On a multivariate IFR and positively dependent lifetime model induced by multiple shot-noise processes

Abstract

In reliability and survival analysis, the shot-noise processes have been intensively employed as a useful tool for modeling the impact of a dynamic environment (in the form of the point process of stresses) on survival characteristics of systems and organisms. However, the studies on the shot-noise processes have been mostly focused on the univariate analysis. In this paper, we develop a class of multivariate stochastic failure models based on a dynamic shock model induced by multiple shot-noise processes and study its properties. Explicit parametric forms for the multivariate survival functions are suggested. Multivariate ageing properties and dependence structures of the class are discussed as well.

Appendix

Proof of Theorem 1

First suppose that x₁ ≤ x₂ ≤···≤ x_n. In this case, the ordered set of {x₁, x₂,…,x_n} is given by

$$ \{ x_{(1)} ,\; x_{(2)} , \ldots , x_{(n)} \} = \left\{ {x_{1} ,\;x_{2} , \ldots ,x_{n} } \right\}. $$

Observe that

$$ \begin{aligned} & P(X_{1} > x_{1} ,X_{2} > x_{2} , \ldots ,X_{n} > x_{n} |N(s),0 \le s \le x_{n} ,D_{1} ,D_{2} , \ldots ,D_{{N(x_{n} )}} ) \\ & \quad = \exp \left\{ { - \int\limits_{0}^{{x_{1} }} {r_{1} (s)ds} } \right\}\exp \left\{ { - \int\limits_{0}^{{x_{2} }} {r_{2} (s)ds} } \right\} \times \cdots \times \exp \left\{ { - \int\limits_{0}^{{x_{n} }} {r_{n} (s)ds} } \right\} \\ & \qquad \times \left[ {\prod\limits_{{j = 1}}^{{N(x_{1} )}} {(1 - P( \cup _{{i = 1}}^{n} A_{i} (T_{j} )))} } \right] \cdot \prod\limits_{{i = 1}}^{n} {\exp \left\{ { - k_{i} \sum\limits_{{j = 1}}^{{N(x_{1} )}} {D_{j} H_{i} (x_{i} - T_{j} )} } \right\}} \\ & \qquad \times \left[ {\prod\limits_{{j = N(x_{1} ) + 1}}^{{N(x_{2} )}} {(1 - P( \cup _{{i = 2}}^{n} A_{i} (T_{j} )))} } \right] \cdot \prod\limits_{{i = 2}}^{n} {\exp \left\{ { - k_{i} \sum\limits_{{j = N(x_{1} ) + 1}}^{{N(x_{2} )}} {D_{j} H_{i} (x_{i} - T_{j} )} } \right\}} \\ & \qquad \times \left[ {\prod\limits_{{j = N(x_{2} ) + 1}}^{{N(x_{3} )}} {(1 - P( \cup _{{i = 3}}^{n} A_{i} (T_{j} )))} } \right] \cdot \prod\limits_{{i = 3}}^{n} {\exp \left\{ { - k_{i} \sum\limits_{{j = N(x_{2} ) + 1}}^{{N(x_{3} )}} {D_{j} H_{i} (x_{i} - T_{j} )} } \right\}} \\ & \qquad \times \cdots \times \left[ {\prod\limits_{{j = N(x_{{n - 1}} ) + 1}}^{{N(x_{n} )}} {(1 - P( \cup _{{i = n}}^{n} A_{i} (T_{j} )))} } \right] \cdot \exp \left\{ { - k_{n} \sum\limits_{{j = N(x_{{n - 1}} ) + 1}}^{{N(x_{n} )}} {D_{j} H_{n} (x_{i} - T_{j} )} } \right\} \\ & \quad = \exp \left\{ { - \sum\limits_{{i = 1}}^{n} {\int\limits_{0}^{{x_{i} }} {r_{i} (s)ds} } } \right\} \\ & \qquad \times \left[ {\prod\limits_{{j = 1}}^{{N(x_{1} )}} {(1 - P( \cup _{{i = 1}}^{n} A_{i} (V_{{1j}} )))} } \right] \cdot \prod\limits_{{i = 1}}^{n} {\exp \left\{ { - k_{i} \sum\limits_{{j = 1}}^{{N(x_{1} )}} {D_{j} H_{i} (x_{i} - V_{{1j}} )} } \right\}} \\ & \qquad \times \left[ {\prod\limits_{{j = 1}}^{{N(x_{2} ) - N(x_{1} )}} {(1 - P( \cup _{{i = 2}}^{n} A_{i} (V_{{2j}} )))} } \right] \cdot \prod\limits_{{i = 2}}^{n} {\exp \left\{ { - k_{i} \sum\limits_{{j = 1}}^{{N(x_{2} ) - N(x_{1} )}} {D_{{N(x_{1} ) + j}} H_{i} (x_{i} - V_{{2j}} )} } \right\}} \\ & \qquad \times \left[ {\prod\limits_{{j = 1}}^{{N(x_{3} ) - N(x_{2} )}} {(1 - P( \cup _{{i = 3}}^{n} A_{i} (V_{{3j}} )))} } \right] \cdot \prod\limits_{{i = 3}}^{n} {\exp \left\{ { - k_{i} \sum\limits_{{j = 1}}^{{N(x_{3} ) - N(x_{2} )}} {D_{{N(x_{2} ) + j}} H_{i} (x_{i} - V_{{3j}} )} } \right\}} \\ & \qquad \times \cdots \times \left[ {\prod\limits_{{j = 1}}^{{N(x_{n} ) - N(x_{{n - 1}} )}} {(1 - P( \cup _{{i = n}}^{n} A_{i} (V_{{nj}} )))} } \right] \cdot \exp \left\{ { - k_{n} \sum\limits_{{j = 1}}^{{N(x_{n} ) - N(x_{{n - 1}} )}} {D_{{N(x_{{n - 1}} ) + j}} H_{n} (x_{n} - V_{{nj}} )} } \right\} \\ & \quad = \exp \left\{ { - \sum\limits_{{i = 1}}^{n} {\int\limits_{0}^{{x_{i} }} {r_{i} (s)ds} } } \right\} \\ & \qquad \times \prod\limits_{{j = 1}}^{{N(x_{1} )}} {\left[ {(1 - P( \cup _{{i = 1}}^{n} A_{i} (V_{{1j}} )))\exp \left\{ { - \sum\limits_{{i = 1}}^{n} {k_{i} D_{j} H_{i} (x_{i} - V_{{1j}} )} } \right\}} \right]} \\ & \qquad \times \prod\limits_{{j = 1}}^{{N(x_{2} ) - N(x_{1} )}} {\left[ {(1 - P( \cup _{{i = 2}}^{n} A_{i} (V_{{2j}} )))\exp \left\{ { - \sum\limits_{{i = 2}}^{n} {k_{i} D_{{N(x_{1} ) + j}} H_{i} (x_{i} - V_{{2j}} )} } \right\}} \right]} \\ & \qquad \times \prod\limits_{{j = 1}}^{{N(x_{3} ) - N(x_{2} )}} {\left[ {(1 - P( \cup _{{i = 3}}^{n} A_{i} (V_{{3j}} )))\exp \left\{ { - \sum\limits_{{i = 3}}^{n} {k_{i} D_{{N(x_{2} ) + j}} H_{i} (x_{i} - V_{{3j}} )} } \right\}} \right]} \\ & \qquad \times \cdots \times \prod\limits_{{j = 1}}^{{N(x_{n} ) - N(x_{{n - 1}} )}} {\left[ {(1 - P( \cup _{{i = n}}^{n} A_{i} (V_{{nj}} )))\exp \left\{ { - k_{n} D_{{N(x_{{n - 1}} ) + j}} H_{n} (x_{n} - V_{{nj}} )} \right\}} \right]} \\ & \quad = \exp \left\{ { - \sum\limits_{{i = 1}}^{n} {\int\limits_{0}^{{x_{i} }} {r_{i} (s)ds} } } \right\} \\ & \qquad \times \prod\limits_{{k = 1}}^{n} {\left( {\prod\limits_{{j = 1}}^{{N(x_{k} ) - N(x_{{k - 1}} )}} {\left[ {(1 - P( \cup _{{i = k}}^{n} A_{i} (V_{{kj}} )))\exp \left\{ { - \sum\limits_{{i = k}}^{n} {k_{i} D_{{N(x_{{k - 1}} ) + j}} H_{i} (x_{i} - V_{{kj}} )} } \right\}} \right]} } \right)} , \\ \end{aligned} $$

