2.1. Stochastic orders

The likelihood ratio order implies both hazard rate and reversed hazard rate orders. Also, both of the latter imply the usual stochastic order.

Note that the stochastic order is equivalent to $\mathbb{E}( \varphi( X) ) \leq\mathbb{E}( \varphi( Y) ) $ for all increasing function $\varphi$ for which the expectations exist. In addition, the definitions of the hazard rate, reversed hazard rate, and likelihood ratio orders can be rewritten in a more common form for the reliability community.

The above orders compare the locations of random variables. We now introduce orders that can be used for the joint comparison of location and dispersion.

Provided the expectations exist, the monotone convex (respectively concave) order means that $\mathbb{E}(X)\leq\mathbb{E}(Y)$ , plus the fact that X is less (respectively more) variable than Y, in a stochastic sense (see [Reference Shaked and Shanthikumar21, page 182]).

Note that the usual stochastic order implies both increasing convex and concave orders [Reference Kochar13].

2.2. Ageing notions

In this section we provide the definitions and equivalent conditions of some ageing properties.

Henceforth, for the sake of brevity, we will say that X is IFR (or DFR, or DRHR).

Following [Reference Shaked and Shanthikumar21, pages 35 and 42] and [Reference Sengupta and Nanda20], the above definitions are equivalent to the conditions below.

2.3. Results for gamma-distributed random variables

We recall that the gamma distribution $\Gamma_0( a,b) $ with shape and scale parameters $a,b >0$ , respectively, admits the following PDF:

\begin{equation*}f(z) =\dfrac{b^{a}}{\Gamma(a) }z^{a-1}\,{\textrm{e}}^{-bz}\mathds{1}_{\mathbb{R}_{+}}(z)\end{equation*}

with respect to Lebesgue measure, where

\begin{equation*}\Gamma(a) = \int_0^{\infty} s^{a-1}\exp({-}s)\,{\textrm{d}} s\quad \text{for all $a>0$}.\end{equation*}

The corresponding mean and variance are $a/b$ and $a/b^2$ respectively. Note also that if $Z_{1},\ldots,Z_{n}$ are independent gamma-distributed random variables with respective distributions $\Gamma_{0}( a_{1},b),\ldots,\Gamma_{0}( a_{n},b) ,$ where $a_1>0, \ldots, a_n>0, b>0$ , then $bZ_1$ and $\sum_{i=1}^{n}Z_{i}$ are gamma-distributed with parameters $(a_1,1)$ and $(\!\sum_{i=1}^{n}a_{i},b)$ , respectively. Finally, we recall some classic results for gamma distributions.

Lemma 4. ([Reference Yu26].) Let $Z_i\sim\Gamma_0(a_i,b_i)$ , $i=1,\ldots,n$ , be independent gamma-distributed random variables. Let us set

\begin{align*} S_{Z} & =\sum_{i=1}^{n}Z_{i},\quad U_{i}=b_{i}Z_{i}\sim\Gamma_{0}(a_{i},1) \quad\textit{for $i\in\{ 1,\ldots,n\} $,} \\* S_{U} & =\sum_{i=1}^{n}U_{i}\sim\Gamma_{0}\biggl( \sum_{i=1}^{n}a_{i},1\biggr) \quad\textit{and}\quad a_+=\sum_{i=1}^na_i. \end{align*}

Then:

1. (a) $\beta^{-1}\,S_{U}\prec_{\textrm{hr}}S_{Z}$ is equivalent to $\beta^{-1}\,S_{U}\prec_{\textrm{sto}}S_{Z}$ and it is true if and only if

   \begin{equation*} \beta\geq\prod_{i=1}^{n}b_{i}^{{{a_{i}}/{a_{+}}}}.\end{equation*}

2. (b) $\beta^{-1}\,S_{U}\prec_{\textrm{lr}}S_{Z}$ is equivalent to $\beta^{-1}\,S_{U}\prec_{\textrm{rh}}S_{Z}$ and it is true if and only if

   \begin{equation*} \beta\geq\dfrac{1}{a_{+}}\sum_{i=1}^{n}a_{i}b_{i}. \end{equation*}

3.1. Definition

Let $b_{0}$ be a positive constant. Recall that a standard (non-homogeneous) gamma process $\textbf{\textit{Y}}=(Y_{t})_{t \geq 0} \sim \Gamma _{0}(A(\!\cdot\!),b_0)$ , with $A(\!\cdot\!)$ as shape function and $b_0$ as scale parameter, is a stochastic process with independent, non-negative, and gamma-distributed increments:

\begin{equation*}Y_{t}-Y_{s}\sim\Gamma_{0}{ (A(t)-A(s),b_{0})}\quad\text{for all $0<s<t$,}\end{equation*}

and such that $Y_{0}=0$ a.s. The PDF of an increment $Y_{t}-Y_{s}$ is then given by

\begin{equation*} f(x)= \dfrac{b_0^{A(t)-A(s)}}{\Gamma (A(t)-A(s))} x^{A(t)-A(s)-1} \exp ({-b_0 x})\quad\text{for all $ x \geq 0$.}\end{equation*}

The mean and variance of $Y_t$ are given by $\mathbb{E}[Y_t]={A(t)}/{b_0}$ and $ \mathbb{V}[Y_t]={A(t)}/{b_0^2}$ for all $t \geq 0$ , respectively. Hence the variance-to-mean ratio ${\mathbb{V}[Y_t]}/{\mathbb{E}[Y_t]} = b_0$ is constant over time.

3.2. Stochastic ordering and ageing properties for SGPs

We first review ageing properties and stochastic comparisons for the increments of a given SGP, and next provide stochastic comparisons between two SGPs with different parameters.

Based on Section 2.3 (and particularly Lemmas 3 and 5), the three following lemmas are easily obtained.

Finally, we come to present the properties of the crossing time of a given threshold $\ell$ , which can stand for the failure time of a system modeled by an SGP, as mentioned in the Introduction. Note that in the context of an SGP which is a pure jump process, the level $\ell$ is almost surely crossed by a jump, so that the level $\ell$ is almost surely never hit. That is why we call it a crossing time, and not a hitting time, which is the common vocabulary for continuous processes such as Wiener processes. The same remark is valid for an EGP hereafter. Note also that a threshold crossing time can also represent the time for a required preventative maintenance action if $\ell$ stands for a preventative threshold (lower than the failure threshold). In that case, the maintenance action is triggered as soon as the deterioration level is beyond $\ell$ (at time $\sigma_{\ell}$ ), the aim being to avoid failures that could have unsafe or costly consequences.

