2. Preliminaries

Throughout the paper, increasing and decreasing mean non-decreasing and non-increasing, respectively. Furthermore, ratios, derivatives, and conditional distributions are assumed to be well-defined whenever they are used.

In this section we review some notions related to our main results (see [Reference Barlow and Proschan5]). First we give the following definition extracted from [Reference An3].

Definition 1. Let $f\,:\, \mathbb{R} \longmapsto \mathbb{R}_+$ be a Lebesgue-measurable function. Suppose

\[ \{x\,:\, f(x)>0\}=(l,u)\subseteq \mathbb{R}. \]

The function f is said to be log-concave in (l, u) if

\begin{equation*}f(\alpha {x}+(1-\alpha){y})\geq[f({x})]^\alpha [f({y})]^{1-\alpha}\end{equation*}

for all ${x},{y} \in (l,u)$ and $\alpha \in (0,1)$ .

With this definition, f is log-concave if and only if $\log f$ is concave. Moreover, it is known (see e.g. [Reference An3]) that if the function f is continuous, then this definition is equivalent to

(1) \begin{equation}f(x_1+\delta)f(x_2)\geq f(x_1)f(x_2+\delta)\end{equation}

for all $\delta \geq 0$ and all $l\leq x_1\leq x_2\leq x_2+\delta \leq u$ .

Definition 2. Let $\mathcal{X}$ and $\mathcal{Y}$ be subsets of the real line $\mathbb{R}$ . A function $f\,:\, \mathcal{X}\times \mathcal{Y}\rightarrow \mathbb{R}$ is said to be totally positive of order 2 $({\mathrm{TP}}_2)$ if

(2) \begin{equation}f(x_1,y_1)f(x_2,y_2)-f(x_1,y_2)f(x_2,y_1)\geq 0\end{equation}

for all $x_1, x_2\in \mathcal{X}$ and all $y_1,y_2\in\mathcal{Y}$ such that $x_1\leq x_2$ and $y_1\leq y_2$ .

In the next lemma we give a useful characterization for log-concavity of the function $f({{\mathrm{e}}}^x)$ .

Lemma 1. Let $f\,:\, \mathbb{R}_+ \longmapsto \mathbb{R}_+$ be a Lebesgue-measurable function. Suppose

\[ \{x\,:\, f(x)>0\}=(l,u)\subseteq \mathbb{R}_+. \]

Then $f({{\mathrm{e}}}^x)$ is log-concave in $(\!\log l,\log u)\subseteq \mathbb{R}$ if and only if

(3) \begin{equation}f(x)f(y)\geq f(x/\epsilon)f(y \epsilon)\end{equation}

for all $\epsilon \geq 1$ and all $l\leq x \leq y \leq y \epsilon \leq u$ .

Proof. (Necessity) In [Reference Hu, Nanda, Xie and Zhu12] it is pointed out that $f({{\mathrm{e}}}^x)$ is log-concave if and only if $f(s/t)$ is ${\mathrm{TP}}_2$ in $(s,t)\in \mathbb{R}_+\times \mathbb{R}_+$ . So, from (2), $f({{\mathrm{e}}}^x)$ is log-concave in $(\!\log l,\log u)$ if and only if

\begin{equation*}f\biggl(\dfrac{s_1}{t_1}\biggr)f\biggl(\dfrac{s_2}{t_2}\biggr)\geq f\biggl(\dfrac{s_1}{t_2}\biggr)f\biggl(\dfrac{s_2}{t_1}\biggr)\end{equation*}

for any $0< s_1 \leq s_2$ and $0< t_1 \leq t_2$ such that these values are inside the interval $(\!\log l,\log u)$ . Now, for any $\epsilon \geq 1$ such that $l\leq x \leq y \leq y \epsilon \leq u$ , consider the following values: $s_1=x$ , $s_2=\epsilon y$ , $t_1=1$ , and $t_2=\epsilon$ . Thus the desired result follows since $s_1=x\leq y \leq \epsilon y=s_2 $ and $t_1=1\leq \epsilon=t_2$ .