where V_kj denotes the arrival time of the jth shock in the interval (x_k−1, x_k], $x_{k - 1} < V_{k1} \le V_{k2} \le \cdots \le V_{{kN(x_{k} ) - N(x_{k - 1} )}} \le x_{k} ,\,k = 1,\;2, \ldots ,n,$ where x₀ ≡ 0. Then, the joint survival function can be obtained by

$$ \begin{aligned} P(X_{1} > x_{1} ,X_{2} > x_{2} , \ldots ,X_{n} > x_{n} ) & = \exp \left\{ { - \sum\limits_{{i = 1}}^{n} {\int\limits_{0}^{{x_{i} }} {r_{i} (s)ds} } } \right\} \\ & \quad \times E\left[ {\prod\limits_{{k = 1}}^{n} {\left( {\prod\limits_{{j = 1}}^{{N(x_{k} ) - N(x_{{k - 1}} )}} {(1 - P( \cup _{{i = k}}^{n} A_{i} (V_{{kj}} )))} \cdot \exp \left\{ { - \sum\limits_{{i = k}}^{n} {k_{i} D_{{N(x_{{k - 1}} ) + j}} H_{i} (x_{i} - V_{{kj}} )} } \right\}} \right)} } \right] \\ & = \exp \left\{ { - \sum\limits_{{i = 1}}^{n} {\int\limits_{0}^{{x_{i} }} {r_{i} (s)ds} } } \right\} \\ & \quad \times E\left[ {E\left[ {\left. {\prod\limits_{{k = 1}}^{n} {\left( {\prod\limits_{{j = 1}}^{{N(x_{k} ) - N(x_{{k - 1}} )}} {(1 - P( \cup _{{i = k}}^{n} A_{i} (V_{{kj}} )))} \cdot \exp \left\{ { - \sum\limits_{{i = k}}^{n} {k_{i} D_{{N(x_{{k - 1}} ) + j}} H_{i} (x_{i} - V_{{kj}} )} } \right\}} \right)} } \right|N(x_{1} ),N(x_{2} ) - N(x_{1} ), \ldots ,N(x_{n} ) - N(x_{{n - 1}} )} \right]} \right], \\ \end{aligned} $$

where, due to the independent increments property of NHPP,

$$ \begin{aligned} & E\left[ {\left. {\prod\limits_{{k = 1}}^{n} {\left( {\prod\limits_{{j = 1}}^{{N(x_{k} ) - N(x_{{k - 1}} )}} {(1 - P( \cup _{{i = k}}^{n} A_{i} (V_{{kj}} )))} \cdot \exp \left\{ { - \sum\limits_{{i = k}}^{n} {k_{i} D_{{N(x_{{k - 1}} ) + j}} H_{i} (x_{i} - V_{{kj}} )} } \right\}} \right)} } \right|N(x_{1} ),N(x_{2} ) - N(x_{1} ), \ldots ,N(x_{n} ) - N(x_{{n - 1}} )} \right] \\ & \quad = \prod\limits_{{k = 1}}^{n} {E\left[ {\left. {\prod\limits_{{j = 1}}^{{N(x_{k} ) - N(x_{{k - 1}} )}} {(1 - P( \cup _{{i = k}}^{n} A_{i} (V_{{kj}} ))} ) \cdot \exp \left\{ { - \sum\limits_{{i = k}}^{n} {k_{i} D_{{N(x_{{k - 1}} ) + j}} H_{i} (x_{i} - V_{{kj}} )} } \right\}} \right|N(x_{k} ) - N(x_{{k - 1}} )} \right].} \\ \end{aligned} $$

If λ(0 +) > 0, then for all k = 1, 2,…,n, the joint distribution of $V_{k1} ,\;V_{k2} , \ldots ,V_{{kN(x_{k} ) - N(x_{k - 1} )}} ,$ given N(x_k) − N(x_k-1) = m_k, is identical with the joint distribution of the order statistics $V^{\prime}_{k(1)} \le V^{\prime}_{k(2)} \le \cdots \le V^{\prime}_{{k(m_{k} )}}$ of i.i.d. random variables $V^{\prime}_{k1} ,V^{\prime}_{k2} , \cdots ,V^{\prime}_{{km{}_{k}}} ,$ where the pdf of the common distribution of V′_kj’s is given by λ(t)/[Λ(x_k) − Λ(x_k−1)], x_k−1 < t ≤ x_k (see, e.g., Cha and Mi 2007; Cha and Finkelstein 2011). Thus,