The results below can be found in [Reference Sangüesa, Bada and Cha18] and [Reference Kahle, Mercier and Paroissin12].

4.1. Definition

Let $b\colon \mathbb{R}_{+}^*\rightarrow \mathbb{R}_{+}^*$ be a measurable positive function such that

\begin{equation*} \int_{(0,t]}\dfrac{a(s)}{b(s)}\,{\textrm{d}} s<\infty \quad \text{for all $t > 0$.}\end{equation*}

Following [Reference Çinlar4] and [Reference Dykstra and Laud9], the process $\textbf{\textit{X}}=(X_{t})_{t\geq0}$ is said to be an EGP with shape function $A(\!\cdot\!)$ and scale function $b(\!\cdot\!)$ if $X_{t}$ can be represented as a stochastic integral with respect to an SGP $\textbf{\textit{Y}}$ with $Y_{t}\sim\Gamma_{0}(A(t),1)$ :

\begin{equation*} X_{t}=\int_{(0,t]}\dfrac{{\textrm{d}} Y_{s}}{b(s)} \quad \text{for all $t >0$,}\end{equation*}

and $X_{0}=0$ . If $b(\!\cdot\!)$ is constant and equal to $b_{0}$ , the EGP simply reduces to an SGP $\Gamma_{0}(A(t),b_0)$ . An EGP is much more flexible than an SGP and can be proved to have independent increments, and its distribution to be infinitely divisible. It allows us to model many possible behaviors for the system deterioration (see [Reference Al Masry, Mercier and Verdier1] for more precision). Also, an explicit formula is available for the Laplace transform of an increment [Reference Çinlar4], with

\begin{equation*} \mathcal{L}_{X_{t+h}-X_{t}}(\lambda)\,:\!=\mathbb{E}\big( {\textrm{e}}^{-\lambda(X_{t+h}-X_{t})}\big) =\exp\biggl( -\int_{t}^{t+h}\log\biggl(1+\dfrac{\lambda}{b(s)}\biggr) a(s)\,{\textrm{d}} s\biggr)\end{equation*}

for all $t,\lambda\geq0$ and $h>0$ .

Finally, the mean and variance of an EGP are given by

\begin{equation*} \mathbb{E}(X_{t})=\int_{(0,t]}\dfrac{a(s)}{b(s)}\,{\textrm{d}} s\quad \text{and}\quad \mathbb{V}(X_{t})=\int_{(0,t]}\dfrac{a(s)}{b(s)^{2}}\,{\textrm{d}} s \quad\text{for all $t>0$.}\end{equation*}

4.2. Ageing properties of EGP’s increments

We first explore the ageing properties of an EGP increment, as provided by Lemma 6 in the case of an SGP. We start with the DRHR property. With that aim, we begin with a technical lemma.

Proof. It is a direct consequence of the closure property of DRHR random variables under convolution [Reference Shaked and Shanthikumar21, Corollary 1.B.63], and of the fact that gamma random variables are DRHR (see Lemma 6). □

Next proposition shows that the DRHR property of an SGP increment remains valid for an EGP.

Proof. Note that it is enough to show that $X_{t+h}-X_{t}$ is DRHR. The proof is divided into two parts. Following some arguments developed in [Reference Al Masry, Mercier and Verdier1], we first assume that the scale function b (or the rate function $1/b$ ) is a piecewise constant function given by

(2) \begin{equation}\dfrac{1}{b( v)} =\sum_{i=0}^{m}\dfrac{1}{b_{i}}\,\mathds{1}_{(u_{i},u_{i+1}]}( v) \quad \text{for all }v \geq 0,\end{equation}

where $u_{0}=0<u_{i}<u_{i+1}<u_{m+1}\leq+\infty$ for all $1\leq i\leq m-1$ and $b_{i}>0$ for all $1\leq i\leq m$ . Assuming $u_{i_{t}}<t \leq u_{i_{t}+1}$ , $u_{i_{t+h}}<t+h \leq u_{i_{t+h}+1}$ , and without loss of generality that $u_{i_{t}+1}<u_{i_{t+h}}$ , we have

\begin{align*} X_{t+h}-X_{t} & =\dfrac{1}{b_{i_{t}}}\int_{(t,u_{i_{t}+1}]}\,{\textrm{d}} Y_{v}+\sum_{i=i_{t}+1}^{i_{t+h}-1}\dfrac{1}{b_{i}}\,\int_{(u_{i},u_{i+1}]}\,{\textrm{d}} Y_{v}+\dfrac{1}{b_{i_{t+h}}}\int_{(u_{i_{t+h}},t+h]}\,{\textrm{d}} Y_{v}\notag\\[3pt] & =\dfrac{1}{b_{i_{t}}}( Y_{u_{i_{t}+1}}-Y_{t})+\sum_{i=i_{t}+1}^{i_{t+h}-1}\dfrac{1}{b_{i}}( Y_{u_{i+1}}-Y_{u_{i}})+\dfrac{1}{b_{i_{t+h}}}( Y_{t+h}-Y_{u_{i_{t+h}}}) ,\end{align*}

where

\begin{align*}\dfrac{1}{b_{i_t}}(Y_{u_{i_{t}+1}}-Y_{t}) &\sim\Gamma_{0}(A(u_{i_{t}+1}) -A(t),b_{i_{t}}),\\[3pt]\dfrac{1}{b_{i}}( Y_{u_{i+1}}-Y_{u_{i}}) &\sim\Gamma_{0}(A( u_{i+1}) -A( u_{i}),b_{i}),\\[3pt] \text{and}\quad \dfrac{1}{b_{i_{t+h}}}( Y_{t+h}-Y_{u_{i_{t+h}}}) &\sim\Gamma_{0}( A( t+h) -A( u_{i_{t+h}}),b_{i_{t+h}}).\end{align*}

The increment $X_{t+h}-X_{t}$ can therefore be written as a convolution of independent gamma random variables. This shows, by Lemma 10, that $X_{t+h}-X_{t}$ is DRHR when $b(\!\cdot\!) $ is piecewise constant.

