2.1. Cumulative inaccuracy measure between non-explosive point processes

A univariate point process over $\mathbb{R}^+$ can be described by an increasing sequence of random variables or by means of its corresponding counting process.

Another equivalent way of describing a univariate point process is through a counting process $N = (N_t)_{t \geq 0}$ with

\begin{equation*} N_t^T(w) = \sum_{k \geq 1} 1_{\{ T_k(w) \leq t \}},\end{equation*}

which is, for each realization w, a right-continuous step function with $N_0(w) = 0$ . As $(N_t)_{t \geq 0}$ and $(T_n)_{n \geq 0}$ carry the same information, the associated counting process is also called a point process.

The mathematical description of our observations is given by the internal family of sub- $\sigma$ -algebras of $\Im$ , denoted by $(\Im_t^T)_{t \geq 0}$ , where

\begin{equation*}\Im_t^T = \sigma \{1_{ \{ T_i >s \}} , i \geq 1, 0 < s < t \}\end{equation*}

satisfies the Dellacherie conditions of right-continuity and completeness.

For a mathematical basis for applied stochastic processes, one may refer to Aven and Jensen [Reference Aven and Jensen3]. In particular, an extended and positive random variable $\tau$ is an $\Im_t^T$ -stopping time if, and only if, $\{ \tau \leq t \} \in \Im_t^T$ , for all $t \geq 0$ ; an $\Im_t^T$ -stopping time $\tau$ is said to be predictable if an increasing sequence $(\tau_n)_{n \geq 0 }$ of $\Im_t^T$ -stopping times, $\tau_n < \tau $ , exists such that $\lim_{n\rightarrow \infty } \tau_n = \tau $ ; an $\Im_t^T$ -stopping time $\tau$ is totally inaccessible if $\mathbb{P}(\tau = \sigma < \infty) = 0$ for every predictable $\Im_t^T$ -stopping time $\sigma$ .

In what follows, we assume that relations between random variables and measurable sets always hold with probability one, which means that the term $\mathbb{P}$ -almost surely can be suppressed.

The point process $(N_t^{ {\bf T} })_{t \geq 0}$ is adapted to $ (\Im_t^{ {\bf T} } )_{t \geq 0}$ and $E[N_t^{ {\bf T} }| \Im_s^{ {\bf T} }] \geq N_s^{ {\bf T} } $ for $s < t$ ; that is, $N_t^{ {\bf T} }$ is an uniformly integrable $\Im_t^{ {\bf T} }$ -submartingale. Then, from the Doob–Meyer decomposition, there exists a unique right-continuous nondecreasing $\Im_t^{ {\bf T} }$ -predictable and integrable process $(A_t^{ {\bf T} })_{t \geq 0}$ , with $A_0^{ {\bf T} }= 0$ , such that $(M_t^{ {\bf T}})_{t \geq 0}$ , with $ N_t^{ {\bf T} } = A_t^{ {\bf T} } + M_t^{ {\bf T} }$ , is a uniformly integrable $\Im_t^{ {\bf T} }$ -martingale. In many cases, the $\Im_t^{ {\bf T} }$ -compensator, $(A_t^{ {\bf T} })_{t \geq 0}$ , of a counting process $(N_t^{ {\bf T} })_{t \geq 0}$ can be represented in the form of an integral as

\begin{equation*}A_t^{ {\bf T} } = \int_0^t \lambda_s^{ {\bf T} } ds\end{equation*}

for some non-negative ( $\Im_t^{ {\bf T} }$ -progressively measurable) stochastic process $(\lambda_t^{ {\bf T} })_{t \geq 0}$ , called the $\Im_t^{ {\bf T}}$ -intensity of $(N_t^{ {\bf T} })_{t \geq 0}$ .

The compensator process is expressed in terms of conditional probabilities, given the available information, and it generalizes the classical notion of hazard. Intuitively, it corresponds to whether the failure is going to occur now, on the basis of all observations available up to but not including the present time.

Following Aven and Jensen [Reference Aven and Jensen3], the compensator process is given by the following theorem.

Theorem 1. Let $(N_t^{ {\bf T} })_{t \geq 0}$ be an integrable point process and $(\Im_t^{ {\bf T} })_{t \geq 0}$ its internal history. Suppose that for each n there exists a regular conditional distribution of $T_{n+1}- T_n$ , given the past $ \Im_{T_n}^{ {\bf T} },$ of the form

\begin{equation*} G_n(w, A) = \mathbb{P}(T_{n+1} - T_n \in A| \Im_{T_n}^{ {\bf T} })(w) = \int_{A}g_n(w, s)ds,\end{equation*}

where $g_n(w, s)$ is a measurable function. Then the process given by $\lambda_ t^{{\bf T}} =\Sigma_{n=0}^{\infty} \lambda_t^n$ , where

\begin{equation*} \lambda_t^n = \frac{ g_n(t - T_n)}{G_n([t - T_n, \infty))}1_{\{ T_n < t \leq T_{n+1} \}} = \frac{ g_n(t - T_n)}{1 - \int_0^{t - T_n} g_n(s) ds} 1_{\{ T_n < t \leq T_{n+1} \}}, \end{equation*}

is called the $\Im_t^{ {\bf T} } $ -intensity of $N_t^{ {\bf T} }$ , and

\begin{equation*} N_t^{{\bf T}} - \int_0^t \lambda_s^{{\bf T}} ds \end{equation*}

is an $\Im_t^T$ -martingale.

We note that the compensator process

\begin{equation*} A_t^T = A_{T_n}^T + \int_0^{t - T_n} \lambda_s^n ds\end{equation*}

is defined by its parts.

Our aim now is to define a cumulative residual inaccuracy measure between two independent non-explosive point processes, T and S, and then to proceed to using a superposition process.

Now, we define

\begin{equation*}N_t(0) = \sum_{n=1}^{\infty} 1_{ \{U_n =0\}} 1{\{ V_n \leq t\}}\end{equation*}

as the number of occurrences of the process T and

\begin{equation*}N_t(1) = \sum_{n=1}^{\infty} 1_{ \{ U_n =1\}} 1{\{ V_n \leq t\}}\end{equation*}

as the number of occurrences of the process S on the superposition process.

The observed history is thus

\begin{equation*}\Im_t^V = \sigma \{ N_s(0), N_s(1), 0 \leq s < t\} = \sigma\{ (V_n, U_n), V_n <t\}. \end{equation*}

Theorem 2. Let $(V_n, U_n)_{n \geq 1}$ be a marked point process, the superposition of two univariate point processes ${\bf T} = (T_n)_{n \geq 0}$ and ${\bf S} = (S_n)_{n \geq 0}$ , defined in a complete probability space $(\Omega, \Im, \mathbb{P})$ with $\Im_t^V $ -compensator processes $(A_t)_{t \geq 0}$ and $(B_t)_{t \geq 0}$ , respectively. Furthermore, let $N_t(0)$ and $N_t(1)$ be the $\Im_t$ -submartingales defined as the number of occurrences of the processes T and S, respectively. Then $N_t(0)$ has $\Im_t$ -compensator

\begin{equation*}\displaystyle\sum_{n=1}^{\infty} \int_0^t 1_{\{ U_n = 0 \}} dA_s^n, \end{equation*}

and $N_t(1)$ has $\Im_t$ -compensator

\begin{equation*}\displaystyle\sum_{n=1}^{\infty} \int_0^t 1_{\{ U_n = 1 \}} dB_s^n .\end{equation*}

Proof. To prove this theorem, we use the known result that the integration of an $\Im_t$ -predictable process with respect to an integrable $\Im_t$ -martingale of bounded variation is an $\Im_t$ -martingale.

