2. Notation and preliminary results

From [Reference Barlow and Proschan1] a (two-state) system is a Boolean function $\psi\colon \{0,1\}^n\to \{0,1\}$ , where $\psi(x_1,\ldots,\break x_n)$ represents the state of the system (1 means that it works and 0 that it does not work) which is completely determined by the component states $x_1,\ldots,x_n\in \{0,1\}$ . A system $\psi$ is semi-coherent if it is increasing, $\psi(0,\ldots,0)=0$ and $\psi(1,\ldots,1)=1$ . We say that the jth component is relevant for the system $\psi$ if there exist $x_1,\ldots,x_{j-1},$ $x_{j+1},\ldots,x_n\in\{0,1\}$ such that

\begin{equation*}0=\psi (x_1,\ldots,x_{j-1},0, x_{j+1},\ldots,x_n)<\psi(x_1,\ldots,x_{j-1},1,x_{j+1},\ldots,x_n)=1.\end{equation*}

A system $\psi$ is coherent if it is increasing and all the components are relevant. Clearly, if $\psi$ is coherent, then $\psi$ is semi-coherent. However, for example, $\psi(x_1,x_2)=x_1$ is semi-coherent but it is not coherent (since the second component is irrelevant for the system).

Let T be the lifetime of a coherent system with component lifetimes $T_1,\ldots,T_n$ , and let $T_{1:n},\ldots,T_{n:n}$ be the associated ordered component lifetimes. Here $T_{k:n}$ represents the lifetime of the k-out-of-n system. It is well known that the system lifetime T is equal to one of these component lifetimes. Moreover, Samaniego [Reference Samaniego22] proved that, if $T_1,\ldots,T_n$ are IID with a continuous reliability (survival) function $\bar F(t)=\mathbb{P}(T_j>t)$ , then the system reliability function $\bar F_T(t)=\mathbb{P}(T>t)$ can be represented as

(1) \begin{equation}\bar F_T(t)=s_1\bar F_{1:n}(t)+\dots+s_n\bar F_{n:n}(t)\end{equation}

for any time t, where $\bar F_{i:n}(t)=\mathbb{P}(T_{i:n}>t)$ and $s_i=\mathbb{P}(T=T_{i:n})$ for $i=1,\ldots,n$ . The vector with these coefficients $\mathbf{s}=(s_1,\ldots,s_n)$ depends only on the structure $\psi$ of the system and it is called the signature of the system (see [Reference Samaniego22] and [Reference Samaniego23]). From the theory of order statistics, $\bar F_{1:n},\ldots,\bar F_{n:n}$ can be calculated from $\bar F$ . So the signature-based mixture representation (1) is a useful tool for computing the system reliability (see [Reference Samaniego23]). It can also be used to compare stochastically systems with different structures (see [Reference Kochar, Mukerjee and Samaniego7], [Reference Navarro, Samaniego, Balakrishnan and Bhattacharya16], [Reference Rychlik, Navarro and Rubio21], and [Reference Samaniego23]).

Representation (1) (with the same coefficients) was extended in [Reference Navarro, Samaniego, Balakrishnan and Bhattacharya16] (see also [Reference Navarro and Rychlik13]) to the case in which the random vector $(T_1,\ldots,T_n)$ of component lifetimes has an EXC joint reliability function ${\bar{\mathbf{F}}}(t_1,\ldots,t_n)=\mathbb{P}(T_1>t_1, \ldots, T_n>t_n),$ that is,

\begin{equation*}{\bar{\mathbf{F}}}(t_1,\ldots,t_n)={\bar{\mathbf{F}}}(t_{\sigma(1)},\ldots,t_{\sigma(n)})\end{equation*}

holds for any permutation $\sigma$ . In this case, the same property holds for the joint distribution function and $(T_1,\ldots,T_n)$ is invariant in law under permutations.

From the copula theory (see e.g. [3, p. 33] or [17, p. 32]), we know that ${\bar{\mathbf{F}}}$ can be written as

\begin{equation*}{\bar{\mathbf{F}}}(t_1,\ldots,t_n)=\hat{C}(\bar F_1(t_1),\ldots,\bar F_n(t_n))\end{equation*}

for an n-dimensional copula function $\hat C$ (called the survival copula) and for the marginal (component) reliability functions $\bar F_i(t_i)=\mathbb{P}(T_i>t_i)$ , $i=1,\ldots,n$ . A copula is an n-dimensional distribution function with uniform marginals over the interval (0,1).

It is easy to prove that ${\bar{\mathbf{F}}}$ is EXC if and only if $\hat C$ is EXC and the component lifetimes are ID, i.e. $\bar F_1=\dots=\bar F_n$ . As above, in the ID case, the common component reliability function will be represented simply as $\bar F$ .