We now consider a general function $b\colon \mathbb{R}_{+}^{\ast}\rightarrow\mathbb{R}_{+}^{\ast}$ . Then there exists an increasing sequence $( \beta^{( n) }(\!\cdot\!) ) _{n\in\mathbb{N}}$ of piecewise constant functions of the shape (2) that converges to ${{1}/{b}}$ . Let $( X_{t}^{( n) }) _{t\geq0}\sim\Gamma(A(\!\cdot\!),b^{( n) }(\!\cdot\!))$ with $b^{( n) }(t)={{1}/{(\beta^{( n) }}(t))}$ . We have

\begin{equation*}\lim_{n\rightarrow+\infty}\mathcal{L}_{X_{t+h}^{( n) }-X_{t}^{( n) }}(\lambda) =\mathcal{L}_{X_{t+h}-X_t}( \lambda) \quad \text{for all $\lambda>0$}\end{equation*}

by the monotone convergence theorem, so that

\begin{equation*}X_{t+h}^{(n)}-X_{t}^{( n) }\overset{\mathcal{D}}{\rightarrow}X_{t+h}-X_t\quad \text{when $n\rightarrow+\infty$}\end{equation*}

for all $t>0$ , where the symbol $\overset{\mathcal{D}}{\rightarrow}$ stands for convergence in distribution. The result is then easily derived from the piecewise constant case and letting $n\rightarrow+\infty$ (as the DRHR property is preserved by convergence in distribution; see [Reference Shaked and Shanthikumar21]). □

We now come to the DFR property of an EGP increment, in the case where $A(t+h)-A(t)$ is smaller than 1. Here again we begin with a technical lemma.

Proof. The DFR property is known not to be stable through convolution in general, so the proof of Lemma 10 for the DRHR case should be adapted. Let us set

\begin{align*}S_{Z} & =\sum_{i=1}^{n}Z_{i},\quad U_{i}=b_{i}Z_{i}\sim\Gamma_{0}(a_{i},1) \quad\text{for $i\in\{ 1,\ldots,n\} $}\\*S_{U} & =\sum_{i=1}^{n}U_{i}\sim\Gamma_{0}\biggl( \sum_{i=1}^{n}a_{i},1\biggr) .\end{align*}

Using an argument similar to that in the proof of Theorem 3 of [Reference Yu26], we may write

\begin{equation*}S_{Z}=\sum_{i=1}^{n}\dfrac{1}{b_{i}}U_{i}=V\,S_{U}\end{equation*}

with

\begin{equation*}V=\sum_{i=1}^{n}\dfrac{1}{b_{i}}\dfrac{U_{i}}{S_{U}}.\end{equation*}

The random vector $( {{U_{1}}/{S_{U}}},\ldots,{{U_{n}}/{S_{U}}}) $ is known to be independent on $S_{U}$ , so V is also independent on $S_{U}$ . Then

\begin{equation*}f_{S_{Z}}( z) =\int_{\mathbb{R}_{+}}f_{VS_{U}|V=v}( z)f_{V}( v) \,{\textrm{d}} v\end{equation*}

with $f_{VS_{U}|V=v}$ the conditional PDF of $VS_{U}$ given that $V=v$ . The distribution of $S_{Z}$ therefore appears as a mixture of the $vS_{U}$ . As $\sum_{i=1}^{n}a_{i}\leq1$ , we know that $S_{U}$ is DFR and $vS_{U}$ as well. Then the distribution of $S_{Z}$ appears as a mixture of DFR distributions, and since the DFR property is closed under mixture [Reference Marshall and Olkin15], the proof is completed. □

Proof. We can use similar arguments as for the proof of Proposition 1 as the DFR property is also preserved by convergence in distribution; see [Reference Shaked and Shanthikumar21]. □

We now explore the possibly IFR property in the case where $A(t+h)-A(t)$ is larger than 1, and for that, a numerical example is considered.

Example 1. Let us consider $A( t)=t$ (which is both a concave and convex function) and let

\begin{equation*} b( t) =10\,\mathds{1}_{[0,0.75)}( t) +0.1\,\mathds{1}_{[0.75,\infty)}( t) \end{equation*}

(decreasing function). The function $\log(\bar{F}_{X_{1}})$ is plotted in Figure 1 for $t=1$ (and hence $A(t)=1$ ). As can be seen, $\log(\bar{F}_{X_{1}})$ is not a concave function, which shows that $X_{1}$ is not IFR.

Considering the increasing function

\begin{equation*} b( t) =0.1\,\mathds{1}_{[0,0.25)}( t) +10\,\mathds{1}_{[0.25,\infty)}( t) \end{equation*}

provides the same distribution for $X_{1}$ (convolution of the gamma distributions $\Gamma_{0}( 0.25,0.1) $ and $\Gamma_{0}( 0.75,10) $ ) and the same conclusion.

Based on the above example, we can see that even considering a convex or a concave shape function and a monotone scale function, an increment $X_{t+h}-X_t$ need not be IFR when the shape parameter $A(t+h)-A(t)$ is larger than 1. Hence it seems difficult to find sufficient conditions to ensure the IFR property of an increment. As a by-product, we observe that not all ageing properties provided in Lemma 6 for SGPs remain true for an EGP.

4.3. Monotonicity of an EGP and stochastic comparison between increments

Here we explore how to adapt the results of Lemma 7 in the case of an EGP. We first look at the possible monotonicity of an EGP with respect to the likelihood ratio order. For that, a numerical example is considered.

Example 2. Let us consider $A( t) =t$ and $b( t) =0.1\,\mathds{1}_{[0,0.25)}( t) +10\,\mathds{1}_{[0.25,\infty)}( t)$ . The function $r( x) ={\bar{F}_{X_{1}}( x) }/{ \bar{F}_{X_{0.25}}( x) }$ is plotted in Figure 2. We can observe that the function r(x) is not monotone with respect to x. Based on Definition 1, we can derive that $X_{0.25}$ and $X_{1}$ are not comparable with respect to the hazard rate order. Hence they are not comparable with respect to the likelihood ratio order either.

Figure 1:

Plot of $\log(\bar{F}_{X_{1}}(t)) $ with respect to t for $A( t)=t$ and $b( t) =10\,\mathds{1}_{[0,0.75)}( t) +0.1\,\mathds{1}_{[0.75,\infty)}( t) $ .

Figure 2:

Plot of $r( x) ={\bar{F}_{X_{1}}( x) }/{ \bar{F}_{X_{0.25}}( x) }$ as a function of x for $A( t) =t$ and $b( t) =0.1\,\mathds{1}_{[0,0.25)}( t) +10\,\mathds{1}_{[0.25,\infty)}( t)$ .

Even though, under specific conditions, we were not able to conclude the possible likelihood-ratio monotonicity of an EGP, we now show its monotonicity with respect to the reversed hazard rate order under any conditions.