Observe that the deterministic process

\begin{equation*} 1_{\{ U_n = 0 \}}(w,s) = 1_{\{ U_n = 0 \}}(w)\end{equation*}

is left-continuous and, therefore, $\Im_t$ -predictable, implying that

\begin{equation*} \int_0^t 1_{\{ U_n = 0 \}}(s) dM_s^n,\end{equation*}

where $ M_t^n = 1_{ \{ T_n \leq t\}} - A_t^n$ , is an $\Im_t$ -martingale.

As the sum of $\Im_t$ -martingales is an $\Im_t$ -martingale, we readily have that

\begin{equation*} \displaystyle \sum_{n=1}^{\infty} \int_0^t 1_{\{ U_n = 0 \}} dM_s^n =\displaystyle \sum_{n=1}^{\infty} \int_0^t 1_{\{ U_n = 0 \}} d1_{\{ T_n \leq s \}} -\displaystyle\sum_{n=1}^{\infty} \int_0^t 1_{\{ U_n = 0 \}} dA_s^n\end{equation*}

is an $\Im_t$ -martingale. As the compensator is unique, the proof is readily completed. The proof for the $N_t(1)$ process follows in an analogous manner.

\begin{equation*} B_t = B_{S_n} + \int_0^{t-S_n} dB_s = B_{V_n^{{*}}} + \int_0^{t-V_n^{*}} dB_s, \ \ V_n < t \leq V_{n+1},\end{equation*}

where $V_n^{{*}} = \max_{1 \leq j < n} \{ U_j \cdot V_j \},$ and

\begin{equation*} A_t = A_{T_n} + \int_0^{t-T_n} dA_s = A_{V_n^{{*}}} + \int_0^{t-V_n^{*}} dA_s, \ \ V_n < t \leq V_{n+1},\end{equation*}

where $V_n^{{*}} = \max_{1 \leq j < n} \{ (1- U_j)\cdot V_j \}.$

In view of Definition 1, in the introduction, we define a cumulative inaccuracy measure between two univariate point processes at any $\Im_t$ -stopping time $\tau$ , in particular, at time t, as follows.

Definition 4. Let $ {\bf T} = (T_n)_{n \geq 0}$ and ${\bf S} = (S_n)_{n \geq 0}$ be univariate point processes with $\Im_t^V $ -compensator processes $(A_t)_{t \geq 0}$ and $(B_t)_{t \geq 0}$ , respectively, defined in a complete probability space $(\Omega, \Im, \mathbb{P})$ . Let $(V_n)_{n \geq 0}$ be their superposition process. Then the cumulative residual inaccuracy measure at time t between $ {\bf T}$ and $ {\bf S}$ is given by

\begin{equation*} CRI_t(N^{ {\bf T} }, N^{ {\bf S} }) = \mathbb{E}\left[ \int_0^t \displaystyle\sum_{n=1}^{\infty} \int_0^s 1_{\{ U_n = 0 \}} dA_u^n ds + \int_0^t \displaystyle\sum_{n=1}^{\infty} \int_0^s 1_{\{ U_n = 1 \}} dB_u^n ds \right].\end{equation*}

It is important to observe that, under Theorem 2.2, the indicator process is essential in Definition 4.

An interpretation of Definition 4 is given by the following theorem.

Theorem 3. Let $ {\bf T} = (T_n)_{n \geq 0}$ and ${\bf S} = (S_n)_{n \geq 0}$ be two non-explosive univariate point processes, and let ${\bf V} = (V_n)_{n \geq 0}$ be their superposition process. Then

\begin{equation*} \int_0^t A_s ds + \int_0^t B_s ds = \Sigma_{n=1}^{N_t^{{\bf V} }} [ 1_{ \{ U_n = 1 \}} + 1_{ \{ U_n = 0\}}] |V_n - V_{n-1}| = \Sigma_{n=1}^{N_t^{{\bf V} }} |V_n - V_{n-1}|,\end{equation*}

where

\begin{equation*} \{U_k = 1 \} = \bigcup_{j=1}^{\infty} \{ T_j \leq S_{k-1}\wedge t \} \bigcup\{ T_{j-1} < S_k \leq T_j \wedge t \},\end{equation*}

\begin{equation*} \{U_k = 0 \} = \bigcup_{k=1}^{\infty} \{S_k \leq T_{j-1}\wedge t \} \bigcup\{ S_{k-1} < T_j \leq S_k \wedge t \}.\end{equation*}

Proof. We let $(\tau_n^{ \bf T})_{n \geq 0}$ be an increasing sequence of $\Im_t$ -stopping times as the localizing sequence of the stopped martingale $( N_{t\wedge { \tau_n^{ \bf T}}}^{ \bf T)} - A_{t \wedge { \tau_n^{ \bf T} }} ) _{t \geq 0} $ , and let $(\tau_n^{ \bf S})_{n \geq 0}$ be an increasing sequence of $\Im_t$ -stopping times as the localizing sequence of the stopped martingale $( N_{t\wedge {\tau_n^{ \bf S} }}^{ \bf S} - B_{t \wedge {\tau_n^{ \bf S}}} ) _{t \geq 0}$ . We then apply the optimal sampling theorem; see Aven and Jensen [Reference Aven and Jensen3].

Note that $\tau_n = \tau_n^{ \bf T} \vee \tau_n^{ \bf S}$ is also an $\Im_t$ -stopping time and that the point process $(S_k)_{k \geq 0}$ defines a partition of $ \mathbb{\Re^+}$ , that is, $ \mathbb{\Re^+ }= \cup_{k=0}^{\infty} (S_{k-1}, S_k]$ . Therefore, we can write

\begin{equation*} \int_0^{\tau_n} A_s ds = \sum_{k=1}^{N_{\tau_n}^{{\bf T} }} \int_{S_{k-1}}^{S_k \wedge {\tau_n}} A_s ds = \sum_{k=1}^{N_{\tau_n}^{{\bf T}}} \int_{S_{k-1}}^{S_k \wedge {\tau_n}} \left( \int_0^s dA_u \right) ds \end{equation*}

\begin{equation*} = \sum_{k=1}^{N_{\tau_n}^{{\bf T} }} \left[ \int_0^{S_{k-1}} \left(\int_{S_{k-1}}^{S_k \wedge {\tau_n}} ds\right) dA_u + \int_{S_{k-1}}^{S_k \wedge {\tau_n}} \left( \int_u^{S_k \wedge t}ds\right) dA_u \right] \end{equation*}

\begin{equation*} = \sum_{k=1}^{N_{\tau_n}^{{\bf T} }} \left[ \int_0^{S_{k-1}} ({S_k \wedge {\tau_n}} - S_{k-1}) dA_u + \int_{S_{k-1}}^{S_k \wedge {\tau_n}} ( {S_k \wedge {\tau_n}} - u) dA_u \right].\end{equation*}