Example 5.1 in [Reference Navarro, Samaniego, Balakrishnan and Bhattacharya16] proves that representation (1) does not necessarily hold when the component lifetimes are independent but not ID. Therefore the ID assumption (included in the EXC case) cannot be relaxed if we want to get (1). However, in the following section we will prove that the other assumption, ‘ $\hat C$ is EXC’, can be relaxed.

For this purpose we need the following representation for the system reliability which is valid in the general case, that is, for any joint reliability function ${\bar{\mathbf{F}}}$ . From [1, p. 12], we know that the system lifetime T can be written as

\begin{equation*}T=\max_{i=1,\ldots,r}T_{P_i},\end{equation*}

where $T_{P_i}=\min_{j\in P_i}T_j$ is the lifetime of the series system with components in $P_i$ for $i=1,\ldots,r$ , and $P_1,\ldots,P_r$ are the minimal path sets of the system. A path set is a set $P\subseteq\{1,\ldots,n\}$ of components such that the system works when all the components in P work (i.e. $\psi(x_1,\ldots,x_n)=1$ when $x_i=1$ for all $i\in P$ ). A path set is a minimal path set if it does not contain other path sets. Then, by using the inclusion–exclusion formula, the system reliability can be written as

(2) \begin{align} \bar F_T(t)&=\mathbb{P}(T>t) \notag \\[2pt] &=\mathbb{P}\Bigl(\max_{i=1,\ldots,r}T_{P_i}>t\Bigr) \notag \\[2pt] &=\mathbb{P}\Biggl(\bigcup_{i=1}^r \{T_{P_i}>t\}\Biggr)\notag\\[2pt] &=\sum_{i=1}^{r}\bar F_{P_i}(t)-\sum_{i=1}^{r-1}\sum_{j=i+1}^r \bar F_{P_i\cup P_j}(t)+\dots+(-1)^{r+1}\bar F_{P_1\cup\dots\cup P_r}(t)\end{align}

for all t, where we use the notation $\bar F_P(t)=\mathbb{P}(T_P>t)$ for the reliability function of the series system with components in the set P. Expression (2) proves that the reliability function of the system $\bar F_T$ is a linear combination of the reliability functions of series systems. However, it is not a mixture representation since it may contain some negative coefficients. Expression (2) can also be used to compute the system reliability and to compare systems (see the review in [Reference Navarro11]).

3. Main results

Let us start with a definition extracted from [Reference Okolewski18]. From now on we will use the following notation. For any set $I\subseteq \{1,\ldots,n\}$ , $\mathbf{u}_I\,:\!=(u_1,\ldots,u_n)$ denotes the vector with $u_i=u$ for $i\in I$ and $u_i=1$ if $i\notin I$ . The cardinality of the set I is denoted by $|I|$ .

Definition 1. An n-dimensional copula C is said to be diagonal-dependent (denoted by DD) if

\begin{equation*} C(\mathbf{u}_P)=C(\mathbf{u}_Q)\quad \text{for all } P,Q\subseteq\{1,\ldots,n\} \quad \text{with } |P|=|Q|.\end{equation*}

The function $\delta(u)=C(u,\ldots,u)$ is called the diagonal section of the copula C. Hence note that C is DD if and only if

(3) \begin{equation}C(\mathbf{u}_P)=\delta_m(u)\quad \text{for all } P\subseteq\{1,\ldots,n\} \quad \text{with } |P|=m\end{equation}

for $m=1,\ldots,n$ , where

\begin{align*}\delta_m(u)\,:\!=C\left(\underbrace{u,\ldots,u}_{\text{\textit{m}-times}},\underbrace{1,\ldots,1}_{\text{$(n-m)$-times}}\right)\\[-15pt]\end{align*}

is the diagonal section for the copula of the marginal distribution of the first m-variables. Clearly, $\delta_n(u)=C(u,\ldots,u)=\delta(u)$ and $\delta_1(u)=u$ for all $u\in[0,1]$ (since all the univariate marginals have a uniform distribution over the interval (0,1)). So we just need to check (3) for $m=2,\ldots,n-1$ .

In particular, a copula C is DD when all the marginals of dimension m have the same copula for all $1<m<n$ . Of course, all the EXC copulas are, in particular, DD. The reverse is not true (see Proposition 1 and Example 1 below).

Now we are ready to state the main result of the paper.