$$ \begin{aligned} & \prod\limits_{{k = 1}}^{n} {E\left[ {\left. {\prod\limits_{{j = 1}}^{{N(x_{k} ) - N(x_{{k - 1}} )}} {(1 - P( \cup _{{i = k}}^{n} A_{i} (V_{{kj}} ))} ) \cdot \exp \left\{ { - \sum\limits_{{i = k}}^{n} {k_{i} D_{{N(x_{{k - 1}} ) + j}} H_{i} (x_{i} - V_{{kj}} )} } \right\}} \right|N(x_{k} ) - N(x_{{k - 1}} ) = m_{k} } \right]} \\ & \quad = \prod\limits_{{k = 1}}^{n} {E\left[ {\prod\limits_{{j = 1}}^{{m_{k} }} {(1 - P( \cup _{{i = k}}^{n} A_{i} (V^{\prime}_{{k(j)}} )))} \cdot \exp \left\{ { - \sum\limits_{{i = k}}^{n} {k_{i} D_{j} H_{i} (x_{i} - V^{\prime}_{{k(j)}} )} } \right\}} \right]} \\ & \quad = \prod\limits_{{k = 1}}^{n} {E\left[ {\prod\limits_{{j = 1}}^{{m_{k} }} {(1 - P( \cup _{{i = k}}^{n} A_{i} (V^{\prime}_{{kj}} )))} \cdot \exp \left\{ { - \sum\limits_{{i = k}}^{n} {k_{i} D_{j} H_{i} (x_{i} - V^{\prime}_{{kj}} )} } \right\}} \right]} \\ & \quad = \prod\limits_{{k = 1}}^{n} {\left( {E\left[ {(1 - P( \cup _{{i = k}}^{n} A_{i} (V^{\prime}_{{k1}} ))) \cdot \exp \left\{ { - \sum\limits_{{i = k}}^{n} {k_{i} D_{1} H_{i} (x_{i} - V^{\prime}_{{k1}} )} } \right\}} \right]} \right)} ^{{m_{k} }} . \\ \end{aligned} $$

Since

$$ \begin{aligned} & E\left[ {\left. {(1 - P( \cup_{i = k}^{n} A_{i} (V^{\prime}_{k1} ))) \cdot \exp \left\{ { - \sum\limits_{i = k}^{n} {k_{i} D_{1} H_{i} (x_{i} - V^{\prime}_{k1} )} } \right\}} \right|V^{\prime}_{k1} = s\,} \right] \\ & \quad = E\left[ {(1 - P( \cup_{i = k}^{n} A_{i} (s))) \cdot \exp \left\{ { - \sum\limits_{i = k}^{n} {k_{i} D_{1} H_{i} (x_{i} - s)} } \right\}\,} \right] \\ & \quad = \int\limits_{0}^{\infty } {\left( {1 - P\left( { \cup_{i = k}^{n} A_{i} (s)} \right)} \right)} \cdot \exp \left\{ { - \sum\limits_{i = k}^{n} {k_{i} uH_{i} } \left( {x_{i} - s} \right)} \right\}f_{D} (u)du, \\ \end{aligned} $$

the unconditional expectation is obtained by

$$ \begin{aligned} & E\left[ {(1 - P( \cup _{{i = k}}^{n} A_{i} (V^{\prime}_{{k1}} ))) \cdot \exp \left\{ { - \sum\limits_{{i = k}}^{n} {k_{i} D_{1} H_{i} (x_{i} - V^{\prime}_{{k1}} )} } \right\}} \right] \\ & \quad = \int\limits_{{x_{{k - 1}} }}^{{x_{k} }} {\int\limits_{0}^{\infty } {(1 - P( \cup _{{i = k}}^{n} A_{i} (s))) \cdot \exp \left\{ { - \sum\limits_{{i = k}}^{n} {k_{i} uH_{i} (x_{i} - s)} } \right\}f_{D} (u)du} } \frac{{\lambda (s)}}{{\Lambda (x_{k} ) - \Lambda (x_{{k - 1}} )}}ds. \\ \end{aligned} $$

Finally, the joint survival function is derived by

$$ \begin{aligned} P(X_{1} > x_{1} ,X_{2} > x_{2} , \ldots ,X_{n} > x_{n} ) & \quad = \exp \left\{ { - \sum\limits_{{i = 1}}^{n} {\int\limits_{0}^{{x_{i} }} {r_{i} (s)ds} } } \right\} \times \sum\limits_{{m_{n} = 0}}^{\infty } {\sum\limits_{{m_{{n - 1}} = 0}}^{\infty } { \cdots \sum\limits_{{m_{2} = 0}}^{\infty } {\sum\limits_{{m_{1} = 0}}^{\infty } {} } } } \\ & \qquad \prod\limits_{{k = 1}}^{n} {\left[ {\left( {\int\limits_{{x_{{k - 1}} }}^{{x_{k} }} {\int\limits_{0}^{\infty } {(1 - P( \cup _{{i = k}}^{n} A_{i} (s))) \cdot \exp \left\{ { - \sum\limits_{{i = k}}^{n} {k_{i} uH_{i} (x_{i} - s)} } \right\}f_{D} (u)du} \frac{{\lambda (s)}}{{\Lambda (x_{k} ) - \Lambda (x_{{k - 1}} )}}} ds} \right)^{{m_{k} }} \times \frac{{[\Lambda (x_{k} ) - \Lambda (x_{{k - 1}} )]^{{m_{k} }} }}{{m_{k} !}}\exp \{ - [\Lambda (x_{k} ) - \Lambda (x_{{k - 1}} )\} } \right]} \\ & \quad = \exp \left\{ { - \sum\limits_{{i = 1}}^{n} {\int\limits_{0}^{{x_{i} }} {r_{i} (s)ds} } } \right\} \\ & \qquad \times \exp \left\{ { - \int\limits_{0}^{{x_{1} }} {\left( {1 - \int\limits_{0}^{\infty } {(1 - P( \cup _{{i = 1}}^{n} A_{i} (s)))\exp \left\{ { - \sum\limits_{{i = 1}}^{n} {k_{i} uH_{i} (x_{i} - s)} } \right\}f_{D} (u)du} } \right)\lambda (s)} ds} \right\} \\ & \quad \times \exp \left( { - \int\limits_{{x_{1} }}^{{x_{2} }} {\left( {1 - \int\limits_{0}^{\infty } {(1 - P( \cup _{{i = 2}}^{n} A_{i} (s)))\exp \left\{ { - \sum\limits_{{i = 2}}^{n} {k_{i} uH_{i} (x_{i} - s)} } \right\}f_{D} (u)du} } \right)} \lambda (s)ds} \right) \\ & \qquad \times \exp \left( { - \int\limits_{{x_{2} }}^{{x_{3} }} {\left( {1 - \int\limits_{0}^{\infty } {(1 - P( \cup _{{i = 3}}^{n} A_{i} (s)))\exp \left\{ { - \sum\limits_{{i = 3}}^{n} {k_{i} uH_{i} (x_{i} - s)} } \right\}f_{D} (u)du} } \right)} \lambda (s)ds} \right) \\ & \qquad \times \cdots \times \exp \left( { - \int\limits_{{x_{{n - 1}} }}^{{x_{n} }} {\left( {1 - \int\limits_{0}^{\infty } {(1 - P( \cup _{{i = n}}^{n} A_{i} (s)))\exp \left\{ { - k_{n} uH_{n} (x_{n} - s)} \right\}f_{D} (u)du} } \right)} \lambda (s)ds} \right) \\ & \quad = \exp \left\{ { - \sum\limits_{{i = 1}}^{n} {\int\limits_{0}^{{x_{i} }} {r_{i} (s)ds} } } \right\} \\ & \qquad \times \exp \left\{ { - \int\limits_{0}^{{x_{{(1)}} }} {\left( {1 - \int\limits_{0}^{\infty } {(1 - P( \cup _{{i \in S(x_{{(0)}} )}} A_{i} (s)))\exp \left\{ { - \sum\limits_{{i \in S(x_{{(0)}} )}} {k_{i} uH_{i} (x_{i} - s)} } \right\}f_{D} (u)du} } \right)\lambda (s)} ds} \right\} \\ & \qquad \times \exp \left( { - \int\limits_{{x_{{(1)}} }}^{{x_{{(2)}} }} {\left( {1 - \int\limits_{0}^{\infty } {(1 - P( \cup _{{i \in S(x_{{(1)}} )}} A_{i} (s)))\exp \left\{ { - \sum\limits_{{i \in S(x_{{(1)}} )}} {k_{i} uH_{i} (x_{i} - s)} } \right\}f_{D} (u)du} } \right)} \lambda (s)ds} \right) \\ & \qquad \times \exp \left( { - \int\limits_{{x_{{(2)}} }}^{{x_{{(3)}} }} {\left( {1 - \int\limits_{0}^{\infty } {(1 - P( \cup _{{i \in S(x_{{(1)}} ,x_{{(2)}} )}} A_{i} (s)))\exp \left\{ { - \sum\limits_{{i \in S(x_{{(1)}} ,x_{{(2)}} )}} {k_{i} uH_{i} (x_{i} - s)} } \right\}f_{D} (u)du} } \right)} \lambda (s)ds} \right) \times \cdots \\ & \qquad \times \exp \left( { - \int\limits_{{x_{{(n - 1)}} }}^{{x_{{(n)}} }} {\left( {1 - \int\limits_{0}^{\infty } {(1 - P( \cup _{{i \in S(x_{{(1)}} ,x_{{(2)}} , \ldots ,x_{{(n - 1)}} )}} A_{i} (s)))\exp \left\{ { - \sum\limits_{{i \in S(x_{{(1)}} ,x_{{(2)}} , \ldots ,x_{{(n - 1)}} )}} {k_{i} uH_{i} (x_{i} - s)} } \right\}f_{D} (u)du} } \right)} \lambda (s)ds} \right). \\ \end{aligned} $$