(Sufficiency) Let us assume now that (3) holds and let $g(x)=\log(f({{\mathrm{e}}}^x))$ . Since f is Lebesgue-measurable, hence so is g. Letting $x=y$ in inequality (3), we have

\[(f(y))^2 \geq f(y/\epsilon) f(y\epsilon)\]

for all $\epsilon \geq 1$ such that $l\leq y\leq y\epsilon \leq u$ . Taking $y={{\mathrm{e}}}^x$ and $\epsilon={{\mathrm{e}}}^c$ , we get

\[2 g(x) \geq g(x-c) + g(x+c)\]

for all real numbers x and c such that $c>0$ and $\log l\leq x-c \leq x+c\leq \log u$ . Let $a=x-c$ and $b=x+c$ . Then

\[ g\biggl(\dfrac{a+b}2\biggr) \geq \dfrac{ g(a)+g(b)}2\]

for all real numbers a and b. So, g is mid-concave (for the definition and properties of mid-concavity, we refer the readers to [Reference Roberts and Varberg22]). As g is mid-concave and Lebesgue-measurable, then it is continuous (see e.g. [Reference Roberts and Varberg22, p. 221]). Now, since mid-concavity with continuity implies concavity, g is concave and thus $f({{\mathrm{e}}}^x)$ is log-concave.

Let $X_1,\dots,X_n$ be a random sample of size n formed with i.i.d. random variables with a common cumulative distribution function (CDF) $F(x)=\mathbb{P}(X\le x)$ and survival (or reliability) function (SF) $\bar{F}(x)=\mathbb{P}(X>x)$ . We denote the SF of its order statistics $X_{1:n}\leq \dots \leq X_{n:n}$ by $\bar{F}_{k:n}$ for $k=1,\ldots,n$ . From expression (2.2.15) in [Reference Arnold, Balakrishnan and Nagaraja4], the SF of the kth-order statistic can be written as $\bar F_{k:n}(x)= G_{k:n}(\bar F(x))$ , where

(4) \begin{equation}G_{k:n}(x)=k\binom{n}{k}\int_{0}^{x} y^{n-k}(1-y)^{k-1} {{\mathrm{d}}} y\end{equation}

is the CDF of a beta distribution. To get the proof of our main result, we would also need the following technical lemma.

Proof. The function $G_{k:n}({{\mathrm{e}}}^x)$ is log-concave in $({-}\infty,0)$ if and only if $\psi(x)=\log G_{k:n}({{\mathrm{e}}}^x)$ is concave in $({-}\infty,0)$ . This property holds if

\[\psi^{\prime}(x)=\dfrac{ {{\mathrm{e}}}^x g_{k:n}({{\mathrm{e}}}^x)}{ G_{k:n}({{\mathrm{e}}}^x)}=\alpha({{\mathrm{e}}}^x)\]

is decreasing in $({-}\infty,0)$ , where

(5) \begin{equation}g_{k:n}(x)=G^{\prime}_{k:n}(x)=k\binom{n}{k} x^{n-k}(1-x)^{k-1}\end{equation}

and $\alpha(u)=ug_{k:n}(u)/G_{k:n}(u)$ . A straightforward calculation shows that the function $\alpha$ is decreasing in (0,1) for $k=1,\dots, n$ .

Finally we include here the basic definitions and properties of systems needed for our results. For more properties we refer the reader to [Reference Barlow and Proschan5] and [Reference Navarro20]. A system with n components is a Boolean function $\phi\,:\, \{0,1\}^n\to\{0,1\}$ , where $x_i=1$ (resp. 0) indicates that the ith component works (does not work) and where the state of the system $\phi(x_1,\dots,x_n)$ just depends on the state of these components. A system is semi-coherent if $\phi$ is increasing, $\phi(0,\dots,0)=0$ and $\phi(1,\dots,1)=1$ . A system is coherent if $\phi$ is increasing and it is strictly increasing in each variable for at least one point (see [Reference Barlow and Proschan5]).

Another well-known property is that the lifetime T of a system is also a function of the lifetimes $X_1,\dots,X_n$ of its components. Moreover, if these component lifetimes have a common SF function $\bar F(t)=\mathbb{P}(X_i>t)$ , then the system SF $\bar F_T(t)=\mathbb{P}(T>t)$ can be written as (see e.g. Theorem 2.11 of [Reference Navarro20])

(6) \begin{equation}\bar F_T(t)=\bar q(\bar F(t))\end{equation}

for all t, where $\bar q\,:\, [0,1]\to[0,1]$ is a distortion function, that is, it is continuous, increasing, and satisfies $\bar q(0)=0$ and $\bar q(1)=1$ . This function $\bar q$ does not depend on $\bar F$ . In Theorem 4.1 of [Reference Navarro20] it is proved that if $\bar F$ is absolutely continuous and $\bar q$ is differentiable in (0,1), then the IFR (DFR) class is preserved if and only if the function $\alpha(u)=u\bar q^{\prime}(u)/\bar q(u)$ is decreasing (increasing) in (0,1). It can be proved that this alpha function is decreasing for $T=X_{k:n}$ and i.i.d. components, that is, the IFR class is preserved under the formation of k-out-of-n in the i.i.d. continuous case. However, the DFR class is not preserved in these systems except in the case of series systems.