Proof. From (2) we know that the system reliability function $\bar F_T$ can be written as a linear combination of the reliability functions of the series systems. If the component lifetimes are ID with a reliability function $\bar F$ and a DD survival copula $\hat C$ , then

(4) \begin{equation}\bar F_P(t)=\mathbb{P}\Bigl(\min_{j\in P}T_j>t\Bigr)=\hat C_P (\bar F(t),\ldots,\bar F(t))=\hat \delta_{m}(\bar F(t))\end{equation}

holds for all t and all $P\subseteq\{1,\ldots,n\}$ , where $\hat C_P (u_1,\ldots,u_n)\,:\!=\hat C (u_1^P,\ldots,u_n^P)$ and $u_i^P=u_i$ if $i\in P$ and $u_i^P=1$ if $i\notin P$ , $m=|P|$ and $\hat \delta_m$ is defined as

\begin{align*}\hat \delta_m(u)\,:\!=\hat C\left(\underbrace{u,\ldots,u}_{\text{\textit{m}-times}},\underbrace{1,\ldots,1}_{\text{$(n-m)$-times}}\right)\\[-15pt]\end{align*}

for all $u\in[0,1]$ and $m=1,\ldots,n$ . Hence all the series systems with the same number of components m have the same reliability function given by (4). Therefore the general representation (2) can be reduced to

(5) \begin{equation}\bar F_T(t)=a_1 \hat \delta_1(\bar F(t))+\dots+a_n \hat \delta_n(\bar F(t)),\end{equation}

where $a_1,\ldots,a_n$ are some coefficients that depend only on the system structure (i.e. the minimal path sets).

The preceding representation (5) holds for any coherent system structure (with the appropriate coefficients $a_1,\ldots,a_n$ ). For example, the series system with n components has just one minimal path set $P_1=\{1,\ldots,n\}$ and lifetime $T_{1:n}=\min(T_1,\ldots,T_n)$ . Hence

(6) \begin{equation}\bar F_{1:n}(t)=\mathbb{P}(T_1>t,\ldots,T_n>t)=\hat C(\bar F(t),\ldots,\bar F(t))= \hat \delta_n(\bar F(t))\end{equation}

for all t.

Analogously, the minimal path sets of $T_{2:n}$ are all the subsets with $n-1$ elements. So there are $n=\binom n {n-1}$ minimal path sets and, from (2),

(7) \begin{equation}\bar F_{2:n}(t)= n \hat \delta_{n-1}(\bar F(t))-(n-1) \hat \delta_{n}(\bar F(t))\end{equation}

holds for all t. The last coefficient in the preceding expression is $n-1$ because the coefficients in (5) sum up to 1 (take $t\to-\infty$ ).

In general, $T_{i:n}$ has $\binom n {n-i+1}$ minimal path sets and, from (2), its reliability function can be written as

(8) \begin{equation}\bar F_{i:n}(t)= a_{i, n-i+1} \hat \delta_{n-i+1}(\bar F(t))+\dots+a_{i,n} \hat \delta_{n}(\bar F(t))\end{equation}

for some coefficients $a_{i, n-i+1},\ldots,a_{i, n}$ such that $a_{i,n-i+1}+\dots +a_{i,n}=1$ and $a_{i, n-i+1}=\binom n {n-i+1}$ for $i=1,\ldots,n$ .

Thus, if we define the column vectors

\begin{equation*}\mathbf r(t)=(\bar F_{1:n}(t),\ldots,\bar F_{n:n}(t))'\quad\text{and}\quad\mathbf d(t)=(\hat \delta_{1}(\bar F(t)),\ldots,\hat \delta_{n}(\bar F(t)))',\end{equation*}

(8) proves that $\mathbf r(t)=A \mathbf d(t)$ for a triangular real-valued matrix $A=(a_{i,j})$ such that $a_{i, n-i+1}=\binom n {n-i+1}$ and $a_{i, j}=0$ for $i=1,\ldots,n$ and $j=1,\ldots, n-i$ . Hence A is not singular and so we can write $\mathbf d(t)=A^{-1} \mathbf r(t)$ , where $A^{-1}$ is the inverse matrix of A. Moreover, note that (5) can be rewritten as

\begin{equation*}\bar F_T(t)=(a_1,\ldots,a_n)\mathbf d(t).\end{equation*}

Then

\begin{equation*}\bar F_T(t)=(a_1,\ldots,a_n)A^{-1} \mathbf r(t)=(c_1,\ldots,c_n)\mathbf r(t)=c_1\bar F_{1:n}(t)+\dots+c_n\bar F_{n:n}(t)\end{equation*}

for all t, where $(c_1,\ldots,c_n)=(a_1,\ldots,a_n)A^{-1}$ are some coefficients which depend only on the structure of the system. Therefore these coefficients should be the same as that obtained in the IID continuous case, i.e. $c_i=s_i$ for $i=1,\ldots,n$ . So (1) holds with the same coefficients for systems with ID component lifetimes and DD survival copulas. $\Box$