The results for the other cases can be obtained symmetrically.

Proof of Theorem 5

Due to Proposition 1, it is sufficient to show that the proposed class of distributions possesses multivariate IFR property on each OPS. Without loss of generality, we consider only the case when x₁ ≤ x₂ ≤···≤ x_n. Note that

$$ \begin{aligned} & \frac{{P(X_{1} > x_{1} + t,\;X_{2} > x_{2} + t, \ldots ,X_{n} > x_{n} + t)}}{{P(X_{1} > x_{1} ,\;X_{2} > x_{2} , \ldots ,X_{n} > x_{n} )}} \\ & \quad = \exp \left\{ { - \sum\limits_{{i = 1}}^{n} {\int\limits_{{x_{i} }}^{{x_{i} + t}} {r_{i} (s)ds} } } \right\} \\ & \quad \quad \times \exp \left\{ { - \int\limits_{0}^{\infty } {\left[ {\int\limits_{0}^{{x_{1} + t}} {\left( {1 - \left( {1 - \sum\limits_{{i = 1}}^{{n + 1}} {p_{i} (s)} } \right)\exp \left\{ { - \sum\limits_{{i = 1}}^{n} {k_{i} uH_{i} (x_{i} + t - s)} } \right\}} \right)\lambda (s)ds} } \right.} } \right. \\ & \left. {\left. { \quad \quad - \int\limits_{0}^{{x_{1} }} {\left( {1 - \left( {1 - \sum\limits_{{i = 1}}^{{n + 1}} {p_{i} (s)} } \right)\exp \left\{ { - \sum\limits_{{i = 1}}^{n} {k_{i} uH_{i} (x_{i} - s)} } \right\}} \right)\lambda (s)} ds} \right]f_{D} (u)du} \right\} \\ & \quad \quad \times \prod\limits_{{k = 2}}^{n} {\exp \left\{ { - \int\limits_{0}^{\infty } {\left[ {\int\limits_{{x_{{k - 1}} + t}}^{{x_{k} + t}} {\left( {1 - \left( {1 - \sum\limits_{{i = k}}^{{n + 1}} {p_{i} (s)} } \right)\exp \left\{ { - \sum\limits_{{i = k}}^{n} {k_{i} uH_{i} (x_{i} + t - s)} } \right\}} \right)\lambda (s)ds} } \right.} } \right.} \\ & \left. {\left. { \quad \quad - \int\limits_{{x_{{k - 1}} }}^{{x_{k} }} {\left( {1 - \left( {1 - \sum\limits_{{i = k}}^{{n + 1}} {p_{i} (s)} } \right)\exp \left\{ { - \sum\limits_{{i = k}}^{n} {k_{i} uH_{i} (x_{i} - s)} } \right\}} \right)\lambda (s)} ds} \right]f_{D} (u)du} \right\}. \\ \end{aligned} $$

From the condition that r_i(t), i = 1, 2,…,n are increasing for t ≥ 0, it is clear that $\exp \left\{ { - \sum\nolimits_{i = 1}^{n} {\int_{{x_{i} }}^{{x_{i} + t}} {r_{i} (s)ds} } } \right\}$ is decreasing in x₁, x₂,…,x_n ≥ 0 for all t ≥ 0. Let

$$ \begin{aligned} & h(x_{1} ,\;x_{2} , \ldots ,x_{n} ;\;t) \equiv \int\limits_{0}^{{x_{1} + t}} {\left( {1 - \left( {1 - \sum\limits_{i = 1}^{n + 1} {p_{i} (s)} } \right)\exp \left\{ { - \sum\limits_{i = 1}^{n} {k_{i} uH_{i} (x_{i} + t - s)} } \right\}} \right)\lambda (s)} ds \\ & \quad - \int\limits_{0}^{{x_{1} }} {\left( {1 - \left( {1 - \sum\limits_{i = 1}^{n + 1} {p_{i} (s)} } \right)\exp \left\{ { - \sum\limits_{i = 1}^{n} {k_{i} uH_{i} (x_{i} - s)} } \right\}} \right)\lambda (s)} ds \\ & \quad + \sum\limits_{k = 2}^{n} {\left[ {\int\limits_{{x_{k - 1} + t}}^{{x_{k} + t}} {\left( {1 - \left( {1 - \sum\limits_{i = k}^{n + 1} {p_{i} (s)} } \right)\exp \left\{ { - \sum\limits_{i = k}^{n} {k_{i} uH_{i} (x_{i} + t - s)} } \right\}} \right)\lambda (s)} ds} \right.} \\ & \left. {\quad - \int\limits_{{x_{k - 1} }}^{{x_{k} }} {\left( {1 - \left( {1 - \sum\limits_{i = k}^{n + 1} {p_{i} (s)} } \right)\exp \left\{ { - \sum\limits_{i = k}^{n} {k_{i} uH_{i} (x_{i} - s)} } \right\}} \right)\lambda (s)} ds} \right]. \\ \end{aligned} $$