3. Preservation of the IFR class in the discrete case

Let us suppose in this section that X is a discrete random variable with ordered support $S_X=\{x_i\}_{i\in I}$ , with $x_i<x_{i+1}$ for all $i,i+1\in I$ and probability mass function (PMF) $p(x)=\mathbb{P}(X=x)$ . The failure rate of a discrete distribution is (see [Reference Johnson, Kemp and Kotz13, p. 45])

\begin{align*}h(x_i)&=\mathbb{P}(X=x_i\mid X \geq x_i)=\dfrac{\mathbb{P}(X=x_i)}{\mathbb{P}(X \geq x_i)}=\dfrac{p(x_i)}{\bar{F}(x_{i-1})}=1-\dfrac{\bar{F}(x_{i})}{\bar{F}(x_{i-1})},\quad i \in I.\end{align*}

Then we say that X is IFR (DFR) if and only if $h(x_i)$ is increasing (decreasing) in i, which is equivalent to $\bar{F}(x_{i})/\bar{F}(x_{i-1})$ is decreasing (increasing) in i. This property holds if and only if

\begin{equation*}(\bar{F}(x_i))^2 \geq (\leq)\bar{F}(x_{i-1})\bar{F}(x_{i+1})\quad \text{for all $i-1,i,i+1\in I$.}\end{equation*}

Let us now study the preservation of the IFR/DFR ageing classes in k-out-of-n systems. First we note that it is not difficult to see that the DFR class is not preserved under the formation of k-out-of-n systems in the discrete case by constructing a parallel system with two components having a common geometric distribution which are both DFR and IFR. Both IFR and DFR classes are closed under the formation of series systems with n i.i.d. components because their survival functions are $(\bar F(x))^n$ for $n=1,2,\dots$ .

Now we focus on the preservation of the IFR property in the discrete case.

Proof. We first note that X is IFR if and only if

(7) \begin{equation}\bar F(x_i)\bar F(x_i)\geq \bar F(x_{i-1})\bar F(x_{i+1})\end{equation}

for all $i-1,i,i+1\in I$ . Analogously, $X_{k:n}$ is IFR if and only if

(8) \begin{equation}G_{k:n}(\bar F(x_i))G_{k:n}(\bar F(x_i))\geq G_{k:n}(\bar F(x_{i-1}))G_{k:n}(\bar F(x_{i+1}))\end{equation}

for all $i-1,i,i+1\in I$ , where $G_{k:n}$ is the function defined in (4). Then, from Lemmas 1 and 2, we have that

(9) \begin{equation}G_{k:n}(x)G_{k:n}(y)\geq G_{k:n}(x/\epsilon)G_{k:n}(y\epsilon)\end{equation}

holds for any $\epsilon \geq 1$ and any $0< x \leq y\leq y \epsilon < 1$ . Hence, from (9) with $x=y=\bar F(x_i)$ and $\epsilon= \bar F(x_i)/\bar F(x_{i+1})\geq 1$ , we get

\[ G_{k:n}(\bar F(x_i))G_{k:n}(\bar F(x_i))\geq G_{k:n}(\bar F(x_{i+1}))G_{k:n}\biggl(\dfrac{\bar F^2(x_i)}{\bar F(x_{i+1})}\biggr).\]

Finally, by using (7) and the fact that $G_{k:n}$ is non-negative and increasing, we deduce that (8) holds, which concludes the proof.

The same technique could be applied to other system structures with i.i.d. components or to the case of dependent i.d. components when the common distribution is a discrete distribution $\bar F$ with ordered support $\{x_i\}_{i\in I}$ by using (6). Thus it can be proved that the IFR class is preserved when the function $\alpha$ defined in the preceding section (by using the distortion function of the system) is decreasing. We include a general result in this sense and an example in the following section.

4. General case

First we give a general definition for the IFR ageing class.