However, the compensator differential $dA_u$ is defined by parts and can be written as

\begin{equation*}dA_u = \sum_{j=1}^{\infty} 1{\{ T_{j-1} < u \leq T_j \}} dA_u(j),\end{equation*}

where $dA_u(j)$ is the differential compensator of $ 1_{\{ T_j \leq t\}}$ defined in $( T_{j-1}, T_j]$ , and 0 otherwise. It then follows that

\begin{equation*} \int_0^t A_s ds = \sum_{k=1}^{N_{\tau_n}^{{\bf T} }} \sum_{j=1}^{N_{\tau_n}^{{\bf S} }} \left[ \int_{T_{j-1}}^{S_{k-1} \wedge T_j} ({S_k \wedge {\tau_n}} - S_{k-1}) dA_u(j) + \int_{S_{k-1} \vee T_{j-1}}^{{S_k \wedge {\tau_n}} \wedge T_j} ( {S_k \wedge {\tau_n}} - u) dA_u(j) \right] \end{equation*}

\begin{equation*} = \sum_{k=1}^{N_{\tau_n}^{{\bf T} }} \sum_{j=1}^{N_{\tau_n}^{{\bf S} }} \left[ ({S_k \wedge {\tau_n}} - S_{k-1})1_{\{ T_j \leq S_{k-1} \wedge T_j \}} + ( {S_k \wedge {\tau_n}} - T_j)1_{\{S_{k-1} \vee T_{j-1} < T_j \leq {S_k \wedge {\tau_n}} \wedge T_j \}} \right] \end{equation*}

\begin{equation*} = \sum_{k=1}^{N_{\tau_n}^{{\bf T} }} \sum_{j=1}^{N_{\tau_n}^{{\bf S} }} [ ({S_k \wedge t} - S_{k-1})1_{\{ T_j \leq S_{k-1}\}} + ( {S_k \wedge {\tau_n}} - T_j) 1_{\{ S_{k-1} < T_j \leq {S_k \wedge {\tau_n}}\}} ] .\end{equation*}

Using similar arguments we can prove that

\begin{equation*} \int_0^{\tau_n} B_s ds = \sum_{k=1}^{N_{\tau_n}^{{\bf T} }} \sum_{j=1}^{N_{\tau_n}^{{\bf S} }} [ (T_j\wedge {\tau_n} - T_{j-1})1_{\{ S_k \leq T_{j-1}\}} + ( T_j \wedge {\tau_n} - S_k \wedge t) 1_{\{ T_{j-1} < S_k \leq T_j \wedge {\tau_n}\}} ] .\end{equation*}

Hence we have

\begin{equation*}\int_0^{\tau_n} A_s ds + \int_0^{\tau_n} B_s ds \end{equation*}

\begin{equation*} = \sum_{k=1}^{N_{\tau_n}^{{\bf T} }} \sum_{j=1}^{N_{\tau_n}^{{\bf S} }} [ ({S_k \wedge t} - S_{k-1})1_{\{ T_j \leq S_{k-1}\}} + ( {S_k \wedge {\tau_n}} - T_j) 1_{\{ S_{k-1} < T_j \leq {S_k \wedge {\tau_n}}\}} ] \end{equation*}

\begin{equation*} + \sum_{k=1}^{N_{\tau_n}^{{\bf T} }} \sum_{j=1}^{N_{\tau_n}^{{\bf S} }} [ (T_j\wedge {\tau_n} - T_{j-1})1_{\{ S_k \leq T_{j-1}\}} + ( T_j \wedge {\tau_n} - S_k \wedge t) 1_{\{ T_{j-1} < S_k \leq T_j \wedge {\tau_n}\}} ] \end{equation*}

\begin{equation*} = \Sigma_{n=1}^{N_{\tau_n}^{{\bf V} }} [ 1_{ \{ U_n = 1 \}} + 1_{ \{ U_n = 0\}}]|V_n - V_{n-1}| = \Sigma_{n=1}^{N_{\tau_n}^{{\bf V} }} |V_n - V_{n-1}|,\end{equation*}

which is the sum of the inter-arrival times of the superposition process, where

\begin{equation*} \{U_k = 1 \} = \bigcup_{j=1}^{\infty} \{ T_j \leq S_{k-1}\wedge {\tau_n} \} \bigcup\{ T_{j-1} < S_k \leq T_j \wedge {\tau_n}\},\end{equation*}

\begin{equation*} \{U_k = 0 \} = \bigcup_{k=1}^{\infty} \{S_k \leq T_{j-1}\wedge {\tau_n} \} \bigcup\{ S_{k-1} < T_j \leq S_k \wedge {\tau_n} \}.\end{equation*}

As $( N_{t\wedge \tau}^{\bf V}) _{t \geq 0} $ is uniformly integrable, we let $\lim_{n \rightarrow \infty} \tau_n = \infty$ , and provided that we identify random variables that are equal almost everywhere, the quantity $CRI_{\infty}(N^{ {\bf T} }, N^{ {\bf S} }) = \Sigma_{k=1}^{\infty} |V_k - V_{k-1}| $ , as t goes to infinity, can be seen as a dispersion measure in the $L^1$ space of sequences of random variables when using the point process S, which represents the information asserted by the experimenter about the true point process ${ \bf T}$ .

2.2. Application to minimal repair point processes

A repair is minimal if the intensity $\lambda_t^T$ is not affected by the occurrence of failures, or, in other words, if we cannot determine the failure time points from observation of $\lambda_t^T$ . Formally, we have the following definition.

If T is a non-homogeneous Poisson process, $\lambda_t = \lambda(t)$ is a time-dependent deterministic function, and this means that the age does not get changed as the result of a failure. Here, $ \Im_t^{\lambda^T} = \{ \Omega, \emptyset \}$ for all $ t \in \mathbb{R^+}$ , and the failure times $T_n$ are not $\Im_t^{\lambda^T}$ -stopping times.

In practice, we consider the ordered lifetimes $T_1, \ldots ,T_n $ with a conditional reliability function given by

\begin{equation*} \overline{G_i}(t_i|t_1,\ldots ,t_{i-1}) = \exp{ \left[{-} \left( \frac{t_i}{\theta_1}\right)^{\beta} + \left( \frac{t_{i-1}}{\theta_1}\right)^{\beta} \right] } \end{equation*}

for $0 \leq t_{i-1} < t_i$ , where the $t_i$ are the ordered observations. The $\Im^T$ -compensator process is then

\begin{equation*}A_t = \sum_{j=1}^n \left[\left(\frac{t_j}{\theta_1}\right)^{\beta} - \left(\frac{t_{j-1}}{\theta_1}\right)^{\beta} \right] + \left[\left(\frac{t}{\theta_1}\right)^{\beta} - \left(\frac{t_n}{\theta_1}\right)^{\beta} \right] = \left(\frac{t}{\theta_1}\right)^{\beta}, \ \ t_n \leq t < t_{n+1}.\end{equation*}

Furthermore, with respect to $ (S_n)_{n \geq 0 }$ , the $\Im^S$ -compensator process is

\begin{equation*} B_t = \sum_{j=1}^n \left[ \left(\frac{s_j}{\theta_2}\right)^{\beta} - \left( \frac{s_{j-1}}{\theta_2}\right)^{\beta} \right] + \left[\left(\frac{t}{\theta_2}\right)^{\beta} - \left(\frac{s_n}{\theta_2}\right)^{\beta} \right] = \left( \frac{t}{\theta_2}\right)^{\beta} , \ \ s_n \leq t < s_{n+1}.\end{equation*}

Note that the compensator process in a minimal repair point process is independent of the occurrence times, in which case the indicator process does not apply. Therefore, the cumulative inaccuracy measure at time t is given by

\begin{equation*} CRI_t(N^T, N^S) = \mathbb{E}\left[ \int_0^t A_s ds + \int_0^t B_s ds \right] = \mathbb{E}\left[ \int_0^t \left(\frac{s}{\theta_1}\right)^{\beta} ds + \int_0^t \left(\frac{s}{\theta_2}\right)^{\beta} ds \right] \end{equation*}

\begin{equation*} = \frac{t^{\beta+1}}{\beta+1} \left(\frac{\theta_1^{\beta} + \theta_2^{\beta}}{\theta_1^{\beta} \theta_2^{\beta}}\right).\end{equation*}

Example 2. (Application to a coherent system minimally repaired at component level.) Suppose we observe, as in Barlow and Proschan [Reference Barlow and Proschan4], the lifetimes of a system with three components, $U_1$ , $U_2$ , and $U_3$ , which are independent and identically exponentially distributed with parameter $\lambda$ , through the filtration

\begin{equation*} \Im_t = \sigma \{ 1_{\{U_1>s\}}, 1_{\{U_2>s\}}, 1_{\{U_3>s\}} , 0 \leq s \leq t \}.\end{equation*}

The system with lifetime $T_1 = U_1 \wedge ( U_2 \vee U_3 )$ has intensity $\lambda_t^{T_1} = \lambda+ \lambda 1_{\{ U_2 \wedge U_3 \leq t \}}$ , and clearly $T_1$ is not an $\Im_t^{\lambda^{T_1}}$ -stopping time.