Remark 1. The usual copula representation for the joint distribution function of $(T_1,\ldots,T_n)$ is

\begin{equation*}\mathbf{F}(t_1,\ldots,t_n)\,:\!=\mathbb{P}(T_1\leq t_1, \ldots, T_n\leq t_n)=C(F_1(t_1),\ldots,F_n(t_n) ),\end{equation*}

where $F_i(t_i)=\mathbb{P}(T_i\leq t_i)$ , $i=1,\ldots,n$ , are the univariate marginal distribution functions and C is the distributional copula. Both copulas C and $\hat C$ determine the dependence structure of $(T_1,\ldots,T_n)$ . So C determines $\hat C$ and vice versa. Moreover, it is easy to see that $\hat C$ is DD if and only if C is DD. Hence we can obtain an alternative proof of Theorem 1 by using copula C and the representation of the system lifetime in terms of its minimal cut sets (see [Reference Barlow and Proschan1, p. 12]).

Remark 2. If $T_{1:j}=\min(T_1,\ldots,T_j)$ , then $\bar F_{1:j}(t)\,:\!=\mathbb{P}(T_{1:j}>t)=\hat \delta_j(\bar F(t))$ for all t and $j=1,\ldots,n$ . Hence, under the assumptions of the preceding theorem, expression (5) can also be written as

\begin{equation*}\bar F_T(t)=a_1 \bar F_{1:1}(t)+\dots+a_n \bar F_{1:n}(t).\end{equation*}

The vector $(a_1,\ldots,a_n)$ with these coefficients is called the minimal signature (or the domination coefficients) of the system (see e.g. [Reference Navarro, Ruiz and Sandoval15] and [23, p. 77]). Hence the representation based on minimal signatures obtained in [Reference Navarro, Ruiz and Sandoval15] for systems with EXC component lifetimes can also be extended to systems with ID component lifetimes and DD survival copulas. From the comments given in the preceding remark, the same can be applied to the representation based on parallel systems and the maximal signature (see [Reference Navarro, Ruiz and Sandoval15]).

As an immediate consequence of the main theorem we obtain several properties that can be used to compare stochastically systems with different structures. In the following theorem we state the results for the (usual) stochastic order ( $\leq_{st}$ ), the hazard rate order ( $\leq_{\text{hr}}$ ), the mean residual life order ( $\leq_{\text{mrl}}$ ), and the likelihood ratio order ( $\leq_{\text{lr}}$ ), extending the comparison results obtained in [Reference Kochar, Mukerjee and Samaniego7] (IID case) and [Reference Navarro, Samaniego, Balakrishnan and Bhattacharya16] (EXC case). Similar results can be obtained for the reversed hazard rate and mean inactivity time orders extending that given in [Reference Navarro and Rubio12]. For the formal definitions of these orders, their basic properties, and their main applications, we refer the reader to [Reference Li and Li8], [Reference Müller and Stoyan10], and [Reference Shaked and Shanthikumar24]. We simply note here that the relationships between these orders are as follows:

\begin{equation*} \begin{tabular}{cc c c c } $X\leq_{\text{lr}}Y$& \quad $\Rightarrow$ & \quad $X\leq_{\text{hr}}Y$ & \quad $\Rightarrow$ & $X\leq_{\text{mrl}}Y$ \\ &&$\Downarrow$ & & $\Downarrow$ \\ & & $X\leq_{st}Y$ & \quad $\Rightarrow$ & \quad $\mathbb{E}(X)\leq \mathbb{E}(Y)$. \end{tabular} \end{equation*}

Because the signature vector $\mathbf s=(s_1,\ldots,s_n)$ of a system can be seen as the probability mass function of a discrete random variable with support contained in the set $\{1,\ldots,n\}$ , these stochastic orders can also be applied to compare two signature vectors (as discrete distributions). Thus we can state the following theorem.

The proof is immediate from (1), Theorem 1, and Theorems 1.A.6, 1.B.14, 2.A.15, and 1.C.17 in [Reference Shaked and Shanthikumar24], respectively. Note that in (iii) we need the hr ordering between the signatures and the mrl ordering in (10), to get the mrl ordering between the system lifetimes. The mrl ordering between the signatures is not enough. In the IID case, if we just assume $\mathbf s\leq_{\text{mrl}}\mathbf s^*$ , then we need some extra conditions (see [Reference Lindqvist, Samaniego and Wang9]).