Definition 3. Let X be a random variable with SF $\bar F$ and left-hand end point of the support $\ell=\inf\{x\,:\, \bar F(x)<1\}$ . Then we say that X (or $\bar F$ ) is IFR if

(10) \begin{equation}\bar F(z+s)\bar F(t)\geq \bar F(z+t) \bar F(s)\quad \text{for all $ \ell\leq s\leq t$,}\ z\geq 0.\end{equation}

Note that if $\bar F(t)>0$ , then the condition in (10) implies that $X_s\geq_{\textrm{ST}}X_t$ for all $s\leq t$ , where $\geq_{\textrm{ST}}$ denotes the usual stochastic order (see Chapter 1 of [Reference Shaked and Shanthikumar26]) and where $X_z=(X-z \mid X>z)$ represents the residual lifetime of X at age z. This is the property used in Theorem 1.A.30 of [Reference Shaked and Shanthikumar26] to characterize the IFR class in the continuous case. Thus the meaning of the IFR property is clear: the younger units are ST-better (more reliable) than the older ones. It also implies that if X has a discrete distribution, then its hazard rate function is increasing in the mass points (see Section 3). If X has an absolutely continuous distribution, then it implies that $\bar F$ is log-concave and that there exists a probability density function such that its hazard rate function is increasing. Note that the condition ‘ $\bar F$ is log-concave’ stated in [Reference Shaked and Shanthikumar26, p. 1] cannot be used in discrete models. However, the general definition given in (10) could be applied to any kind of model (for example, in mixtures of discrete and continuous models). It may also be applied to random variables that can take negative values.

However, we must note that the DFR ageing class cannot be extended to the general case by reversing the inequality in (10). The reason is that if X has a discrete distribution and $x_i$ and $x_{i+1}$ are two consecutive mass points, then $X_t$ is ST-strictly decreasing in t in the interval $(x_i,x_{i+1})$ . For example, the geometric distribution does not satisfy this reverse inequality. So the reverse inequality in (10) can only be used in the continuous case.

Recall from (6) that if T is the lifetime of a semi-coherent system with i.d. components, then the system SF function can be obtained as $\bar F_T(t)=\bar q(\bar F(t))$ , where $\bar q$ is a distortion function. If $\bar q$ is strictly increasing and differentiable, then we can state the following general result for the preservation of the IFR class in semi-coherent systems with i.d. components. Note that this result can also be applied to general distorted distributions and for random variables with negative values.

Proof. (i) $\Rightarrow$ (ii) If $X_1$ has a standard exponential model, then it is absolutely continuous and IFR and, from (i), so is T. Hence its hazard rate $h_T$ is increasing. As the system SF function is $\bar F_T(t)=\bar q(\bar F(t))=\bar q({{\mathrm{e}}}^{-t})$ for all $t\geq 0$ , then its probability density function can be obtained as

\[f_T(t)=-\bar F^{\prime}_T(t)={\bar q} ^{\prime}(\bar F(t)) f(t)=\bar q^{\prime}({{\mathrm{e}}}^{-t}) {{\mathrm{e}}}^{-t} \]

for all $t\geq 0$ , where $f(t)=-\bar F^{\prime}(t)= {{\mathrm{e}}}^{-t}$ is the probability density function of the standard exponential. Hence the system hazard rate function is

\[h_T(t)=\dfrac{f_T(t)}{\bar F_T(t)}=\alpha({{\mathrm{e}}}^{-t})\]

for all $t\geq 0$ . Then $\alpha(u)=h_T({-}\log(u))$ for $u\in (0,1)$ . Therefore, as $h_T$ is increasing in $(0,\infty)$ , $\alpha $ is decreasing in (0,1).

(i) $\Leftarrow$ (ii) Now we assume that $\alpha $ is decreasing. Then we consider the function $\psi(x)=\log\bar q({{\mathrm{e}}}^x)$ for $x\in({-}\infty,0)$ . Its derivative is

\[\psi^{\prime}(x)=\dfrac{{{\mathrm{e}}}^x\bar q^{\prime}({{\mathrm{e}}}^x)}{\bar q({{\mathrm{e}}}^x)}=\alpha({{\mathrm{e}}}^x).\]

Hence $\psi^{\prime}$ is decreasing in $({-}\infty,0)$ . Therefore $\psi$ is concave and $\bar q({{\mathrm{e}}}^x)$ is log-concave. So we can apply Lemma 2 to the function $\bar q$ in the interval (0,1), that is,

(11) \begin{equation}\bar q(x)\bar q(y)\geq \bar q(x/\epsilon)\bar q(y\epsilon)\end{equation}

for any $\epsilon \geq 1$ such that $0< x \leq y \leq y\epsilon < 1$ . If X is IFR with left-hand end point of the support $\ell$ , $\ell\leq s\leq t$ and $z\geq 0$ , then

\[\bar F(z+s)\bar F(t)\geq \bar F(z+t) \bar F(s).\]

From (6), as $\bar q$ is strictly increasing, we find that the left-hand end point of the support of T satisfies $\ell_T=\ell$ .