At system failure $T_1$ , the component that causes the system to fail is repaired minimally. As the component lifetimes are independent and identically distributed, the additional lifetime, given by the lifetime $U_4$ , is independent of and distributed identically to $U_1$ , $U_2$ , and $U_3$ , and the repaired system then has lifetime $T_2 = T_1 + U_4.$

We allow for repeated minimal repairs considering a sequence of random variables, $(U_n)_{n \geq 1}$ , that are independent and identically exponentially distributed with parameter $\lambda$ . Then $ T_1 = U_1 \wedge (U_2 \vee U_3) $ and $ T_{n+1} = T_n + U_{n+3},$ $ n \geq 2$ , successively, constituting a minimal repair point process with compensator

\begin{equation*}A_t = \int_0^t\lambda_s^T ds = \int_0^t \left[ \lambda + \lambda 1_{\{ U_2 \wedge U_3 \leq s \}}\right] ds = 2 \lambda t - \lambda ( U_2 \wedge U_3 ) \ \ \text{if} \ \ t \leq T, \end{equation*}

where T is the actual system lifetime.

Let us now consider another minimally repaired coherent system, $S_1$ , asserted by the experimenter, with the same structure function, but component lifetimes $V_1$ , $V_2$ , and $V_3$ , which are independent and identically exponentially distributed with parameter $\lambda^*$ and compensator process

\begin{equation*}B_t = \int_0^t\lambda_s^S ds = 2 \lambda^* t - \lambda^* ( V_2 \wedge V_3 ) \ \ \text{if} \ \ t \leq S, \end{equation*}

where S is the actual system lifetime.

Then, in the set $\{ t \leq T \wedge S \}$ , the cumulative inaccuracy measure at time t is given by

\begin{equation*} CRI_t(N^T, N^S)= \mathbb{E}\left[ \int_0^t A_s ds + \int_0^t B_s ds \right] = \end{equation*}

\begin{equation*} \mathbb{E}\left[ \int_0^t [2 \lambda s - \lambda ( U_2 \wedge U_3 ) ds + \int_0^t [ 2 \lambda^* s - \lambda^* ( V_2 \wedge V_3 )] ds\right] \end{equation*}

\begin{equation*} = \lambda t^2 + \lambda t E[ U_2 \wedge U_3] + \lambda^* t^2 + \lambda t E[ V_2 \wedge V_3] = (\lambda + \lambda^*) t^2 - t. \end{equation*}

Clearly, the expression for $ CRI_t(N^T, N^S)$ can be negative. However, we observe that, always, the superposition process is minimally repaired with an exponential lifetime with mean $\frac{1}{\lambda + \lambda^*}$ . We then consider $t \geq \frac{1}{\lambda + \lambda^*}$ , resulting in a positive $ CRI_t(N^T, N^S)$ .

Also, in the minimally repaired coherent system, the compensator process is independent of occurrence times, in which case the indicator process does not apply.

3.1. Inaccuracy measure between occurrence times in Markov chains

Let $(X_t)_{t \geq 0}$ be an E-valued process defined in a probability space $(\Omega,\Im, \mathbb{P})$ and adapted to some history $(\Im_t)_{t \geq 0}$ . The observations are through its internal history

\begin{equation*}\Im_t^X = \sigma\{ X_s, s \leq t\}\end{equation*}

for all $t \geq 0$ , and $ \Im_t^X \subseteq \Im_t$ for all $t \geq 0$ . Then $\Im_{\infty}^X $ records all the events linked to the process $(X_t)_{t \geq 0}$ .

The process $(X_t)_{t \geq 0}$ is said to be an $\Im_t$ -Markov process if, and only if, for all $t \geq 0$ , $ \sigma(X_s, s > t)$ and $\Im_t$ are independent, given $ X_t$ . In particular, if E is the set of natural numbers ${\bf N^+}$ , we call $(X_t)_{t \geq 0}$ an $\Im_t$ -Markov chain. In what follows, we consider a right-continuous Markov chain with left limits.

The $\Im_t$ -Markov chain is associated with a sequence of its sojourn times $ (T_{n+1} - T_n)_{n \geq 0}$ , with $ T_0 = 0$ , and its infinitesimal characteristics $Q = [q_{i,j}]$ , $(i,j) \in {\bf N^+} \times {\bf N^+}$ . If, for each natural number i, we have

\begin{equation*}q_i = \sum_{j \neq i} q_{i,j} < \infty,\end{equation*}

then the chain is said to be stable and conservative. We set $q_{i,i} = - q_i$ .

The interpretation of the terms $q_i$ and $q_{i,j}$ is as follows:

\begin{equation*}\mathbb{P}(T_{n+1} - T_n > t | \Im_{T_n}^X) = e^{-t q_{X_{T_n}}}, \ \ t > 0,\end{equation*}

\begin{equation*}\mathbb{P}(X_{T_{n+1}} = i, T_{n+1} - T_n > t | \Im_{T_n}^X) = e^{-t q_{X_{T_n}}}\left( \frac{q_{X_{T_n},i}}{q_{X_{T_n}}}\right), \ \ t > 0.\end{equation*}

We are now interested in the cumulative inaccuracy process between point processes related to Markov chain occurrence times, with

• • $N_t^X(k, l)$ denoting the number of transitions from state k to state l in the interval (0, t],

• • $N_t^X(l)$ denoting the number of transitions into state l during the interval (0, t], and

• • $N_t^X$ denoting the number of transitions in (0, t].

Now, the occurrence observation times are through the internal family of sub- $\sigma$ -algebras of  $\Im$ ,

\begin{equation*}\Im_t^{\bf T} = \sigma \{ N_s^{\bf T}, 0 \leq s < t \} \subseteq \Im_t^X \subseteq \Im_t, \end{equation*}

and satisfy the Dellacherie conditions of right-continuity and completeness. This family of counts leads to the same information as the sequence of occurrence times $ {\bf T}$ and hence provides an equivalent point process description. Clearly, from the sojourn times interpretation, the $ T_n,$ $n > 0$ , are totally inaccessible $\Im_t$ -stopping times. In a certain way, absolutely continuous lifetimes are totally inaccessible $\Im_t^{ {\bf T} }$ -stopping times.

Let $(X_t)_{t \geq 0}$ be a right-continuous ${\bf N^+}$ -valued $\Im_t$ -Markov chain with left-hand limits which is stable and conservative, with associated sequence of occurrence times ${\bf T} = (T_n)_{n \geq 1}$ and matrix of infinitesimal characteristics $ Q = [q_{i,j}], i, j \in {\bf N^+} $ . Then a martingale property of this process is given in the following theorem.

Theorem 4. Let $(X_t)_{t \geq 0}$ be a right-continuous ${\bf N^+}$ -valued $\Im_t$ -Markov chain with left-hand limits, which is stable and conservative with associated matrix $ Q = [q_{i,j}], i, j \in {\bf N^+}$ . Let f be a non-negative function from ${\bf N^+} \times {\bf N^+}$ to $\Re_+$ , with

\begin{equation*}\mathbb{E}\left[\int_0^t \sum_{j \neq X_u}q_{X_u,j}|f(X_u,j)| du \right] < \infty.\end{equation*}

Then

\begin{equation*} \sum_{0 < u \leq t} f(X_{u^-},X_u)- \int_0^t \sum_{j \neq X_u}q_{X_u,j}f(X_u,j) du \end{equation*}

is an $\Im_t$ -martingale.