Example 2 shows how to use the above theorem to obtain (distribution-free) comparisons results for systems with different structures. The signature vectors and the ordering relationships between all the systems (signatures) with 1–4 components are given in Table 1 and Figures 1–3 of [Reference Navarro, Samaniego, Balakrishnan and Bhattacharya16] (see also Figures 1–3 in [Reference Navarro11]). These orderings can be extended to systems satisfying the assumptions of Theorems 1 and 2. Note that Theorem 2 can also be applied to mixed systems (i.e. mixtures of coherent systems) and, in particular, to semi-coherent systems since they can be written as mixed systems (from Theorem 1).

Remark 3. Expression (8) can be used jointly with the results for distorted distributions obtained in [Reference Navarro, del Águila, Sordo and Suárez-Llorens14] (see also Theorem 4 in [Reference Navarro11]) to check if (9), (10), and (11) hold for a given DD survival copula $\hat C$ . We show this procedure in Example 2. These properties depend on the survival copula. For example, from (6) and (7), $T_{1:n}\leq_{\text{hr}}T_{2:n}$ holds for all $\bar F$ if and only if the function

\begin{equation*}\dfrac{n\hat \delta_{n-1}(u)-(n-1)\hat \delta_{n}(u)}{\hat\delta_{n}(u)}\end{equation*}

is decreasing in (0,1), that is, if and only if the function

\begin{equation*}\dfrac{\hat \delta_{n}(u)}{\hat \delta_{n-1}(u)}=\dfrac{\hat C(u,\ldots,u,u)}{\hat C(u,\ldots,u,1)}\end{equation*}

is increasing in (0,1).

Now we introduce a new family of copulas which are DD but not EXC.

Proposition 1. Let D be the absolutely continuous n-dimensional copula which has the following probability density function:

\begin{equation*} d(\mathbf{u})=\begin{cases} 2^{n-1}-\alpha & for\ \mathbf{u}\in I_{0}\times \dots \times I_{0},\\[4pt] 2^{n-1}-\alpha & for\ \mathbf{u}\in I_{1}\times \dots \times I_{1},\\[4pt] \dfrac{\alpha}{2^{n-1}-1} & for\ \mathbf{u}\in I_{i_1}\times \dots \times I_{i_n}, i_j=0,1,\ but\ i_1=\dots=i_n\ does\ not\ hold,\end{cases}\end{equation*}

where

\begin{equation*}\mathbf{u}=(u_1,\ldots,u_n),\quad I_0=[0,1/2),\quad I_1=[1/2,1], \quad 1/2< \alpha <2^{n-1}-1.\end{equation*}

Let $g_1,\ldots,g_n\colon [0,1]\to[0,1]$ be different absolutely continuous functions such that $g_i(0)=g_i(1)=0$ and $-1\leq 2g^{\prime}_i(u)\leq 1$ for all $i=1,\ldots,n$ and all $u\in[0,1]$ such that this derivative exists. Then

(12) \begin{equation}C(u_1,\ldots,u_n)=D(u_1,\ldots,u_n)+g_1(u_1)\dots g_n(u_n)\end{equation}

is a non-EXC DD copula with $\delta_m(u)\neq u^m$ for all $m=2,\ldots,n-1$ . Moreover, if for an $i\in\{1,\ldots,n\}$ , $g_i(1/2)=0$ , then $\delta_n(u)\neq u^n$ .

Proof. First we note that D is an EXC n-dimensional absolutely continuous copula since $2^{n-1}-\alpha>0$ and

\begin{equation*} 2\dfrac{2^{n-1}-\alpha}{2^n}+(2^n-2)\dfrac{\alpha}{2^n(2^{n-1}-1)}=\dfrac{2^{n}-2\alpha}{2^n}+\dfrac{2\alpha}{2^{n}}=1.\end{equation*}

Moreover, its diagonal section satisfies

(13) \begin{equation}\delta_D(1/2)=D(1/2,\ldots,1/2)=\dfrac{2^{n-1}-\alpha}{2^n}> \dfrac 1{2^n}\end{equation}

(since $\alpha <2^{n-1}-1$ ).