Then, to prove that $\frac{{P(X_{1} > x_{1} + t,\;X_{2} > x_{2} + t, \ldots ,X_{n} > x_{n} + t)}}{{P(X_{1} > x_{1} ,\;X_{2} > x_{2} , \ldots ,X_{n} > x_{n} )}}$ is decreasing in x₁, x₂,…,x_n ≥ 0 for all t ≥ 0, we need to show that h(x₁, x₂,…,x_n; t) is increasing in x₁, x₂,…,x_n ≥ 0, for each fixed u, t ≥ 0. Observe that

$$ \begin{aligned} & h(x_{1} ,x_{2} , \ldots ,x_{n} ;t) = \int\limits_{{ - t}}^{{x_{1} }} {\left( {1-\left( {1 - \sum\limits_{{i = 1}}^{{n + 1}} {p_{i} (s + t)} } \right)1 - \exp \left\{ { - \sum\limits_{{i = 1}}^{n} {k_{i} uH_{i} (x_{i} - s)} } \right\}} \right)\lambda (s + t)} ds \\ & \quad - \int\limits_{0}^{{x_{1} }} {\left( {1 - \left( {1 - \sum\limits_{{i = 1}}^{{n + 1}} {p_{i} (s)} } \right)\exp \left\{ { - \sum\limits_{{i = 1}}^{n} {k_{i} uH_{i} (x_{i} - s)} } \right\}} \right)\lambda (s)} ds \\ & \quad + \sum\limits_{{k = 2}}^{n} {\left[ {\int\limits_{{x_{{k - 1}} }}^{{x_{k} }} {\left( {1 - \left( {1 - \sum\limits_{{i = k}}^{{n + 1}} {p_{i} (s + t)} } \right)\exp \left\{ { - \sum\limits_{{i = k}}^{n} {k_{i} uH_{i} (x_{i} - s)} } \right\}} \right)\lambda (s + t)} ds} \right.} \\ & \left. {\quad - \int\limits_{{x_{{k - 1}} }}^{{x_{k} }} {\left( {1 - \left( {1 - \sum\limits_{{i = k}}^{{n + 1}} {p_{i} (s)} } \right)\exp \left\{ { - \sum\limits_{{i = k}}^{n} {k_{i} uH_{i} (x_{i} - s)} } \right\}} \right)\lambda (s)} ds} \right] \\ & \quad = - \int\limits_{0}^{{ - t}} {\left( {1 - \left( {1 - \sum\limits_{{i = 1}}^{{n + 1}} {p_{i} (s + t)} } \right)\exp \left\{ { - \sum\limits_{{i = 1}}^{n} {k_{i} uH_{i} (x_{i} - s)} } \right\}} \right)\lambda (s + t)} ds \\ & \quad + \int\limits_{0}^{{x_{1} }} {\left( {1 - \left( {1 - \sum\limits_{{i = 1}}^{{n + 1}} {p_{i} (s + t)} } \right)\exp \left\{ { - \sum\limits_{{i = 1}}^{n} {k_{i} uH_{i} (x_{i} - s)} } \right\}} \right)\lambda (s + t)} ds \\ & \quad - \int\limits_{0}^{{x_{1} }} {\left( {1 - \left( {1 - \sum\limits_{{i = 1}}^{{n + 1}} {p_{i} (s)} } \right)\exp \left\{ { - \sum\limits_{{i = 1}}^{n} {k_{i} uH_{i} (x_{i} - s)} } \right\}} \right)\lambda (s)} ds \\ & \quad + \sum\limits_{{k = 2}}^{n} {\left[ {\int\limits_{{x_{{k - 1}} }}^{{x_{k} }} {\left( {1 - \left( {1 - \sum\limits_{{i = k}}^{{n + 1}} {p_{i} (s + t)} } \right)\exp \left\{ { - \sum\limits_{{i = k}}^{n} {k_{i} uH_{i} (x_{i} - s)} } \right\}} \right)\lambda (s + t)} ds} \right.} \\ & \quad \left. { - \int\limits_{{x_{{k - 1}} }}^{{x_{k} }} {\left( {1 - \left( {1 - \sum\limits_{{i = k}}^{{n + 1}} {p_{i} (s)} } \right)\exp \left\{ { - \sum\limits_{{i = k}}^{n} {k_{i} uH_{i} (x_{i} - s)} } \right\}} \right)\lambda (s)} ds} \right] \\ & \quad = \int\limits_{0}^{t} {\left( {1 - p_{0} (t - s)\exp \left\{ { - \sum\limits_{{i = 1}}^{n} {k_{i} uH_{i} (x_{i} + s)} } \right\}} \right)\lambda (t - s)} ds \\ & \quad + \int\limits_{0}^{{x_{1} }} {\{ \lambda (s + t) - \lambda (s)\} - \left[ {p_{0} (s + t)\lambda (s + t) - p_{0} (s)\lambda (s)} \right]\exp \left\{ { - \sum\limits_{{i = 1}}^{n} {k_{i} uH_{i} (x_{i} - s)} } \right\}} ds \\ & \quad + \sum\limits_{{k = 2}}^{n} {\int\limits_{{x_{{k - 1}} }}^{{x_{k} }} {\{ \lambda (s + t) - \lambda (s)\} - \left[ {\left( {1 - \sum\limits_{{i = k}}^{{n + 1}} {p_{i} (s + t)} } \right)\lambda (s + t) - \left( {1 - \sum\limits_{{i = k}}^{{n + 1}} {p_{i} (s)} } \right)\lambda (s)} \right]\exp \left\{ { - \sum\limits_{{i = k}}^{n} {k_{i} uH_{i} (x_{i} - s)} } \right\}} ds} . \\ \end{aligned} $$

Let

$$ h_{0} (x_{1} ,\;x_{2} , \ldots , x_{n} ,\;s;\;t) \equiv \left( {1 - p_{0} (t - s)\exp \left\{ {\sum\limits_{i = 1}^{n} {k_{i} uH_{i} \left( {x_{i} + s} \right)} } \right\}} \right)\lambda (t - s), $$

and

$$ \begin{aligned} h_{k} (x_{1} ,x_{2} , \ldots ,x_{n} ,s;t) & \equiv \{ \lambda (s + t) - \lambda (s)\} \\ & \quad - \left[ {\left( {1 - \sum\limits_{{i = k}}^{{n + 1}} {p_{i} (s + t)} } \right)\lambda (s + t) - \left( {1 - \sum\limits_{{i = k}}^{{n + 1}} {p_{i} (s)} } \right)\lambda (s)} \right] \\ & \quad \times \exp \left\{ { - \sum\limits_{{i = k}}^{n} {k_{i} uH_{i} (x_{i} - s)} } \right\},\quad k = 1,2, \ldots ,n. \\ \end{aligned} $$