Proof. The argument to be used is the Lévy formula, which states that if f is a non-negative function from $ {\bf N^+} \times {\bf N^+}$ to $\mathbb{R}^+$ , then for any $0 \leq s \leq t$ we have

\begin{equation*} \mathbb{E}\left[ \sum_{s< u \leq t} f(X_{u^-},X_u) | \sigma(X_s)\right] = \mathbb{E}\left[\int_s^t \sum_{j \neq X_u}q_{X_u,j}f(X_u,j) du | \sigma(X_s)\right].\end{equation*}

By the $\Im_t$ -Markov property of $X_t$ , i.e., the conditional independence of $\Im_s$ and $(X_u, u > s)$ given $X_s$ , conditioning on $\sigma(X_s)$ in the above equation becomes equivalent to conditioning with respect to $\Im_s$ , and so

\begin{equation*} \mathbb{E}\left[ \sum_{s< u \leq t} f(X_{u^-},X_u) | \Im_s\right] = \mathbb{E}\left[\int_s^t \sum_{j \neq X_u}q_{X_u,j}f(X_u,j) du | \Im_s \right].\end{equation*}

Now, if $f\,:\, {\bf N^+} \times {\bf N^+} \rightarrow \mathbb{R}^+$ is such that

\begin{equation*}\mathbb{E}\left[\int_0^t \sum_{j \neq X_u}q_{X_u,j}|f(X_u,j)| du\right] < \infty,\end{equation*}

we have

\begin{equation*} \mathbb{E}\left[ \sum_{0 < u \leq t} f(X_{u^-},X_u) | \Im_s\right] - \mathbb{E}\left[ \sum_{0 < u \leq s} f(X_{u^-},X_u) | \Im_s \right] \end{equation*}

\begin{equation*} = \mathbb{E}\left[ \sum_{s< u \leq t} f(X_{u^-},X_u) | \Im_s\right] = \mathbb{E}\left[\int_s^t \sum_{j \neq X_u}q_{X_u,j}f(X_u,j) du | \Im_s \right] \end{equation*}

\begin{equation*} = \mathbb{E}\left[\int_0^t \sum_{j \neq X_u}q_{X_u,j}f(X_u,j) du | \Im_s\right] - \mathbb{E}\left[\int_0^s \sum_{j \neq X_u}q_{X_u,j}f(X_u,j) du | \Im_s\right].\end{equation*}

As $ \int_0^s \sum_{j \neq X_u}q_{X_u,j}f(X_u,j) du $ is $\Im_s$ -measurable, we can conclude that

\begin{equation*} \mathbb{E}\left[ \sum_{0 < u \leq t} f(X_{u^-},X_u)- \int_0^t \sum_{j \neq X_u}q_{X_u,j}f(X_u,j) du | \Im_s\right] \end{equation*}

\begin{equation*} = \sum_{0 < u \leq s} f(X_{u^-},X_u)- \int_0^s \sum_{j \neq X_u}q_{X_u,j}f(X_u,j) du;\end{equation*}

that is,

\begin{equation*} \sum_{0 < u \leq t} f(X_{u^-},X_u)- \int_0^t \sum_{j \neq X_u}q_{X_u,j}f(X_u,j) du \end{equation*}

is an $\Im_t$ -martingale.

From Theorem 3.1, we have that

\begin{equation*}N_t(n,n+1)-\lambda_n \int_0^t 1_{\{ X_s = n\}}ds \end{equation*}

is an $\Im_t$ -martingale.

As $\sum_{n=0}^{\infty} \int_0^t \lambda_n \mathbb{P}(X_s = n) ds < \infty $ , we have that

\begin{equation*} \sum_{n=0}^{\infty} N_t(n,n+1) - \sum_{n=0}^{\infty} \lambda_n \int_0^t 1_{\{ X_s = n\}}ds\end{equation*}

is an $\Im_t$ -martingale.

It then follows from Definition 4 that the cumulative inaccuracy measure at time t between $ N_t^+$ and $N_t^{+*} $ is

\begin{equation*} CRI_t( N_t^+), N_t^{+*}) = E\left[ \sum_{n=0}^{\infty} \int_0^t \left( \lambda_n \int_0^s 1_{\{U_{X_u} = 0 \}} 1_{\{ X_u = n\}}du \right) ds \right] \end{equation*}

\begin{equation*} + E\left[\sum_{n=0}^{\infty} \int_0^t \lambda_n^* \left( \int_0^s 1_{\{U_{X_u} = 1 \}} 1_{\{ X_u = n\}}du \right) ds \right].\end{equation*}

Furthermore, the number of downward jumps in (0, t] is $N_t^- = \sum_{n=0}^{\infty} N_t(n,n-1)$ , associated with the $\Im_t$ -martingale

\begin{equation*} N_t(n,n-1) - \mu_n \int_0^t 1_{\{ X_s = n \}} ds.\end{equation*}

As $\mathbb{E}[X_0] < \infty$ and $\mathbb{E}[ \sum_{n=0}^{\infty} N_t(n,n+1)] < \infty,$ we have

\begin{equation*}\mathbb{E}\left[ \sum_{n=1}^{\infty} N_t(n, n-1)\right] \leq \mathbb{E}\left[ \sum_{n=0}^{\infty} N_t(n,n+1)\right] + \mathbb{E}[X_0] < \infty. \end{equation*}

So we have

\begin{equation*} \sum_{n=1}^{\infty} N_t(n, n-1) - \sum_{n=1}^{\infty} \mu_n \int_0^t 1_{\{ X_s = n \}} ds \end{equation*}

as an $\Im_t$ -martingale.

It then follows from the definition that the cumulative inaccuracy measure at time t between $ N_t^-$ and $N_t^{-*}$ is

\begin{equation*} CRI_t( N_t^-), N_t^{-*}) = \mathbb{E}\left[ \sum_{n=1}^{\infty} \int_0^t \mu_n \left( \int_0^s 1_{\{U_{X_u} = 0 \}} 1_{\{ X_u = n \}} du \right) ds \right] \end{equation*}

\begin{equation*} + \mathbb{E}\left[\sum_{n=1}^{\infty} \int_0^t\sum_{n=1}^{\infty} \mu_n^* \left( \int_0^s 1_{\{U_{X_u} = 1 \}} 1_{\{ X_u = n \}} du \right) ds \right].\end{equation*}

The cumulative inaccuracy measure at time t of the birth-and-death process is

\begin{equation*} CRI_t( N_t, N_t^*) = CRI_t( N_t^+, N_t^{+*}) + CRI_t( N_t^-, N_t^{-*}).\end{equation*}

A birth-and-death process is a pure birth process if $\mu_n = 0$ for all n. If the birth rate $\lambda_n = \lambda$ , for all n, the pure birth process is simply a Poisson process.

As above, we have that

\begin{equation*}N_t = \sum_{n=0}^{\infty} N_t(n,n+1) - \sum_{n=0}^{\infty} \lambda\int_0^t 1_{\{ X_s = n\}}ds\end{equation*}

is an $\Im_t$ -martingale.