Proposition 3. Let $\textbf{\textit{X}}=(X_{t})_{t\geq0}\sim\Gamma(A(t),b(t))$ be an EGP. Then, for all $s<t$ , we have

\begin{equation*}X_{s}\prec_{\textrm{rh}}X_{t}.\end{equation*}

Proof. Let $s<t$ . As $X_{t}-X_{s}\geq 0$ , it is easy to check that

\begin{equation*}0\prec_{\textrm{rh}}X_{t}-X_{s}.\end{equation*}

Since $X_s$ is DRHR (see Proposition 1), we derive from [Reference Shaked and Shanthikumar21, Lemma 1.B.44] that $X_{s}=0+X_{s}\prec_{\textrm{rh}}X_{t}=( X_{t}-X_{s}) +X_{s}$ , which achieves the proof. □

We now provide comparison results between increments in the reversed hazard rate ordering. We begin with a technical lemma.

Lemma 12. Let $Z_i^{(1)}\sim \Gamma_0(a_i,b_i)$ and $Z_i^{(2)}\sim \Gamma_0(c_i,d_i)$ , $i=1,\ldots,n$ , such that $a_i\leq c_i$ and $b_i\geq d_i$ for all i. Then

\begin{equation*}\sum_{i=1}^{n}Z_i^{(1)}\prec_{\textrm{rh}}\sum_{i=1}^{n}Z_i^{(2)}.\end{equation*}

Proof. From Lemma 3 we have

\begin{equation*}Z_i^{(1)}\prec_{\textrm{lr}}Z_i^{(2)}\quad\text{for all $i=1,\ldots,n$,}\end{equation*}

so that

\begin{equation*}Z_i^{(1)}\prec_{\textrm{rh}}Z_i^{(2)}\quad \text{for all $i=1,\ldots,n$.}\end{equation*}

As the reversed hazard rate order is stable under the convolution of DRHR random variables [Reference Shaked and Shanthikumar21, Theorem 1.B.45], the result follows. □

Proposition 4. Let $\textbf{\textit{X}}=(X_{t})_{t\geq0}\sim\Gamma(A(t),b(t))$ be an EGP. Assume that

1. • $A(\!\cdot\!) $ is convex,

2. • $b(\!\cdot\!) $ is decreasing.

Then, for all $s\leq t$ and all $h>0$ ,

\begin{equation*}X_{s+h}-X_{s}\prec_{\textrm{rh}}X_{t+h}-X_{t}.\end{equation*}

Proof. Let $s\leq t\leq t+h$ . Here again we first consider the case where $b(\!\cdot\!) $ is piecewise constant (and decreasing). Then (0, h] is the union of a finite number of disjoint intervals $(u_{i},u_{i+1}]$ , $0\leq i\leq n$ such that $b( s+z) $ is constant for all $z\in(u_{i},u_{i+1}]$ and all $0\leq i\leq n$ . Cutting each of these intervals into pieces when necessary, we can construct disjoint intervals $(v_{j},v_{j+1}]$ for $0\leq j\leq m$ with $v_{0}=0$ and $v_{m+1}=h$ , such that $\bigcup_{j=0}^{m}(v_{j},v_{j+1}]=(0,h]$ and on which both $b( s+z) $ and $b( t+z) $ are constant for all $z\in(v_{j},v_{j+1}]$ and all $0\leq j\leq m$ . This provides

\begin{align*}X_{t+h}-X_{t} & =\sum_{j=0}^{m}\dfrac{1}{b( t+v_{j+1}) }(Y_{t+v_{j+1}}-Y_{t+v_{j}})=\sum_{j=0}^mZ_j^{(t)} ,\\*X_{s+h}-X_{s} & =\sum_{j=0}^{m}\dfrac{1}{b( s+v_{j+1}) }(Y_{s+v_{j+1}}-Y_{s+v_{j}})=\sum_{j=0}^mZ_j^{(s)} .\end{align*}

For all $j=1,\ldots,m$ , we have

\begin{equation*}Z_j^{(z)}\sim\Gamma_{0}( A(z+v_{j+1}) -A( z+v_{j}) ,b(z+v_{j+1})) \quad\text{for $z\in\{s,t\} $}\end{equation*}

with

\begin{equation*}A( s+v_{j+1}) -A( s+v_{j}) \leq A(t+v_{j+1}) -A( t+v_{j})\end{equation*}

and

\begin{equation*}b(s+v_{j+1})\geq b(t+v_{j+1})\end{equation*}

because A is convex and b is decreasing. From Lemma 12 we then obtain

\begin{equation*}X_{s+h}-X_{s}\prec_{\textrm{rh}}X_{t+h}-X_{t}.\end{equation*}

For a general decreasing function b, we proceed as in the proof of Proposition 1, and the result follows since $\prec_{\textrm{rh}}$ is preserved by convergence in distribution.□

Reversing the assumptions, we obtain a reversed inequality, as provided in the following proposition. The proof is similar and is omitted.

Proposition 5. Let $\textbf{\textit{X}}=(X_{t})_{t\geq0}\sim\Gamma(A(t),b(t))$ be an EGP. Assume that

1. • $A(\!\cdot\!) $ is concave,

2. • $b(\!\cdot\!) $ is increasing.

Then, for all $s\leq t$ and all $h>0$ ,

\begin{equation*}X_{s+h}-X_{s}\succ_{\textrm{rh}}X_{t+h}-X_{t}.\end{equation*}

Remark 3. In Propositions 4 and 5, the monotonicity of $b(\!\cdot\!)$ is essential. Indeed, it is not difficult to provide a counterexample to the previous results when the monotonicity assumption is no longer satisfied. Let $A(t)=t$ and $b(t)=1/2\,\mathds{1}_{[0,1)}(t)+1/3\,\mathds{1}_{[1,2)}(t)+1\,\mathds{1}_{[2,3)}(t) +1/4\,\mathds{1}_{[3,\infty)}(t)$ (which is not monotonic). Then

\begin{equation*}X_2-X_0\overset{\mathcal{D}}{=}2Y_1+3Y_2\,\,\textrm{and}\,\,X_4-X_2\overset{\mathcal{D}}{=}Y_3+4Y_4,\end{equation*}

where $\overset{\mathcal{D}}{=}$ means ‘is identically distributed as’ and where $Y_i\sim\Gamma(1,1)$ for $i=1,\ldots,4$ . As noted by Diaconis and Perlman [Reference Diaconis and Perlman6], we have $V[X_2-X_0]<V[X_4-X_2]$ , so the distributions of the two increments are not identical. However, they share the same expectation ( $E[X_2-X_0]=E[X_4-X_2]$ ), which entails that the sign of $F_{X_2-X_0}(x)-F_{X_4-X_2}(x)$ changes at least once on $(0,\infty)$ (in fact exactly once; see [Reference Diaconis and Perlman6, Proposition 2.1]). As a consequence, the two increments are not comparable with respect to the usual stochastic order, and hence they are not comparable with respect to the hr, rh, and lr orders either.