Then,

$$ \begin{aligned} h(x_{1} ,\;x_{2} , \ldots ,x_{n} ;\;t) & = \int\limits_{0}^{t} {h_{0} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds \\ & \quad + \sum\limits_{k = 1}^{n} {\int\limits_{{x_{k - 1} }}^{{x_{k} }} {h_{k} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} } ds,\end{aligned} $$

where x₀ ≡ 0, and it is clear that h₀(x₁, x₂,…,x_n, s; t) is increasing in x₁, x₂,…,x_n ≥ 0, for each fixed s, t ≥ 0. It is now necessary to verify that

$$ \varphi (x_{1} ,\;x_{2} , \ldots ,x_{n} ;\;t) \equiv \sum\limits_{k = 1}^{n} {\int\limits_{{x_{k - 1} }}^{{x_{k} }} {h_{k} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} } ds $$

is increasing in x₁, x₂,…,x_n ≥ 0. Note that if p_i(t), i = 1, 2,…,n + 1, and λ(t) are increasing for t ≥ 0, and the condition (11) holds, then, for each fixed s, t ≥ 0,

$$ \begin{aligned} & \left[ {\left( {1 - \sum\limits_{i = k}^{n + 1} {p_{i} (s + t)} } \right)\lambda (s + t) - \left( {1 - \sum\limits_{i = k}^{n + 1} {p_{i} (s)} } \right)\lambda (s)} \right] \\ & \quad = \left[ {\left( { \sum\limits_{i = 0}^{k - 1} {p_{i} (s + t)} } \right)\lambda (s + t) - \left( { \sum\limits_{i = 0}^{k - 1} {p_{i} (s)} } \right)\lambda (s)} \right] \ge 0,\quad {\text{for}}\;{\text{all}}\;k = 1,\;2, \ldots , n. \\ \end{aligned} $$

Then, under the given conditions, it is obvious that h_i(x₁, x₂,…,x_n, s; t), i = 1, 2,…,n are increasing in x₁, x₂,…,x_n ≥ 0, for each fixed s, t ≥ 0. In addition, it can be shown that h_i(x₁, x₂,…,x_n, s; t) ≥ h_i+1(x₁, x₂,…,x_n, s; t), for i = 1, 2,…,n − 1, for all s, t ≥ 0, for each x₁, x₂,…,x_n ≥ 0. Furthermore, under the given conditions, we have that h_i(x₁, x₂,…,x_n, s; t) ≥ 0, i = 2, 3,…,n, for all s, t ≥ 0, for each x₁, x₂,…,x_n ≥ 0, because

$$ \begin{aligned} h_{k} (x_{1} ,x_{2} , \ldots ,x_{n} ,s;t) & \ge \{ \lambda (s + t) - \lambda (s)\} - \left[ {\left( {1 - \sum\limits_{{i = k}}^{{n + 1}} {p_{i} (s + t)} } \right)\lambda (s + t) - \left( {1 - \sum\limits_{{i = k}}^{{n + 1}} {p_{i} (s)} } \right)\lambda (s)} \right] \\ & = \left( {\sum\limits_{{i = k}}^{{n + 1}} {p_{i} (s + t)} } \right)\lambda (s + t) - \left( {\sum\limits_{{i = k}}^{{n + 1}} {p_{i} (s)} } \right)\lambda (s) \\ \end{aligned} $$

and

$$ \frac{\lambda (s + t)}{\lambda (s)} \ge \frac{{\sum\nolimits_{i = k}^{n + 1} {p_{i} (s)} }}{{\sum\nolimits_{i = k}^{n + 1} {p_{i} (s + t)} }},\quad k = 2,\;3, \ldots ,n,\quad {\text{for}}\;{\text{all}}\;s,\;t \ge 0. $$

Choose $x_{1}^{\prime},\,x_{2}^{\prime}, \ldots ,x_{n}^{\prime}\ge 0$ such that $x_{1}^{\prime}\le x_{2}^{\prime}\le\cdots\le x_{n}^{\prime}$ and $x_{i}\le x_{i}^{\prime},$ i = 1, 2,…,n. Now, to complete the proof, we need to show that the equality

$$ \varphi (x_{1} ,\;x_{2} , \ldots ,x_{n} ;\;t) \le \phi (x^{\prime}_{1} ,\;x^{\prime}_{2} , \ldots ,x^{\prime}_{n} ;\;t) $$

(12)

holds for each fixed t ≥ 0. For this, we will use the principle of mathematical induction as follows. Let

$$ \varphi_{m} (x_{1} ,\;x_{2} , \ldots ,x_{n} ;\;t) \equiv \sum\limits_{k = 1}^{m} {\int\limits_{{x_{k - 1} }}^{{x_{k} }} {h_{k} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} } ds,\quad m = 2,\;3, \ldots ,n. , $$

Then inequality (12) is equivalent to φ_n(x₁, x₂,…,x_n; t) ≤ φ_n$(x_{1}^{\prime},\,x_{2}^{\prime},\ldots,x_{n}^{\prime}; t ).$

Step 1 we prove that φ₂(x₁, x₂,…,x_n; t) ≤ φ₂(x ^′₁ , x ^′₂ ,…,x ^′_n ; t), holds (for the case m = 2).

(i) First consider the case when x₁ ≤ x ^′₁  ≤ x₂ ≤ x ^′₂ . Then we have

$$ \begin{aligned} & \phi_{2} (x_{1} ,\;x_{2} , \ldots ,x_{n} ;\;t) = \int\limits_{0}^{{x_{1} }} {h_{1} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds + \int\limits_{{x_{1} }}^{{x_{2} }} {h_{2} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)ds} \\ & \quad = \int\limits_{0}^{{x_{1} }} {h_{1} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds + \int\limits_{{x_{1} }}^{{x^{\prime}_{1} }} {h_{2} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds + \int\limits_{{x^{\prime}_{1} }}^{{x_{2} }} {h_{2} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)ds} \\ & \quad \le \int\limits_{0}^{{x_{1} }} {h_{1} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds + \int\limits_{{x_{1} }}^{{x^{\prime}_{1} }} {h_{1} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)ds} + \int\limits_{{x^{\prime}_{1} }}^{{x_{2} }} {h_{2} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds + \int\limits_{{x_{2} }}^{{x^{\prime}_{2} }} {h_{2} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)ds} \\ & \quad = \int\limits_{0}^{{x^{\prime}_{1} }} {h_{1} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds + \int\limits_{{x^{\prime}_{1} }}^{{x^{\prime}_{2} }} {h_{2} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds \\ & \quad \le \int\limits_{0}^{{x^{\prime}_{1} }} {h_{1} (x^{\prime}_{1} ,\;x^{\prime}_{2} , \ldots ,x^{\prime}_{n} ,\;s;\;t)} ds + \int\limits_{{x^{\prime}_{1} }}^{{x^{\prime}_{2} }} {h_{2} (x^{\prime}_{1} ,\;x^{\prime}_{2} , \ldots ,x^{\prime}_{n} ,\;s;\;t)} ds = \phi_{2} (x^{\prime}_{1} ,\;x^{\prime}_{2} , \ldots ,x^{\prime}_{n} ;\;t). \\ \end{aligned} $$