To evaluate the cumulative inaccuracy measure at time t between the counting process $ N_t^*$ related to the Markov chain $(X^*_t)_{t \geq 0}$ with parameter $\lambda^*$ —which represents the experimenter’s information about the true Markov chain $(X_t)_{t \geq 0}$ having parameter $ \lambda$ —and the counting process $ N_t$ which corresponds to the latter, using Definition 4 we obtain

\begin{align*}& CRI_t( N_t, N_t^*) \\& \qquad = \mathbb{E}\left[ \sum_{n=0}^{\infty} \int_0^t \left( \int_0^s 1_{ \{U_n = 0 \}} \lambda 1_{\{ X_u = n \}} du \right) ds\right] +\mathbb{E}\left[ \sum_{n=0}^{\infty} \int_0^t \left( \int_0^s 1_{ \{U_n = 1 \}} \lambda^* 1_{\{ X_u = n \}} du \right) ds\right] \\ & \qquad = \sum_{n=0}^{\infty} \int_0^t\left( \int_0^s \frac{\lambda}{\lambda + \lambda^* } \lambda \mathbb{P}(X_u = n) du \right) ds +\sum_{n=0}^{\infty} \int_0^t\left( \int_0^s \frac{\lambda^*}{\lambda + \lambda^* } \lambda^* \mathbb{P}(X_u = n) du \right) ds \\& \qquad = \sum_{n=0}^{\infty} \int_0^t \left( \int_0^s \frac{\lambda}{\lambda + \lambda^*} \lambda e^{-\lambda u} \frac{(\lambda u)^n}{n!} du \right) ds + \sum_{n=0}^{\infty} \int_0^t \left( \int_0^s \frac{\lambda*}{\lambda + \lambda^*} \lambda^* e^{-\lambda^* u} \frac{(\lambda^* u)^n}{n!} du\right) ds. \end{align*}

Example 4. If in particular, in Theorem 3.1, we choose f to be $f(i,j) = 1_{\{ i=k \}} 1_{\{ j=l \}},$ then

\begin{equation*} \sum_{0 < u \leq t}f(X_{u_-},X_u) = N_t(k,l)\end{equation*}

is the number of transitions from state k to state l in the interval $(0, t].$ The $\Im_t$ -compensator of $N_t(k,l)$ is $ q_{k,l} \int_0^t 1_{\{ X_u =k\}}du$ , and

\begin{equation*}\mathbb{E}\left[q_{k,l}\int_0^t 1{\{ X_u =k\}}du\right] = q_{k, l} \int_0^t \mathbb{P}(X_u = k) du \leq q_{k,l} t < \infty,\end{equation*}

by hypothesis.

To evaluate the cumulative inaccuracy measure at time t between the counting process $ N_t^*(k,l)$ related to the Markov chain $(X^*_t)_{t \geq 0}$ with infinitesimal matrix $ Q^* = [q_{i,j}^*]$ , $i, j \in {\bf N^+} $ —which represents the experimenter’s information about the true Markov chain $(X_t)_{t \geq 0}$ having infinitesimal matrix $ Q = [q_{i,j}]$ , $i, j \in {\bf N^+}$ —and the counting process $ N_t(k,l)$ which corresponds to the latter, using Definition 4 we obtain

\begin{equation*} CRI_t(N(k,l), N(k,l)^*) = \mathbb{E}\left[ \int_0^t q_{k,l} \left( \int_0^s 1_{\{U_k = 0 \}} 1_{\{ X_u =k\}}du \right) ds\right] \end{equation*}

\begin{equation*} + \mathbb{E}\left[ \int_0^t q_{k,l}^* \left( \int_0^s 1_{\{U_k = 1 \}} 1_{\{ X_u =k\}}du \right) ds \right].\end{equation*}

Once again, as in Remark 2, we observe the importance and essence of the indicator process in Definition 4.

Example 5. Another interesting choice of f is given by $f(i,j) = 1_{\{ j=l\}}$ , in which case

\begin{equation*} \sum_{0 < u \leq t}f(X_{u_-},X_u) = N_t(l)\end{equation*}

is the number of transitions into state l during the interval $(0, t].$ The $\Im_t$ -compensator of $N_t(l)$  is

\begin{equation*}\int_0^t \sum_{i\neq l} q_{i,l} 1_{\{ X_u = i \}} du,\end{equation*}

provided

\begin{equation*}\sum_{i\neq l}\int_0^t q_{i,l} 1_{\{ X_u = i\}} du < \infty. \end{equation*}

To evaluate the cumulative inaccuracy measure at time t between the counting process $ N_t^*(l)$ related to the Markov chain $(X^*_t)_{t \geq 0}$ with infinitesimal matrix $ Q^* = [q_{i,j}^*], i, j \in {\bf N^+}$ , which represents the experimenter’s information about the true Markov chain $(X_t)_{t \geq 0}$ that has infinitesimal matrix $ Q = [q_{i,j}]$ , $i, j \in {\bf N^+}$ , and the counting process $ N_t(l)$ which corresponds to the latter, using Definition 4 we have

\begin{equation*} CRI_t(N(l), N^*(l)) = \mathbb{E}\left[ \int_0^t \left( \int_0^s \sum_{i\neq l} 1_{\{U_i = 0 \}} q_{i,l} 1_{\{ X_u = i \}} du \right) ds\right] \end{equation*}

\begin{equation*}+ \mathbb{E}\left[\int_0^t \left( \int_0^s \sum_{i\neq l} 1_{\{U_i = 1 \}} q_{i,l}^* 1_{\{ X_u = i \}} du \right) ds \right].\end{equation*}

Here again, as in Remark 2, we observe the importance and essence of the indicator process in Definition 4.

3.2. Markov chain characterization through cumulative inaccuracy measure for a proportional risk process

A Markov chain characterization through the cumulative inaccuracy measure for a proportional risk process can be obtained as follows.

Let $(X_t)_{t \geq 0}$ be a right-continuous ${\bf N^+}$ -valued $\Im_t$ -Markov chain that is stable and conservative, with left-hand limits, associated occurrence times ${ \bf T} = (T_n)_{n \geq 0} $ with $T_0 = 0$ , and matrix of infinitesimal characteristics $Q = [q_{i,j}]$ , $(i,j) \in {\bf N^+} \times {\bf N^+}$ . Let the sequence of occurrence times ${ \bf S} = (S_n)_{n \geq 0}$ , with $S_0 = 0$ , and the matrix of infinitesimal characteristics $Q^* = [q_{i,j}^*]$ , $(i,j) \in {\bf N^+} \times {\bf N^+},$ be asserted as information about the true Markov chain. We say that T and S satisfy the proportional risk hazard process if $ q_{i,j}^* = \alpha q_{i,j}$ , $0 < \alpha \leq 1,$ for all $ (i,j) \in {\bf N^+} \times {\bf N^+}$ .

Proof. Let $(X_t)_{t \geq 0}$ be a right-continuous ${\bf N }_+ $ -valued $\Im_t$ -Markov chain with associated occurrence times ${ \bf T}^k$ and infinitesimal characteristics $Q^k = [q_{i,j}^k]$ , $(i,j) \in {\bf N^+} \times {\bf N^+}$ , and let ${ \bf S}^k$ be the occurrence times with infinitesimal characteristics $Q^{k*} = [q_{i,j}^{k*}]$ , $(i,j) \in {\bf N^+} \times {\bf N^+}$ , for $k=1,2.$ We then have

\begin{equation*} CRI_t(N^1, N^{1*}) = CRI_t(N^2, N^{2*}) \end{equation*}

\begin{equation*} \leftrightarrow\mathbb{E}\left[\int_0^t \int_0^s \sum_{j \neq X_u} 1_{\{U_{X_u} = 0 \}} q_{X_u,j}^1 f(X_u,j) du ds \right] \end{equation*}

\begin{equation*} + \mathbb{E}\left[ \int_0^t \int_0^s \sum_{j \neq X_u} 1_{\{U_{X_u} = 1 \}} \alpha^1 q_{X_u,j}^{1* } f(X_u,j) du ds \right] \end{equation*}

\begin{equation*} = \mathbb{E}\left[\int_0^t \int_0^s \sum_{j \neq X_u} 1_{\{U_{X_u} = 0 \}} q_{X_u,j}^2 f(X_u,j) du ds\right] \end{equation*}

\begin{equation*} + \mathbb{E}\left[ \int_0^t \int_0^s \sum_{j \neq X_u} 1_{\{U_{X_u} = 1 \}} \alpha^2 q_{X_u,j}^{2* } f(X_u,j) du ds\right].\end{equation*}