Secondly, let us prove that the function C defined by (12) is a copula. Its probability density function c is given by

\begin{equation*}c(u_1,\ldots,u_n)=d(u_1,\ldots,u_n)+g^{\prime}_1(u_1)\dots g^{\prime}_n(u_n).\end{equation*}

As $1/2<\alpha<2^{n-1}-1$ , then

\begin{equation*}2^{n-1}-\alpha>1>\dfrac{\alpha}{2^{n-1}-1}\end{equation*}

and

\begin{equation*}d(u_1,\ldots,u_n)\geq \dfrac{\alpha}{2^{n-1}-1}>\dfrac{1}{2^{n}-2}.\end{equation*}

Hence, as $-1\leq 2g^{\prime}_i(u)\leq 1$ for $i=1,\ldots,n$ , then

\begin{equation*} c(u_1,\ldots,u_n)\geq \dfrac{\alpha}{2^{n-1}-1}-\dfrac1{2^n}> \dfrac{1}{2^{n}-2}-\dfrac1{2^n}>0\end{equation*}

for all $u_1,\ldots,u_n\in(0,1)$ . Therefore

\begin{equation*}C(x_1,\ldots,x_n)=\int_{[0,x_1]\times \dots \times[0,x_n]} c(u_1,\ldots,u_n)\,{\text{d}} u_1\dots \,{\text{d}} u_n\end{equation*}

is a copula since $g_i(0)=g_i(1)=0$ for $i=1,\ldots,n$ and so

\begin{equation*}C(1,\ldots,1)=D(1,\ldots,1)+g_1(1)\dots g_n(1)=1\end{equation*}

and the other border conditions hold.

Clearly, C is not EXC (since we assume that the functions $g_1,\ldots,g_n$ are different continuous functions). However, it is DD because D is EXC and

\begin{equation*}C_P(u,\ldots,u)=D_P(u,\ldots,u)\end{equation*}

for all $P\subseteq\{1,\ldots,n\}$ with $|P|<n$ (since $g_i(1)=0$ ).

Moreover, for $n=3,\dots$ and $m=2,\ldots,n-1$ , $\delta_m(u)\neq u^m$ since

\begin{align*}\delta_m(1/2)&=C\left(\underbrace{1/2,\ldots,1/2}_{\text{\textit{m}-times}},\underbrace{1,\ldots,1}_{\text{$(n-m)$-times}}\right)\\[4pt] &=\dfrac{2^{n-1}-\alpha}{2^n}+\dfrac{2^{n-m}-1} {2^n}\dfrac{\alpha}{2^{n-1}-1}\\[4pt] &=\dfrac{1}{2}-\alpha \dfrac{2^{-1}-2^{-m}}{2^{n-1}-1}\\[4pt] &>\dfrac{1}{2^m}\end{align*}

because $\alpha<2^{n-1}-1$ . Even more, if for an i, $g_i(1/2)=0$ , then

\begin{equation*}\delta_n(1/2)=C(1/2,\ldots,1/2)=D(1/2,\ldots,1/2)=\dfrac{2^{n-1}-\alpha}{2^n}> \dfrac 1 {2^n}\end{equation*}

from (13). This concludes the proof. $\Box$

In order to characterize the relative size of the class $\mathcal{C}_{\text{EXC}}$ of EXC copulas in the class $\mathcal{C}_{\text{DD}}$ of DD copulas, we will use a topological approach similar to that suggested in [Reference Putter and van Zwet20] working with Baire’s categories (see e.g. [Reference Durante, Fernández-Sánchez and Trutschnig5], [Reference Durante, Fernández-Sánchez and Trutschnig6], and [Reference Oxtoby19]). Let us recall some basic topological definitions. A subset N of a metric space $(\Omega ,d)$ is called nowhere dense if its closure has an empty interior. A subset $A\subseteq \Omega $ is of first category in $(\Omega ,d)$ if it can be expressed as (or covered by) a countable union of nowhere dense sets. The subset A is said to be of second category if it is not of first category. Following [Reference Bruckner, Bruckner and Thomson2], in complete metric spaces, first category sets are ‘small sets’ and nowhere dense sets are ‘very small sets’.

The next property shows that, from a topological viewpoint, the set $\mathcal{C}_{\text{EXC}}$ is very small (i.e. it is nowhere dense) into the set $\mathcal{C}_{\text{DD}}$ . Therefore Theorem 1 provides a relevant extension of signature-based representations from EXC copulas to DD copulas. It is easy to see that $\mathcal{C}_{\text{DD}}$ is a closed set in the set of all the copulas $\mathcal{C}$ . Hence it is compact and complete (see [Reference Durante, Fernández-Sánchez and Sempi4]). Then, from Baire’s theorem, it is a second category set in itself.