4.4. Stochastic comparisons between two EGPs

Here we provide comparison results between two different EGPs.

Proposition 6. Let

\begin{equation*} \textbf{\textit{X}}^{(i)}=\big( X_{t}^{( i) }\big) _{t\geq0}\sim\Gamma (A_{i}(\!\cdot\!) ,b_{i}(\!\cdot\!))\quad\textit{for $i\in\{ 1,2\}$.} \end{equation*}

Assume that

1. • $A_{2}(\!\cdot\!)-A_{1}(\!\cdot\!)$ is increasing,

2. • $b_1(t)\geq b_2(t)$ for all $t\geq0$ .

Then, for all $t\geq0$ and all $h\geq0$ ,

\begin{equation*}X_{t+h}^{(1)}-X_t^{(1)}\prec_{\textrm{rh}}X_{t+h}^{(2)}-X_t^{(2)}.\end{equation*}

As a specific case, for all $t\geq0$ ,

\begin{equation*}X_t^{(1)}\prec_{\textrm{rh}}X_t^{(2)}.\end{equation*}

Proof. Let $t\geq0$ and $h\geq0$ . Here again we first consider the case where the functions $b_1(\!\cdot\!)$ and $b_2(\!\cdot\!)$ are piecewise constant, and we proceed as in the proof of Proposition 4. It is then possible to consider disjoint intervals $(v_{j},v_{j+1}]$ for $0\leq j\leq m$ with $v_{0}=0$ and $v_{m+1}=h$ , such that $\bigcup_{j=0}^{m}(v_{j},v_{j+1}]=(0,h]$ and on which both $b_1(t+z) $ and $b_2( t+z) $ are constant for all $z\in(v_{j},v_{j+1}]$ and all $0\leq j\leq m$ . This provides

\begin{align*}X_{t+h}^{(1)}-X_{t}^{(1)} & =\sum_{j=0}^mZ_j^{(1)} ,\\*X_{t+h}^{(2)}-X_{t}^{(2)} & =\sum_{j=0}^mZ_j^{(2)},\end{align*}

with

\begin{equation*}Z_j^{(i)}\sim\Gamma_{0}( A_i(t+v_{j+1}) -A_i( t+v_{j}) ,b_i(t+v_{j+1})) \quad\text{for $i\in\{ 1,2\} $.}\end{equation*}

By hypothesis, for all $j=1,\ldots,m$ ,

\begin{equation*}A_2(t+v_{j+1})-A_1(t+v_{j+1})\geq A_2( t+v_{j})-A_1( t+v_{j}),\end{equation*}

since $v_{j+1}\geq v_{j}$ , which leads to

\begin{equation*}A_2(t+v_{j+1})-A_2( t+v_{j})\geq A_1(t+v_{j+1})-A_1( t+v_{j}).\end{equation*}

Moreover,

\begin{equation*}b_1(t+v_{j+1})\geq b_2(t+v_{j+1}).\end{equation*}

From Lemma 12 we then obtain

\begin{equation*}X_{t+h}^{(1)}-X_{t}^{(1)}\prec_{\textrm{rh}}X_{t+h}^{(2)}-X_{t}^{(2)},\end{equation*}

which achieves the proof in the case of piecewise constant scale functions.

For general functions $b_i$ , $i=1,2$ , we proceed as in the proof of Proposition 1, and the result follows since $\prec_{\textrm{rh}}$ is preserved by convergence in distribution. □

The above proposition allows us to compare an EGP and an SGP that share the same shape function, as stated in the following corollary. It is an immediate consequence and it is stated without proof.

Corollary 1. Let

\begin{equation*} \textbf{\textit{Y}}=(Y_t)_{t\geq0}\sim\Gamma_0(A(t),\beta)\ \text{(SGP)},\quad \textbf{\textit{X}}=(X_t)_{t\geq0}\sim\Gamma(A(t),b(t))\ \text{(EGP).}\end{equation*}

We have the following results.

1. (a) If $b(t)\leq \beta$ for all t in a given interval $[t_1,t_2]$ , then

   \begin{equation*} Y_{t+h}-Y_{t}\prec_{\textrm{rh}}X_{t+h}-X_{t}, \end{equation*}
   for all $t\in[t_1,t_2[$ and all h such that $t+h\leq t_2$ .

2. (b) In particular, if $b(t)\leq \beta$ for all $t\geq0$ , then

   \begin{equation*} Y_{t}\prec_{\textrm{rh}}X_t. \end{equation*}

3. (c) The inequalities of the two previous points are reversed if $b(t)\geq \beta$ .

By considering more stringent constraints on parameter $\beta$ and function $b(\!\cdot\!)$ , it is possible to give necessary and sufficient conditions for the hr and lr orders.

Proposition 7. Let

\begin{equation*} \textbf{\textit{Y}}=( Y_{t}) _{t\geq0}\sim\Gamma_{0}( A(\!\cdot\!) ,\beta) \ \textit{(SGP)},\quad \textbf{\textit{X}}=( X_{t}) _{t\geq0}\sim \Gamma( A(\!\cdot\!) ,b(\!\cdot\!) ) \ \textit{(EGP)}. \end{equation*}

Then:

1. (a) Assume that there exists $b_{0}>0$ such that $b_{0}\leq b(s) $ for all $s\in[ t,t+h] $ and

   \begin{equation*} \int_{t}^{t+h}\vert\textrm{log}( b( s) ) \vert \,a( s)\,{\textrm{d}} s<+\infty. \end{equation*}
   Then $Y_{t+h}-Y_{t}\prec_{\textrm{hr}}X_{t+h}-X_{t}$ is equivalent to $Y_{t+h}-Y_{t}\prec_{\textrm{sto}}X_{t+h}-X_{t}$ and it is true if and only if
   (3) \begin{equation}\log(\beta) \geq\dfrac{1}{A( t+h) -A( t) }\int_{t}^{t+h}\log( b( s) ) \,a( s) \,{\textrm{d}} s.\end{equation}

2. (b) Assume that

   \begin{equation*} \int_{t}^{t+h}b( s) \,a( s)\,{\textrm{d}} s<+\infty. \end{equation*}
   Then $Y_{t+h}-Y_{t}\prec_{\textrm{lr}}X_{t+h}-X_{t}$ is equivalent to $Y_{t+h}-Y_{t}\prec_{\textrm{rh}}X_{t+h}-X_{t}$ and it is true if and only if
   (4) \begin{equation}\beta\geq\dfrac{1}{A( t+h) -A( t) }\int_{t}^{t+h}b( s)\,a( s) \,{\textrm{d}} s.\end{equation}