However, as the infinitesimal characteristics $q_{X_{T_n^k},X_{T_{n+1}^k}}^k$ are associated with occurrence times ${\bf T}^k$ , we have

\begin{equation*}\sum_{n}q_{X_{T_n^k},X_{T_{n+1}^k}}^k = 1,\end{equation*}

implying that

\begin{equation*} \sum_{n}q_{X_{S_n^k},X_{S_{n+1}^{k*}}} =0 ,\end{equation*}

and

\begin{equation*}q_{X_{S_n^k},X_{S_{n+1}^k}}^k =0 \end{equation*}

for all n and $k=1,2$ ; that is, the process does not jump at time $S_n$ :

\begin{equation*} \int_0^t \int_0^s \sum_{j \neq X_u} 1_{\{U_{X_u} = 1 \}} \alpha^k q_{X_u,j}^k f(X_u,j) du ds] = 0, \ \ k=1,2. \end{equation*}

Hence, the result follows from the equivalence equations given by

\begin{equation*} CRI_t(N^1, N^{1*}) = CRI_t(N^2, N^{2*}) \end{equation*}

\begin{equation*} \leftrightarrow\mathbb{E}\left[\int_0^t \int_0^s \sum_{j \neq X_u} 1_{\{U_{X_u} = 0 \}} q_{X_u,j}^1 f(X_u,j) du ds \right] \end{equation*}

\begin{equation*} = \mathbb{E}\left[\int_0^t \int_0^s \sum_{j \neq X_u} 1_{\{U_{X_u} = 0 \}} q_{X_u,j}^2 f(X_u,j) du ds\right] \end{equation*}

\begin{equation*} \leftrightarrow \mathbb{E}\left[\int_0^t \int_0^s \sum_{j \neq X_u} 1_{\{U_{X_u} = 0 \}} [q_{X_u,j}^1 - q_{X_u,j}^2] f(X_u,j) du ds \right] = 0 \end{equation*}

\begin{equation*} \leftrightarrow \mathbb{E}\left[ \int_0^t \int_0^s \sum_{j \neq X_u} 1_{\{U_{X_u} = 0 \}} [q_{X_u,j}^1 - q_{X_u,j}^2]( 1_{\{q_{X_u,j}^1 > q_{X_u,j}^2 \}}\right] \end{equation*}

\begin{equation*} + \mathbb{E}\left[ 1_{\{q_{X_u,j}^1 \leq q_{X_u,j}^2 \}})f(X_u,j) du ds \right] = 0 \leftrightarrow \end{equation*}

\begin{equation*} \mathbb{E}\left[ \int_0^t \int_0^s \sum_{j \neq X_u} 1_{\{U_{X_u} = 0 \}} [q_{X_u,j}^1 - q_{X_u,j}^2] 1_{\{q_{X_u,j}^1 > q_{X_u,j}^2 \}} f(X_u,j) du ds \right] = \end{equation*}

\begin{equation*}\mathbb{E}\left[\int_0^t \int_0^s \sum_{j \neq X_u} 1_{\{U_{X_u} = 0 \}} [q_{X_u,j}^2 - q_{X_u,j}^1] 1_{\{q_{X_u,j}^1 \leq q_{X_u,j}^2 \}} f(X_u,j) du ds\right] \end{equation*}

\begin{equation*} \leftrightarrow\mathbb{E}\left[\int_0^t \int_0^s \sum_{j \neq X_u} 1_{\{U_{X_u} = 0 \}} |q_{X_u,j}^1 - q_{X_u,j}^2| ( 1_{\{q_{X_u,j}^1 > q_{X_u,j}^2 \}} - 1_{\{q_{X_u,j}^1 \leq q_{X_u,j}^2 \}}) f(X_u,j) du ds \right]= 0. \end{equation*}

As $ \{q_{X_u,j}^1 > q_{X_u,j}^2 \} \cap \{q_{X_u,j}^1 \leq q_{X_u,j}^2 \} = \emptyset$ and the integrand is positive in the above equation, we have $ |q_{X_u,j}^1 - q_{X_u,j}^2| = 0 $ for all $(i,j) \in {\bf N^+} \times {\bf N^+},$ and thus $ Q^1 = Q^2$ characterize the same process $(X_t)_{t \geq 0}$ .

3.3. Cumulative inaccuracy measure between coherent systems

We now consider a coherent system with n independent components, as in Barlow and Proschan [Reference Barlow and Proschan4], subject to failures and repairs according to a birth-and-death process. The state of component i, for $1 \leq i \leq n$ , denoted by $X_t(i)$ , assumes values in the space $\{0, 1\}$ , where 1 represents an operating state and 0 a repair state. Starting from an operating state, component i continues operating for a length of time that is exponentially distributed with parameter $\lambda(i)$ , and starting from a repair state, component i continues in repair for a length of time that is exponentially distributed with parameter $\mu(i)$ . We observe

\begin{equation*} \sigma \{ X_s(i), 1 \leq i \leq s, 0 \leq s < t\}.\end{equation*}

The coherent system is known to depend on its component vector $ {\bf X}_t = (X_t(1), X_t(2), \ldots,X_t(n))$ through its structure function

\begin{equation*} \phi_t = \phi({\bf X}_t) = \phi (X_t(1), X_t(2), \ldots,X_t(n)),\end{equation*}

which is a monotone increasing function, and each component in the system is relevant; that is, there is a time t and a configuration of component states $ {\bf X}_t $ such that

\begin{equation*} \phi(1_i, {\bf X}_t) - \phi(0_i, {\bf X}_t)= 1,\end{equation*}

where a 1 (resp. 0) in position i denotes an operating (resp. repair) state.

If we set $N_t(i)$ to be the number of failed components i up to time t, from the Doob–Meyer decomposition we know that

\begin{equation*} N_t(i) - \int_0^t \lambda(i) X_s(i) ds \end{equation*}

is an $\Im_t$ -martingale. Furthermore, the number of system failures up to t is given by

\begin{equation*}N_t(\phi) = \sum_{i=1}^n \int_0^t[ \phi(1_i, {\bf X}_s) - \phi(0_i, {\bf X}_s)] dN_s(i).\end{equation*}

Let us now consider the process $ \phi(1_i, {\bf X}_{s_-}) - \phi(0_i, {\bf X}_{s_-})$ to be a predictable version of $\phi(1_i, {\bf X}_s) - \phi(0_i, {\bf X}_s)$ . Because at a jump of $N_t(i)$ no other components change their status, we have that

\begin{equation*} N_t(\phi) - \sum_{i=1}^n \int_0^t [ \phi(1_i, {\bf X}_s) - \phi(0_i, {\bf X}_s)] \lambda(i)X_s(i) ds \end{equation*}

\begin{equation*} = N_t(\phi) - \sum_{i=1}^n \int_0^t [ \phi(1_i, {\bf X}_{s_-}) - \phi(0_i, {\bf X}_{s_-})]\lambda(i)X_s(i) ds \end{equation*}

\begin{equation*} = \sum_{i=1}^n \int_0^t [ \phi(1_i, {\bf X}_s) - \phi(0_i, {\bf X}_s)][ dN_s(i) - \lambda(i)X_s(i) ds ]\end{equation*}

is an $\Im_t$ -martingale; that is, the $\Im_t$ -compensator of $N_t(\phi)$ is

\begin{equation*}\sum_{i=1}^n \int_0^t [ \phi(1_i, {\bf X}_s) - \phi(0_i, {\bf X}_s)] \lambda(i) X_s(i) ds.\end{equation*}

