametric families of bivariate distribution functions are also given.

Let us make some remarks on the results:

(i) We have considered here the case of non-negative random variables. The main reason is the applicability of these results in contexts like reliability, survival, and insurance, where the random quantities of interest are, obviously, non-negative. However, with the appropriate modifications the results can be extended to random variables with support not restricted to non-negative values.

(ii) We have only considered the convolution of two random variables. The result can be extended, using the previous techniques, to the case of more than two random variables. The idea is as follows:

Let us consider the case of n non-negative random variables, not necessarily independent, $X_1,X_2,\ldots,X_n$ , and let us denote by $i\in \{1,2,\ldots,n\}$ the index such that

$$r_{X_i}(t \mid Z=y) \text{ is increasing in }y \text{ for all }t\geq 0$$

and

$$r_{X_i}(t \mid Z=y) \text{ is increasing in }t \text{ for all }y\geq 0,$$

where $Z=\sum_{j\neq i} X_j$ ; then, by Theorem 2.1, we get that $r_{\sum_{j=1}^n X_j}(t)\leq r_X(t\!\mid\!Z=t)$ for all $t\geq 0$ .

However, we need to know the distribution of ${(X_i \mid Z=x)}$ , which is a non-trivial task. Therefore, we leave as an open question whether there are some easy-to-check sufficient conditions in the general case.

(iii) This work can be considered as a starting point for the study of some other properties of the hazard and reversed hazard rate functions of the convolution of not necessarily independent random lifetimes, such as the monotonicity or the limiting behaviour.