Proof. Assume first that $b(\!\cdot\!) $ is piecewise constant with

\begin{equation*} b( s) =\sum_{i=1}^{n}b_{i}\textbf{1}_{(t_{i},t_{i+1}]}(s) , \end{equation*}

where $t_{1}=t$ , $t_{n+1}=t+h$ and let $a_{i}=A(t_{i+1}) -A( t_{i}) $ for $i\in\{ 1,\ldots,n\} $ . Then, using the notation of Lemma 4, $X_{t+h}-X_{t}$ and $Y_{t+h}-Y_{t}$ are identically distributed as $S_{Z}$ and $\beta^{-1}\,S_{U}$ , respectively. In that case, we have

\begin{equation*}\dfrac{1}{A( t+h) -A( t) }\int_{t}^{t+h}\log(b( s) ) \,a( s) \,{\textrm{d}} s=\dfrac{1}{a_{+}}\sum_{i=1}^{n}\int_{t_{i}}^{t_{i+1}}\log( b( s) )\,\,a( s) \,{\textrm{d}} s=\dfrac{1}{a_{+}}\sum_{i=1}^{n}a_{i}\log(b_{i})\end{equation*}

and, using Lemma 4, condition (3) is equivalent to

\begin{equation*}\beta\geq {\textrm{e}}^{{{1}/{a_{+}}}\sum_{i=1}^{n}a_{i}\log( b_{i})}=\prod_{i=1}^{n}b_{i}^{{{a_{i}}/{a_{+}}}}.\end{equation*}

This proves the first point in the case of a piecewise constant function $b(\!\cdot\!) $ .

Now let us consider the case of a general function $b(\!\cdot\!) $ and let $( b_{n}(\!\cdot\!) ) _{n\in\mathbb{N}}$ be an increasing sequence of piecewise constant positive functions on $\mathbb{R}_{+}^{\ast}$ , such that $b_{n}( s) $ tends to b(s) when $n\rightarrow+\infty$ for all $s\in\lbrack t,t+h]$ , and let $( X_{t}^{( n) }) _{t\geq0}\sim\Gamma(A(\!\cdot\!) ,b_{n}(\!\cdot\!) ) $ .

Assume that condition (3) is true. Then, as $b_{n}( s)\leq b( s) $ , condition (3) is also true when b(s) is replaced by $b_{n}( s) $ , and hence $Y_{t+h}-Y_{t}\prec_{\textrm{hr}}X_{t+h}^{( n) }-X_{t}^{( n)}$ and $Y_{t+h}-Y_{t}\prec_{\textrm{sto}}X_{t+h}^{( n) }-X_{t}^{(n) }$ . Letting n go to infinity, we get that $Y_{t+h}-Y_{t}\prec_{\textrm{hr}}X_{t+h}-X_{t}$ and $Y_{t+h}-Y_{t}\prec_{\textrm{sto}}X_{t+h}-X_{t}$ .

Conversely, assume that $Y_{t+h}-Y_{t}\prec_{\textrm{sto}}X_{t+h}-X_{t}$ (or $Y_{t+h}-Y_{t}\prec_{\textrm{hr}}X_{t+h}-X_{t}$ ). Observe that as $( b_{n}(\!\cdot\!) ) _{n\in\mathbb{N}}$ increases with n, $X_{t+h}^{( n) }-X_{t}^{( n) }$ decreases in the rh ordering (due to Proposition 6), and hence also in the usual stochastic ordering, when n increases.

Then

\begin{equation*}\bar{F}_{Y_{t+h}-Y_{t}}( x) \leq\bar{F}_{X_{t+h}-X_{t}}(x) =\lim_{m\rightarrow+\infty}\bar{F}_{X_{t+h}^{( m)}-X_{t}^{( m) }}( x) \leq\bar{F}_{X_{t+h}^{(n) }-X_{t}^{( n) }}( x)\end{equation*}

for all $x\in\mathbb{R}$ and all $n\in\mathbb{N}$ . This implies that $Y_{t+h}-Y_{t}\prec_{\textrm{sto}} X_{t+h}^{( n) }-X_{t}^{( n)}$ , and hence condition (3) is true for $b_{n}(\!\cdot\!) $ :

(5) \begin{equation} \log( \beta) \geq\dfrac{1}{A( t+h) -A( t)}\int_{t}^{t+h}\log( b_{n}( s) ) \,a( s)\,{\textrm{d}} s\quad\text{for all $n\in\mathbb{N}^{\ast}$.}\end{equation}

As $\vert \textrm{log}( b_{n}(s) ) \vert \leq\vert \textrm{log}( b_{0})\vert +\vert \textrm{log}( b( s) ) \vert $ , we get by dominated convergence

\begin{equation*}\lim_{n\rightarrow+\infty}\int_{t}^{t+h}\log( b_{n}( s)) \,a( s) \,{\textrm{d}} s=\int_{t}^{t+h}\log( b( s)) \,a( s) \,{\textrm{d}} s.\end{equation*}

Then, letting n go to infinity in (5) tells us that condition (3) is true for $b(\!\cdot\!) $ , and the first point is true. The second point is similar and its proof is omitted. □

Note that when condition (3) is not fulfilled, then $Y_{t+h}-Y_{t}$ and $X_{t+h}-X_{t}$ are not comparable with respect to the hazard rate ordering, as can be easily checked via a counterexample. The same is true for condition (4).

Note also that, under the conditions of Proposition 6, the results obtained for the rh order are also valid for the usual stochastic order, and hence also for both increasing convex and concave orders. We now provide other conditions for the increasing convex and concave orders, which extend some results from Lemma 8 (points (d) and (f)) to EGPs.

Proposition 8. Let

\begin{equation*} \textbf{\textit{X}}^{(i)}=\big( X_{t}^{( i) }\big) _{t\geq0}\sim\Gamma (A_{i}(\!\cdot\!) ,b_{i}(\!\cdot\!))\quad for\ \textit{$i\in\{ 1,2\} $.} \end{equation*}

Assume that

1. • $A_{2}(\!\cdot\!)-A_{1}(\!\cdot\!)$ is decreasing,

2. • ${{A_{2}(\!\cdot\!)}/{b_2(\!\cdot\!)}} - {{A_{1}(\!\cdot\!)}/{b_1(\!\cdot\!)}} $ is increasing,

3. • $b_{1}(\!\cdot\!)$ is decreasing and $b_{2}(\!\cdot\!)$ is increasing.