Proof. As $\mathcal{C}_{\text{EXC}}$ is a closed subset of $\mathcal{C}$ , then so is in $\mathcal{C}_{\text{DD}}$ . Hence we need to prove that $\mathcal{C}_{\text{EXC}}$ does not have interior points. Let us see that for any $C_{\ast }\in \mathcal{C}_{\text{EXC}}$ and any $\varepsilon >0$ , we have

\begin{equation*}B ( C_{\ast },\varepsilon ) =\{ C\in \mathcal{C}_{\text{DD}}\colon d_{\infty }( C,C_{\ast }) <\varepsilon \} \nsubseteqq \mathcal{C}_{\text{EXC}},\end{equation*}

where $d_\infty ( C,C_{\ast }) \,:\!=\sup_{\mathbf{u}\in[0,1]^n} |C(\mathbf{u})-C_{\ast }(\mathbf{u})|$ . If $C_{\ast \ast }\in \mathcal{C}_{\text{DD}}-\mathcal{C}_{\text{EXC}}$ and we define

\begin{equation*}C_{n}=\dfrac{1}{n}C_{\ast \ast }+\dfrac{n-1}{n} C_{\ast },\end{equation*}

then the sequence $\{ C_{n}\} $ converges to $C_{\ast }$ with $ C_{n}\in \mathcal{C}_{\text{DD}}-\mathcal{C}_{\text{EXC}}$ for all n. So, for any $\varepsilon >0$ , there exists $n_{0}$ (depending on $ \varepsilon $ ) such that $C_{n}\in B(C_{\ast },\varepsilon ) $ for all $n\geq n_{0}$ . Therefore $B( C_{\ast},\varepsilon ) \nsubseteqq \mathcal{C}_{\text{EXC}}$ and so $\mathcal{C}_{\text{EXC}}$ is nowhere dense in $\mathcal{C}_{\text{DD}}.$ $\Box$

4 Examples

The first example shows how to use Theorem 1 in a coherent system with ID component lifetimes having a fixed non-EXC DD survival copula.

Example 1. Let us consider the system with lifetime $T=\min(T_1,\max(T_2,T_3)).$ The signature of this coherent system is $(1/3,2/3,0)$ (see e.g. [23, p. 24]). Let us assume that the component lifetimes $T_1,T_2,T_3$ are ID with reliability function $\bar F$ and survival copula

(14) \begin{equation}\hat C(u_1,u_2,u_3)=u_1 C_{FN}(u_2,u_3),\end{equation}

where

\begin{equation*}C_{FN}(u_2,u_3)=\min\biggl(u_2,u_3, \dfrac{u_2^2+u_3^2}2\biggr)\end{equation*}

is the Fredricks–Nelsen copula (see e.g. [3, p. 32]). Clearly the first component is independent from the other components but the second and third components are dependent. So $\hat C$ is not EXC. Therefore the signature-based representations obtained in [Reference Samaniego22] (IID case) and [Reference Navarro, Samaniego, Balakrishnan and Bhattacharya16] (EXC case) cannot be applied to this system. However, $\hat C$ is DD since $\hat \delta_2(u)\,:\!=\hat C(u,u,1)=\hat C(u,1,u)=\hat C(1,u,u)=u^2$ for all $u\in [0,1]$ . Note that $\hat \delta_1(u)\,:\!=\hat C(u,1,1)=\hat C(1,u,1)=\hat C(1,1,u)=u$ (as expected, since the univariate marginal distributions of copulas are uniform) and $\hat \delta_3(u)\,:\!=\hat C(u,u,u)=u^3$ for all $u\in [0,1]$ .

Therefore we can apply Theorem 1, obtaining the following representation for the system reliability function:

\begin{equation*}\bar F_T(t)=\dfrac 1 3 \bar F_{1:3}(t)+\dfrac 2 3 \bar F_{2:3}(t),\end{equation*}

where, from (6) and (7), $\bar F_{1:3}(t)=\hat \delta_3(\bar F(t))=(\bar F(t))^3$ and $\bar F_{2:3}(t)=3\hat \delta_2(\bar F(t))-2\hat \delta_3(\bar F(t))=3(\bar F(t))^2-2(\bar F(t))^3$ for all t. Hence this system (with two dependent components) has the same law (reliability) as the system with the same structure and three IID- $\bar F$ components.

The second example shows that Theorem 2 allows us to compare two coherent systems with different structures (signatures).

Example 2. Let us consider the systems with lifetimes $T=\min(T_1,\max(T_2,T_3))$ (studied in the preceding example) and $T^*=\max(T_1,\min(T_2,T_3))$ . We assume that the component lifetimes are ID with a reliability function $\bar F$ and a survival copula $\hat C$ . The signatures of these coherent systems are $\mathbf{s}=(1/3,2/3,0)$ and $\mathbf{s}^*=(0,2/3,1/3)$ , respectively (see e.g. [23, p. 24]). Therefore, as

\begin{equation*}\dfrac 0{1/3}=0<\dfrac{2/3}{2/3}=1<\dfrac{1/3}0=\infty,\end{equation*}

we have $\mathbf{s}\leq_{\text{lr}}\mathbf{s}^*$ . As the likelihood ratio order is the strongest one, all the signature orderings in Theorem 2 hold. Then $T\leq_{st}T^*$ holds for any $\bar F$ and any DD copula $\hat C$ .