(ii) Now consider the case when $x_{1}\le x_{2}\le x_{1}^{\prime}\le x_{2}^{\prime}.$ In this case,$\begin{aligned} & \phi_{2} (x_{1} ,\;x_{2} , \ldots ,x_{n} ;\;t) = \int\limits_{0}^{{x_{1} }} {h_{1} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds + \int\limits_{{x_{1} }}^{{x_{2} }} {h_{2} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)ds} \\ & \quad \le \int\limits_{0}^{{x_{1} }} {h_{1} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds + \int\limits_{{x_{1} }}^{{x_{2} }} {h_{2} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)ds} + \int\limits_{{x_{2} }}^{{x^{\prime}_{1} }} {h_{2} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)ds} + \int\limits_{{x^{\prime}_{1} }}^{{x^{\prime}_{2} }} {h_{2} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)ds} \\ & \quad \le \int\limits_{0}^{{x_{1} }} {h_{1} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds + \int\limits_{{x_{1} }}^{{x_{2} }} {h_{1} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)ds} + \int\limits_{{x_{2} }}^{{x^{\prime}_{1} }} {h_{1} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)ds} + \int\limits_{{x^{\prime}_{1} }}^{{x^{\prime}_{2} }} {h_{2} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)ds} \\ & \quad \le \int\limits_{0}^{{x^{\prime}_{1} }} {h_{1} (x^{\prime}_{1} ,\;x^{\prime}_{2} , \ldots ,x^{\prime}_{n} ,\;s;\;t)} ds + \int\limits_{{x^{\prime}_{1} }}^{{x^{\prime}_{2} }} {h_{2} (x^{\prime}_{1} ,\;x^{\prime}_{2} , \ldots ,x^{\prime},\;s;\;t)ds} = \phi_{2} (x^{\prime}_{1} ,\;x^{\prime}_{2} , \ldots ,x^{\prime},\;s;\;t). \\ \end{aligned}$ Hence, the equality φ₂(x₁, x₂,…,x_n; t) ≤ φ₂$(x_{1}^{\prime},\,x_{2}^{\prime},\ldots,x_{n}^{\prime}; t )$ holds.

Step 2 assume now that φ_r(x₁, x₂,…,x_n; t) ≤ φ_r$(x_{1}^{\prime},\,x_{2}^{\prime},\ldots,x_{n}^{\prime}; t )$ holds (for the case m = r), where 2 ≤ r ≤ n − 1. Then we prove that φ_r+1(x₁, x₂,…,x_n; t) ≤ φ_r+1$(x_{1}^{\prime},\,x_{2}^{\prime},\ldots,x_{n}^{\prime}; t ),$ holds (for the case m = r + 1).

(i) First consider the case when $x_{r}\le x_{r}^{\prime}\le x_{r+1}\le x_{r+1}^{\prime}.$ In this case, we have

$$ \begin{aligned} & \phi _{{r + 1}} (x_{1} ,\;x_{2} , \ldots ,x_{n} ;\;t) = \sum\limits_{{k = 1}}^{r} {\int\limits_{{x_{{k - 1}} }}^{{x_{k} }} {h_{i} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds} + \int\limits_{{x_{r} }}^{{x_{{r + 1}} }} {h_{{r + 1}} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds \\ & \quad = \sum\limits_{{k = 1}}^{r} {\int\limits_{{x_{{k - 1}} }}^{{x_{k} }} {h_{i} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds} + \int\limits_{{x_{r} }}^{{x^{\prime}_{r} }} {h_{{r + 1}} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds + \int\limits_{{x^{\prime}_{r} }}^{{x_{{r + 1}} }} {h_{{r + 1}} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds \\ & \quad = \sum\limits_{{k = 1}}^{{r - 1}} {\int\limits_{{x_{{k - 1}} }}^{{x_{k} }} {h_{i} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds} + \int\limits_{{x_{{r - 1}} }}^{{x_{r} }} {h_{r} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds \\ & \quad \quad + \int\limits_{{x_{r} }}^{{x^{\prime}_{r} }} {h_{{r + 1}} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds + \int\limits_{{x^{\prime}_{r} }}^{{x_{{r + 1}} }} {h_{{r + 1}} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds \\ & \quad \le \sum\limits_{{k = 1}}^{{r - 1}} {\int\limits_{{x_{{k - 1}} }}^{{x_{k} }} {h_{i} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds} + \int\limits_{{x_{{r - 1}} }}^{{x_{r} }} {h_{r} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds \\ & \quad \quad + \int\limits_{{x_{r} }}^{{x^{\prime}_{r} }} {h_{r} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds + \int\limits_{{x^{\prime}_{r} }}^{{x_{{r + 1}} }} {h_{{r + 1}} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds + \int\limits_{{x_{{r + 1}} }}^{{x^{\prime}_{{r + 1}} }} {h_{{r + 1}} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds \\ & \quad = \sum\limits_{{k = 1}}^{{r - 1}} {\int\limits_{{x_{{k - 1}} }}^{{x_{k} }} {h_{i} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds} + \int\limits_{{x_{{r - 1}} }}^{{x^{\prime}_{r} }} {h_{r} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds + \int\limits_{{x^{\prime}_{r} }}^{{x^{\prime}_{{r + 1}} }} {h_{{r + 1}} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds \\ & \quad \le \sum\limits_{{k = 1}}^{{r - 1}} {\int\limits_{{x_{{k - 1}} }}^{{x_{k} }} {h_{i} (x_{1} , \ldots ,x_{{r - 1}} ,\;x^{\prime}_{r} ,\;x_{{r + 1}} , \ldots ,x_{n} ,\;s;\;t)} ds} + \int\limits_{{x_{{r - 1}} }}^{{x^{\prime}_{r} }} {h_{r} (x_{1} , \ldots ,x_{{r - 1}} ,\;x^{\prime}_{r} ,\;x_{{r + 1}} , \ldots ,x_{n} ,\;s;\;t)} ds \\ & \quad \quad + \int\limits_{{x^{\prime}_{r} }}^{{x^{\prime}_{{r + 1}} }} {h_{{r + 1}} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds \\ & \quad = \phi _{r} (x_{1} , \ldots ,x_{{r - 1}} ,\;x^{\prime}_{r} ,\;x_{{r + 1}} , \ldots ,x_{n} ;\;t) + \int\limits_{{x^{\prime}_{r} }}^{{x^{\prime}_{{r + 1}} }} {h_{{r + 1}} (x_{1} ,\;x_{2} , \ldots ,x_{n} ,\;s;\;t)} ds \\ & \quad \le \phi _{r} (x^{\prime}_{1} , \ldots ,x^{\prime}_{{r - 1}} ,\;x^{\prime}_{r} ,\;x^{\prime}_{{r + 1}} , \ldots ,x^{\prime}_{n} ;\;t) + \int\limits_{{x^{\prime}_{r} }}^{{x^{\prime}_{{r + 1}} }} {h_{{r + 1}} (x^{\prime}_{1} ,\;x^{\prime}_{2} , \ldots ,x^{\prime}_{n} ,\;s;\;t)} ds \\ & \quad = \phi _{{r + 1}} (x^{\prime}_{1} , \ldots ,x^{\prime}_{{r - 1}} ,\;x^{\prime}_{r} ,\;x^{\prime}_{{r + 1}} , \ldots ,x^{\prime}_{n} ;\;t), \\ \end{aligned} $$