Then, for all $t\geq0$ and all $h\geq0$ ,

\begin{equation*}X_{t+h}^{(1)}-X_t^{(1)}\prec_{\textrm{icx}}X_{t+h}^{(2)}-X_t^{(2)}.\end{equation*}

As a specific case, for all $t\geq0$ ,

\begin{equation*}X_t^{(1)}\prec_{\textrm{icx}}X_t^{(2)}.\end{equation*}

Proof. Let $t\geq0$ and $h\geq0$ . Following the proof of Proposition 6, we have

\begin{align*}X_{t+h}^{(1)}-X_{t}^{(1)} & =\sum_{j=0}^mZ_j^{(1)} ,\\*X_{t+h}^{(2)}-X_{t}^{(2)} & =\sum_{j=0}^mZ_j^{(2)},\end{align*}

with

\begin{equation*}Z_j^{(z)}\sim\Gamma_{0}( A_z(t+v_{j+1}) -A_z( t+v_{j}) ,b_z(t+v_{j+1}))\quad\text{for $z\in\{1,2\} $.}\end{equation*}

By hypothesis, for all $j=1,\ldots,m$ ,

\begin{equation*}A_2(t+v_{j+1})-A_1(t+v_{j+1})\leq A_2( t+v_{j})-A_1( t+v_{j}),\end{equation*}

since $v_{j+1}\geq v_{j}$ , which leads to

\begin{equation*}A_2(t+v_{j+1})-A_2( t+v_{j})\leq A_1(t+v_{j+1})-A_1( t+v_{j}).\end{equation*}

Moreover,

\begin{equation*}\dfrac{A_{2}(t+v_{j+1})}{b_2(t+v_{j+1})} - \dfrac{A_{1}(t+v_{j+1})}{b_1(t+v_{j+1})} \geq \dfrac{A_{2}(t+v_{j})}{b_2(t+v_{j})} - \dfrac{A_{1}(t+v_{j})}{b_1(t+v_{j})} .\end{equation*}

This implies that

\begin{align*}\dfrac{A_{2}(t+v_{j+1})-A_{2}(t+v_{j})}{b_2(t+v_{j+1})} &\geq\dfrac{A_{2}(t+v_{j+1})}{b_2(t+v_{j+1})} - \dfrac{A_{2}(t+v_{j})}{b_2(t+v_{j})} \\[3pt]& \geq \dfrac{A_{1}(t+v_{j+1})}{b_1(t+v_{j+1})}- \dfrac{A_{1}(t+v_{j})}{b_1(t+v_{j})}\\[3pt]& \geq\dfrac{A_{1}(t+v_{j+1})-A_{1}(t+v_{j})}{b_1(t+v_{j+1})},\end{align*}

since $b_{1}(\!\cdot\!)$ is decreasing and $b_{2}(\!\cdot\!)$ is increasing.

From Lemma 8, we then have, for all $j=1,\ldots,m$ ,

\begin{equation*}Z_j^{(1)}\prec_{\textrm{icx}}Z_j^{(2)}.\end{equation*}

As the increasing convex order is stable under convolution [Reference Shaked and Shanthikumar21, Theorem 4.A.8], then

\begin{equation*}X_{t+h}^{(1)}-X_{t}^{(1)}\prec_{\textrm{icx}}X_{t+h}^{(2)}-X_{t}^{(2)}.\end{equation*}

We now consider a general function $b\colon \mathbb{R}_{+}^{\ast}\rightarrow\mathbb{R}_{+}^{\ast}$ . Then there exist two increasing sequences $( \beta^{( i,n) }(\!\cdot\!) ) _{n\in\mathbb{N}}$ of piecewise constant functions of the shape (2) that converges to ${1}/{b_i}$ with $i\in\{ 1,2\} $ . Let $( X_{t}^{( i,n) })_{t\geq 0} \sim\Gamma(A_i(\!\cdot\!),b^{( i,n) }(\!\cdot\!))$ with $b^{( i,n) }(t)={1}/{\beta^{( i,n) }(t)}$ .

We have

\begin{equation*}\lim_{n\rightarrow+\infty}\mathcal{L}_{X_{t+h}^{( i,n) }-X_{t}^{( i,n) }}(\lambda) ={ \mathcal{L}_{X_{t+h}^{(i)}-X_t^{(i)}}( \lambda)} \quad \text{for all $\lambda>0$}\end{equation*}

by the monotone convergence theorem, so

\begin{equation*}X_{t+h}^{( i,n) }-X_{t}^{( i,n) }\overset{\mathcal{D}}{\rightarrow}X_{t+h}^{(i)}-X_t^{(i)}\quad\text{when $n\rightarrow+\infty$}\end{equation*}

for $i\in\{1,2\}$ .

Now we show that the previous convergence result also holds in the $L_1$ sense. First note that

(6) \begin{equation}\mathbb{E}\big[\big|\big({X_{t+h}^{(i,n)}}-X_{t}^{(i,n)}\big) -\big({X_{t+h}^{(i)}}-X_{t}^{(i)}\big)\big|\big] \leq \mathbb{E}\big[\big|{X_{t+h}^{(i,n)}}-{X_{t+h}^{(i)}}\big|\big] +\mathbb{E}\big[\big|X_{t}^{(i,n)} -X_{t}^{(i)}\big|\big]\end{equation}

with

\begin{equation*}\mathbb{E}\big[\big|X_{t}^{( i,n)}-X_{t}^{(i)}\big|\big] \leq \mathbb{E}\biggl[\int_{(0,t]} \bigg| \beta^{( i,n)} (s)-\dfrac{1}{b_i (s)}\bigg|\,{\textrm{d}} Y(s)\biggr] \underset{n \rightarrow +\infty}\rightarrow 0\end{equation*}

by dominated convergence, because $| \beta^{( i,n)} (s)-{1}/{b_i (s)}|\leq 2 /{b_i (s)}$ and

\begin{equation*}\mathbb{E}\biggl[\int_{(0,t]} \dfrac{1}{b_i(s)}\,{\textrm{d}} Y(s)\biggr] =\mathbb{E}\big(X_t^{(i)}\big)<+\infty.\end{equation*}

See Section 4.1.