Also, using the same arguments, we can prove that the number of system repairs up to t, given by $M_t(\phi)$ , has $\Im_t$ -compensator

\begin{equation*} \int_0^t \sum_{i=1}^n [ \phi(1_i, {\bf X}_s) - \phi(0_i, {\bf X}_s)] \mu(i) ( 1 - X_s(i)) ds.\end{equation*}

The total operating and repair process, $(N_t(\phi)+ M_t(\phi))_{t \geq 0}$ , has compensator

\begin{equation*}\sum_{i=1}^n \int_0^t [ \phi(1_i, {\bf X}_s) - \phi(0_i, {\bf X}_s)] \lambda(i) X_s(i) ds + \sum_{i=1}^n \int_0^t [ \phi(1_i, {\bf X}_s) - \phi(0_i, {\bf X}_s)] \mu(i) ( 1 - X_s(i)) ds,\end{equation*}

and the cumulative inaccuracy measure at time t between $(N_t(\phi)+ M_t(\phi))_{t \geq 0}$ and $ (N_t^*(\phi)+ M_t^*(\phi))_{t \geq 0}$ , with parameters $\lambda^*$ and $\mu^*$ proposed by the experimenter, is then given by

\begin{equation*} CRI_t( N_t(\phi)+ M_t(\phi), N_t^*(\phi)+ M_t^*(\phi)) \end{equation*}

\begin{equation*} = \mathbb{E}\left[\sum_{i=1}^n \int_0^t \left(\int_0^s [ \phi(1_i, {\bf X}_u) - \phi(0_i, {\bf X}_u)] 1_{\{U_{X_u} = 0 \}} \lambda(i) X_u(i) du \right)ds \right] \end{equation*}

\begin{equation*} + \mathbb{E}\left[\sum_{i=1}^n \int_0^t \left( \int_0^s [ \phi(1_i, {\bf X}_u) - \phi(0_i, {\bf X}_u)] 1_{\{U_{X_u} = 1 \}} \mu(i) ( 1 - X_u(i))du \right)ds \right] \end{equation*}

\begin{equation*} + \mathbb{E}\left[ \sum_{i=1}^n \int_0^t \left(\int_0^s [\phi(1_i, {\bf X}_u) - \phi(0_i, {\bf X}_u)] 1_{\{U_{X_u} = 0 \}} \lambda^*(i) X_u(i) du\right)ds \right] \end{equation*}

\begin{equation*} + \mathbb{E}\left[ \sum_{i=1}^n \int_0^t \left( \int_0^s [ \phi(1_i, {\bf X}_u) - \phi(0_i, {\bf X}_u)] 1_{\{U_{X_u} = 1 \}} \mu^*(i) ( 1 - X_u (i)) du \right)ds\right]. \end{equation*}

The number of system failures up to t is given by

\begin{equation*}N_t(\phi) = \sum_{i=1}^n \int_0^t \phi(1_i, {\bf X}_s) dN_s(i),\end{equation*}

and the $\Im_t$ -compensator of $N_t(\phi)$ is

\begin{equation*}\sum_{i=1}^n \int_0^t \phi(1_i, {\bf X}_s) \lambda X_s(i) ds.\end{equation*}

Also, the number of system repairs up to t, given by $M_t(\phi)$ , has $\Im_t$ -compensator

\begin{equation*}\sum_{i=1}^n \int_0^t \phi(1_i, {\bf X}_s) \mu ( 1 - X_s(i)) ds.\end{equation*}

The total operating and repair process, $(N_t(\phi)+ M_t(\phi))_{t \geq 0}$ , has compensator

\begin{equation*}\sum_{i=1}^n \left[ \int_0^t \phi(1_i, {\bf X}_s) \lambda X_s(i) ds + \int_0^t \phi(1_i, {\bf X}_s) \mu ( 1 - X_s(i)) ds \right],\end{equation*}

and the cumulative inaccuracy measure at time t between $(N_t(\phi)+ M_t(\phi))_{t \geq 0}$ and $(N_t^*(\phi)+ M_t^*(\phi))_{t \geq 0}$ , with parameters $\lambda^*$ and $\mu^*$ proposed by the experimenter, is then given by

\begin{equation*} CRI_t( N_t(\phi)+ M_t(\phi), N_t^*(\phi)+ M_t^*(\phi)) \end{equation*}

\begin{equation*} = \mathbb{E}\left[ \sum_{i=1}^n \int_0^t \int_0^s \phi(1_i, {\bf X}_u) 1_{\{U_{X_u} = 0 \}} \lambda X_u(i) du ds \right]\end{equation*}

\begin{equation*} + \mathbb{E}\left[\int_0^t \int_0^s \phi(1_i, {\bf X}_u) 1_{\{U_{X_u} = 1\}} \mu ( 1 - X_u(i))du ds \right] \end{equation*}

\begin{equation*} + \mathbb{E}\left[ \sum_{i=1}^n \int_0^t \int_0^s \phi(1_i, {\bf X}_u) 1_{\{U_{X_u} = 0 \}} \lambda^* X_u(i) du \right]\end{equation*}

\begin{equation*}+ \mathbb{E}\left[ \int_0^t \int_0^s \phi(1_i, {\bf X}_u) 1_{\{U_{X_u} = 1 \}} \mu^* ( 1 - X_u (i)) duds \right] \end{equation*}

\begin{equation*} = \sum_{i=1}^n \int_0^t \int_0^s \left[ \frac{\lambda}{\lambda + \mu } \lambda \mathbb{P}(X_u(i)=1) + \frac{\mu}{\lambda + \mu } \mu \mathbb{P}(X_u(i) = 0) \right]du ds \end{equation*}

\begin{equation*} + \sum_{i=1}^n \int_0^t \int_0^s \left[ \frac{\lambda^*}{\lambda^* + \mu^* } \lambda^* \mathbb{P}(X_u(i)=1) + \frac{\mu^*}{\lambda^* + \mu^* } \mu^* \mathbb{P}(X_u(i) = 0)\right]du ds.\end{equation*}

If the components are in a stationary state, with $ \mathbb{P}(X_0(i) = 1) = \frac{\lambda}{\lambda + \mu }$ or $ \mathbb{P}(X_0^*(i) = 1) = \frac{\lambda^*}{\lambda^* + \mu^* }$ , we have

\begin{equation*} CRI_t( N_t(\phi)+ M_t(\phi), N_t^*(\phi)+ M_t^*(\phi)) \end{equation*}

\begin{equation*} = \sum_{i=1}^n \int_0^t \int_0^s\left[ \left(\frac{\lambda}{\lambda + \mu }\right)^2 \lambda + \left(\frac{\mu}{\lambda + \mu }\right)^2 \mu \right]du ds \end{equation*}

\begin{equation*} + \sum_{i=1}^n \int_0^t \int_0^s \left[ \left(\frac{\lambda^*}{\lambda^* + \mu^* }\right)^2 \lambda^* + \left(\frac{\mu^*}{\lambda^* + \mu^* }\right)^2 \mu^*\right] du ds \end{equation*}

\begin{equation*} = n \left[\left(\frac{\lambda}{\lambda + \mu }\right)^2 \lambda + \left(\frac{\mu}{\lambda + \mu }\right)^2 + \left(\frac{\lambda^*}{\lambda^* + \mu^* }\right)^2 \lambda^* + \left(\frac{\mu^*}{\lambda^* + \mu^* }\right)^2 \mu^* \right] \frac{t^2}{2}.\end{equation*}