WBF property and stochastical monotonicity of the Markov process associated to Schur-constant survival functions
Let \(T_1, T_2, \dots, T_n\) be the random lifetimes of \(n\) components, and suppose that their joint survival function is Schur-constant, that is, \[ P(T_1 > t_1, T_2 > t_2, \dots, T_n > t_n) = \varphi \left( \sum^n_{i = 1} t_i \right), \quad (t_1, t_2, \dots, t_n) \in \mathbb{R}^n_+, \] for some function \(\varphi\). Let \(\{K_t, t \geq 0\}\) be the process that counts the live components as a function of time, and let \(\{Y_t, t \geq 0\}\) be the process that gives the total time on test as a function of time. Finally define \(Z_t = (K_t, Y_t)\), \(t\geq 0\). The authors note that \(\{Z_t, t \geq 0\}\) is a Markov process, and they obtain its infinitesimal generator. They also obtain necessary and sufficient conditions for the process \(\{Z_t, t \geq 0\}\) to be stochastically increasing. In particular, they show that \(\{Z_t, t \geq 0\}\) is stochastically increasing if, and only if, \(T_1, T_2, \dots,T_n\) are positively dependent in the sense that they are WBF (weakened by failures), see \textit{I. Norros} [Stochastic Processes Appl. 20, 181-196 (1985; Zbl 0578.60082)]. Finally the authors show that each of the \(T_i\)'s has the DFR (decreasing failure rate) property.
scientific article; zbMATH DE number 844626
scientific article