The same reasoning shows that

\begin{equation*}\mathbb{E}\big[\big|{X_{t+h}^{(i,n)}}-{X_{t+h}^{(i)}}\big|\big]\rightarrow 0,\end{equation*}

which allows us to conclude from (6) that

\begin{equation*}X_{t+h}^{( i,n) }-X_{t}^{( i,n) }\overset{{L}^1}{\rightarrow}X_{t+h}^{(i)}-X_t^{(i)}\quad \text{when}\ n\rightarrow+\infty.\end{equation*}

Finally, the result is easily derived from [Reference Shaked and Shanthikumar21, Theorem 4.A.8] and from the piecewise constant case. □

We now provide similar results for the increasing concave order. The proof follows the same steps as in Proposition 8 and is omitted.

Proposition 9. Let $\textbf{\textit{X}}^{(i)}=\big( X_{t}^{( i) }\big) _{t\geq0}\sim\Gamma(A_{i}(\!\cdot\!) ,b_{i}(\!\cdot\!))$ for $i\in\{ 1,2\} $ . Assume that

1. • ${b_1(t )} \leq {b_2(t)} $ for all $t\geq0$ ,

2. • $b_{1}(\!\cdot\!)$ is decreasing and $b_{2}(\!\cdot\!)$ is increasing,

3. • ${{A_{2}(\!\cdot\!)}/{b_2(\!\cdot\!)}} - {{A_{1}(\!\cdot\!)}/{b_1(\!\cdot\!)}} $ is increasing (and hence $A_{2}(\!\cdot\!)-A_{1}(\!\cdot\!)$ is also increasing).

Then, for all $t\geq0$ and all $h\geq0$ ,

\begin{equation*}X_{t+h}^{(1)}-X_t^{(1)}\prec_{\textrm{icv}}X_{t+h}^{(2)}-X_t^{(2)}.\end{equation*}

As a specific case, for all $t\geq0$ ,

\begin{equation*}X_t^{(1)}\prec_{\textrm{icv}}X_t^{(2)}.\end{equation*}

4.5. Ageing properties of threshold crossing times

We finally come to ageing properties of threshold crossing times. We begin with a first result when the threshold is an IFR random variable.

Proposition 10. Let L be an IFR non-negative random variable independent of an EGP $\textbf{\textit{X}}$ with $X_{t}\sim\Gamma(A(t),b(t))$ . Let

\begin{equation*}\sigma_{L}=\inf\{ t\geq0\colon X_{t}\geq L\} .\end{equation*}

Assume that

1. • $A(\!\cdot\!) $ is convex,

2. • $b(\!\cdot\!) $ is decreasing.

Then $\sigma_{L}$ is IFR.

Proof. Let us first note that, by construction, any EGP $\textbf{\textit{X}}$ belongs to the class IPII (independent positive increasing increments) in the vocabulary of [Reference Sangüesa, Bada and Cha18, Definition 2.7]. Also, based on Proposition 4, $\textbf{\textit{X}}$ increases with respect to the rh order. Then the result is a direct consequence of [Reference Sangüesa, Bada and Cha18, Proposition 3.1].

We derive the following corollary, considering a deterministic threshold $\ell$ .

Corollary 2. Let $\ell>0$ and let $\textbf{\textit{X}}$ be an EGP with $X_{t}\sim\Gamma(A(t),b(t))$ . Let

\begin{equation*}\sigma_{\ell}=\inf\{ t\geq0\colon X_{t}\geq\ell\} .\end{equation*}

Assume that

1. • $A(\!\cdot\!) $ is convex,

2. • $b(\!\cdot\!) $ is decreasing.

Then $\sigma_{\ell}$ is IFR.

Proof. For $n \in \mathbb{N}^{*}$ , let $L^{(n)}\sim\Gamma_{0}(n\ell,n)$ . As $L^{(n)}$ is IFR for a sufficiently large n, we know from Proposition 10 that $\sigma_{L^{(n)}}$ is IFR. Also, as $\mathbb{E}((L^{(n)}-\ell)^{2})=\text{var}(L^{(n)})={{\ell}/{n}}\rightarrow 0$ when $n\rightarrow+\infty$ , we have $L^{(n)}\overset{\mathcal{D}}{\rightarrow}\ell$ when $n\rightarrow+\infty$ . We now have

\begin{equation*}\mathbb{P}(\sigma_{L^{(n)}}>t) =\mathbb{P}(X_{t}\leq L^{(n)})\rightarrow \mathbb{P}(X_{t}\leq \ell)=\mathbb{P}(\sigma_{\ell}>t)\end{equation*}

for all t so that $\sigma_{L^{(n)}} \overset{\mathcal{D}}{\rightarrow} \sigma_{\ell}$ . As IFR is stable through convergence in distribution, we derive that $\sigma_{\ell}$ is IFR.□

We next consider the DRHR property. Recall that for an SGP, the concavity of $A(\!\cdot\!)$ is a sufficient condition to ensure that $\sigma_{\ell}$ is DRHR. This is not the case for an EGP, either when $b(\!\cdot\!) $ is decreasing or when it is increasing, whatever the convexity/concavity of the shape function, as shown by the two following examples.

Example 5. Let $A( t)=t$ . We consider a decreasing scale function

\begin{equation*} b( t) =2\,\mathds{1}_{[0,0.25)}( t) +1\,\mathds{1}_{[0.25,\infty)}( t) . \end{equation*}

As can be seen in Figure 3 (a), the function $\log({F}_{\sigma_{2}}) $ is not concave and hence $\sigma_{2}$ is not DRHR. In Figure 3 (b), we can note that the function $\log(\bar{F}_{\sigma_{2}}) $ is concave and hence $\sigma_{2}$ is IFR. This is consistent with the result of Corollary 2.

Figure 3:

Plots of $\log({F}_{\sigma_{2}}) $ and $\log(\bar{F}_{\sigma_{2}}) $ for $A( t)=t$ and $b( t) =2\,\mathds{1}_{[0,0.25)}( t) +1\,\mathds{1}_{[0.25,\infty)}( t) $ .

Example 6. Let $A( t)=t$ . We now consider an increasing scale function

\begin{equation*} b( t) =0.1\,\mathds{1}_{[0,0.25)}( t) +10\,\mathds{1}_{[0.25,\infty)}( t) . \end{equation*}

As can be seen in Figure 4 (a), $\log({F}_{\sigma_{2}}) $ is not concave and hence $\sigma_2$ is not DRHR. In addition, we can observe in Figure 4 (b) that the function $\log(\bar{F}_{\sigma_{2}}) $ is not concave and hence $\sigma_{2}$ is not IFR.

Figure 4:

Plots of $\log({F}_{\sigma_{2}}) $ and $\log(\bar{F}_{\sigma_{2}}) $ for $A( t)=t$ and $b( t) =0.1\,\mathds{1}_{[0,0.25)}( t) +10\,\mathds{1}_{[0.25,\infty)}(t) $ .