Then we have two options. If $t\geq z+s$ , then $\bar F(t)\leq \bar F(z+s)$ . Hence we apply inequality (11) with $x=\bar F(t)$ , $y=\bar F(z+s)$ , and $\epsilon=\bar F(s)/\bar F(z+s)\geq 1$ , obtaining

\[\bar q(\bar F(t))\bar q(\bar F(z+s))\geq \bar q\biggl(\dfrac{\bar F(t)\bar F(z+s)}{\bar F(s)}\biggr)\bar q(\bar F(s))\]

for s such that $\bar F(s)>0$ .

The other option is when $t< z+s$ , then $\bar F(t)\geq \bar F(z+s)$ . Hence we apply (11) with $y=\bar F(t)$ , $x=\bar F(z+s)$ , and $\epsilon=\bar F(s)/\bar F(t)\geq 1$ , obtaining

\[\bar q(\bar F(t))\bar q(\bar F(z+s))\geq \bar q\biggl(\dfrac{\bar F(t)\bar F(z+s)}{\bar F(s)}\biggr)\bar q(\bar F(s))\]

for s such that $\bar F(s)>0$ .

Finally, by using the fact that X is IFR and $\bar q$ is increasing, in both options we get

\[\bar q(\bar F(t))\bar q(\bar F(z+s))\geq \bar q(\bar F(z+t))\bar q(\bar F(s))\]

for all $\ell\leq s\leq t$ and $z\geq 0$ . Note that this inequality also holds if $\bar F(s)=0$ , since it implies that $\bar F(z+s)=0$ for all $z\geq 0$ .

Note that the preceding result can be applied to systems with dependent or independent components when they have a common distribution. This distribution can be continuous, discrete, or a mixture. In the preceding section we mentioned that the function $\alpha$ is always decreasing for k-out-of-n systems with i.i.d. components. Hence, from the preceding theorem, the IFR class is always preserved in k-out-of-n systems with i.i.d. components having a common general reliability function $\bar F$ .

However, it should be noted that the IFR property is not preserved in k-out-of-n systems when the components are dependent and i.d. Thus Example 4.3 of [Reference Navarro20] proves that the IFR class is not preserved in $X_{1:2}$ when the components are dependent with a Clayton copula since $\alpha$ is strictly increasing.

Example 4.2 of [Reference Navarro20] proves that the IFR class can be preserved in other systems (i.e. systems without a k-out-of-n structure). It also shows that there exist systems where neither the IFR nor the DFR classes are preserved.

To conclude, the following example shows how to apply Theorem 2 to a general SF $\bar F$ in a system that does not have a k-out-of-n structure.

Example 1. Let us consider the system with lifetime

\[T=\min(X_1,\max(X_2,X_3))\]

and let us assume that the component lifetimes are i.i.d. with any IFR distribution. Then the distortion function in (6) is $\bar q(u)=2u^2-u^3$ and the associated alpha function $\alpha (u)=(4-3u)/(2-u)$ is decreasing in (0,1) (see [Reference Navarro20, p. 124]). Then the system is also IFR. This result holds for continuous distributions (Theorem 4.1 of [Reference Navarro20]), discrete distributions with ordered support (preceding section), or in the general case by using Definition 3 and Theorem 2. For example, we can consider the SF

\begin{equation*}\bar F(t)=\begin{cases}1 & \text{for $ t<0$,}\\p {{\mathrm{e}}}^{-t}& \text{for $ t\geq 0$,}\end{cases}\end{equation*}

for $0<p<1$ . It is a mixture of a discrete distribution with mass $1-p$ at time $t=0$ and a continuous exponential distribution with mass p in $(0,\infty)$ . A straightforward calculation shows that $\bar F$ is IFR according to Definition 3. Hence so is the system SF $\bar F_T$ .