Analogously, to get $T\leq_{\text{hr}}T^*$ , we need to check if (9) holds. If we choose the survival copula of the preceding example, given in (14), then the distributions of $T_{1:3},T_{2:3},T_{3:3}$ coincide with that obtained in the IID case. Hence (9) holds and we get $T\leq_{\text{hr}}T^*$ for all $\bar F$ . In fact (11) also holds and we have $T\leq_{\text{lr}}T^*$ for all absolutely continuous $\bar F$ . However, if we select the survival copula from the family (12) given by

\begin{equation*} \hat C(u_1,u_2,u_3)=D(u_1,u_2,u_3)+g_1(u_1)g_2(u_2)g_3(u_3),\end{equation*}

where $\alpha\in (1/2,3)$ and $2g_1(u)=3g_2(u)=3g_3(u)=u(1-u)$ , then

\begin{equation*}\hat \delta_2(u)=\hat C(u,u,1)=D(u,u,1)=\begin{cases}\dfrac{6-\alpha}3\ u^2 & \text{for } u\in [0,1/2],\\[8pt] h(u) & \text{for } u\in (1/2,1],\end{cases}\end{equation*}

and $\hat \delta_3(u)=D(u,u,u)+g_1(u)g_2(u)g_3(u),$ where

\begin{align*} h(u) & =\dfrac{6-\alpha}{12}+\dfrac{\alpha}3\biggl(u-\dfrac 1 2\biggr)+\biggl(2-\dfrac{\alpha}3\biggr)\biggl(u-\dfrac 1 2\biggr)^2=1+\dfrac{6-\alpha}3u^2-\dfrac{6-2\alpha}3u-\dfrac{\alpha}3,\\[8pt] D(u,u,u) & =\begin{cases}(4-\alpha)u^3 & \text{for } u\in [0,1/2],\\[5pt] \dfrac {4-\alpha}8+(4-\alpha)\biggl(u-\dfrac 1 2\biggr)^3+\dfrac {\alpha} 3\biggl[u^3-\dfrac 1 8 -\biggl(u-\dfrac 1 2\biggr)^3\biggr] & \text{for } u\in (1/2,1],\end{cases}\end{align*}

and $g_1(u)g_2(u)g_3(u)=u^3(1-u)^3/18$ . Therefore $\hat \delta_3(u)\neq u^3$ .

Note that the reliability functions of $T_{1:3},T_{2:3},T_{3:3}$ are

\begin{equation*}\bar q_{1:3}(\bar F(t)),\quad\bar q_{2:3}(\bar F(t)),\quad\bar q_{3:3}(\bar F(t)),\end{equation*}

respectively, where

\begin{equation*}\bar q_{1:3}(u)=\hat \delta_3(u),\quad\bar q_{2:3}(u)=3\hat \delta_2(u)-2\hat \delta_3(u),\quad\bar q_{3:3}(u)=3u-3\hat \delta_2(u)+\hat \delta_3(u)\end{equation*}

for $u\in[0,1]$ .

Therefore, as per the results given in [Reference Navarro, del Águila, Sordo and Suárez-Llorens14] (or in Theorem 4(ii) of [Reference Navarro11]), $T_{1:3}\leq_{\text{hr}} T_{2:3}$ holds for all $\bar F$ and that survival copula if and only if $\bar q_{2:3}/\bar q_{1:3}$ is decreasing in [0, 1], that is, the ratio $r=\hat \delta_2/\hat \delta_3$ is decreasing in (0, 1). By plotting this ratio we see that the property is not always true. For example, when $\alpha =1$ , we obtain the plot given in Figure 1(a), which is not decreasing. However, when $\alpha =2$ , we obtain the plot given in Figure 1(b), which is decreasing in (0,1). Hence $T_{1:3}\leq_{\text{hr}} T_{2:3}$ holds for all $\bar F$ when $\alpha =2$ . It can be proved analogously that $T_{2:3}\leq_{\text{hr}} T_{3:3}$ holds for all $\bar F$ when $\alpha =2$ . Therefore (9) holds and, from Theorem 2(ii), we get $T\leq_{\text{hr}}T^*$ for all $\bar F$ and that survival copula with $\alpha =2$ .

Figure 1: Plots of the ratio $r=\hat \delta_2/\hat \delta_3$ for the survival copula studied in Example 4 when $\alpha=1$ (a) and $\alpha=2$ (b).