which implies that the equality φ_r+1(x₁, x₂,…,x_n; t) ≤ φ_r+1$(x_{1}^{\prime},\,x_{2}^{\prime},\ldots,x_{n}^{\prime}; t )$ holds.

(ii) Now consider the case when $x_{r}\le x_{r+1}\le x_{r}^{\prime}\le x_{r+1}^{\prime}.$ In this case, we have

$$\begin{aligned} & \phi _{{r + 1}} (x_{1} ,x_{2} , \ldots ,x_{n} ;t) = \sum\limits_{{k = 1}}^{r} {\int\limits_{{x_{{k - 1}} }}^{{x_{k} }} {h_{i} (x_{1} ,x_{2} , \ldots ,x_{n} ,s;t)} ds} + \int\limits_{{x_{r} }}^{{x_{{r + 1}} }} {h_{{r + 1}} (x_{1} ,x_{2} , \ldots ,x_{n} ,s;t)} ds \\ & \quad \le \sum\limits_{{k = 1}}^{r} {\int\limits_{{x_{{k - 1}} }}^{{x_{k} }} {h_{i} (x_{1} ,x_{2} , \ldots ,x_{n} ,s;t)} ds} + \int\limits_{{x_{r} }}^{{x_{{r + 1}} }} {h_{{r + 1}} (x_{1} ,x_{2} , \ldots ,x_{n} ,s;t)} ds \\ & \quad + \int\limits_{{x_{{r + 1}} }}^{{x_{r} '}} {h_{{r + 1}} (x_{1} ,x_{2} , \ldots ,x_{n} ,s;t)} ds + \int\limits_{{x_{r} '}}^{{x_{{r + 1}} '}} {h_{{r + 1}} (x_{1} ,x_{2} , \ldots ,x_{n} ,s;t)} ds \\ & \quad \le \sum\limits_{{k = 1}}^{r} {\int\limits_{{x_{{k - 1}} }}^{{x_{k} }} {h_{i} (x_{1} ,x_{2} , \ldots ,x_{n} ,s;t)} ds} + \int\limits_{{x_{r} }}^{{x_{{r + 1}} }} {h_{r} (x_{1} ,x_{2} , \ldots ,x_{n} ,s;t)} ds \\ & \quad + \int\limits_{{x_{{r + 1}} }}^{{x_{r} '}} {h_{r} (x_{1} ,x_{2} , \ldots ,x_{n} ,s;t)} ds + \int\limits_{{x_{r} '}}^{{x_{{r + 1}} '}} {h_{{r + 1}} (x_{1} ,x_{2} , \ldots ,x_{n} ,s;t)} ds \\ & \quad \le \sum\limits_{{k = 1}}^{r} {\int\limits_{{x_{{k - 1}} }}^{{x_{k} }} {h_{i} (x_{1} , \ldots ,x_{{r - 1}} ,x^{\prime}_{r} ,x_{{r + 1}} , \ldots ,x_{n} ,s;t)} ds} + \int\limits_{{x_{r} }}^{{x_{{r + 1}} }} {h_{r} (x_{1} , \ldots ,x_{{r - 1}} ,x^{\prime}_{r} ,x_{{r + 1}} , \ldots ,x_{n} ,s;t)} ds \\ & \quad + \int\limits_{{x_{{r + 1}} }}^{{x_{r} '}} {h_{r} (x_{1} , \ldots ,x_{{r - 1}} ,x^{\prime}_{r} ,x_{{r + 1}} , \ldots ,x_{n} ,s;t)} ds + \int\limits_{{x_{r} '}}^{{x_{{r + 1}} '}} {h_{{r + 1}} (x_{1} ,x_{2} , \ldots ,x_{n} ,s;t)} ds \\ & \quad \le \phi _{r} (x_{1} , \ldots ,x_{{r - 1}} ,x^{\prime}_{r} ,x_{{r + 1}} , \ldots ,x_{n} ;t) + \int\limits_{{x_{r} '}}^{{x_{{r + 1}} '}} {h_{{r + 1}} (x_{1} , \ldots ,x_{{r - 1}} ,x^{\prime}_{r} ,x_{{r + 1}} , \ldots ,x_{n} ,s;t)} ds \\ & \quad \le \phi _{r} (x^{\prime}_{1} , \ldots ,x^{\prime}_{{r - 1}} ,x^{\prime}_{r} ,x^{\prime}_{{r + 1}} , \ldots ,x^{\prime}_{n} ;t) + \int\limits_{{x_{r} '}}^{{x_{{r + 1}} '}} {h_{{r + 1}} (x^{\prime}_{1} , \ldots ,x^{\prime}_{{r - 1}} ,x^{\prime}_{r} ,x^{\prime}_{{r + 1}} , \ldots ,x^{\prime}_{n} ;s,t)} ds \\ & \quad = \phi _{{r + 1}} (x^{\prime}_{1} , \ldots ,x^{\prime}_{{r - 1}} ,x^{\prime}_{r} ,x^{\prime}_{{r + 1}} , \ldots ,x^{\prime}_{n} ;t), \\ \end{aligned}$$

which implies that the equality φ_r+1(x₁, x₂,…,x_n; t) ≤ φ_r+1(x ^′₁ , x ^′₂ ,…,x ^′_n ; t).

Therefore, by mathematical induction, it holds that φ_m(x₁, x₂,…,x_n; t) ≤ φ_m(x ^′₁ , x ^′₂ ,…,x ^′_n ; t), for all m, m = 2, 3,…,n, which completes the proof. The other cases can be shown symmetrically.

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