Stochastic comparisons of nonhomogeneous processes (Q2731306)

Let \(N(t)\), \(t\geq 0\), be a nonhomogeneous Poisson counting process with NEWLINE\[NEWLINEP(N(t+ dt)= n+ r\mid N(t)= n)= r(t) dt,NEWLINE\]NEWLINE where \(\int^\infty_t r(u) du= \infty\), \(t\geq 0\). The counting epochs are \(T_0= 0\), \(T_1< T_2<\cdots\) and the interepoch intervals are \(X_i= T_i- T_{i-1}\), \(i\geq 1\). Then \(T_1\) has probability density \(f\) and distribution function \(F\), where \(f(t)= r(t)\exp(- \int^t_0 r(u) du)\) so that \(F\) has hazard rate \(r\). Let a second such process have corresonding quantities \(s(t)\), \(g(t)\), \(G(t)\), epochs \(V_i\) and \(Z_i= V_i- V_{i-1}\). The paper studies conditions so that \((V_1,\dots, V_n)\) is stochastically larger than \((T_1,\dots, T_n)\). Also for \(V_n\) and \(T_n\) and for \((Z_1,\dots, Z_n)\) and \((X_1,\dots, X_n)\). Five types of stochastic ordering for \(d\)-dimensional random vectors are considered: the classical one and (as defined for \(d= 1\)) the hazard rate order (\(r(t)> s(t)\) for \(T_1\) and \(V_1\)), the likelihood ratio order (\(r(t)/s(t)\) increases), the dispersive order (\(F^{-1}(b)- F^{-1}(a)\leq G^{-1}(b)- G^{-1}(a)\), \(0< a< b< 1\)) and the mean residual life order \((E[X- t\mid X> t]\leq E[Y- t\mid Y> t])\). Some connections between these orders are given. A typical results: \((T_1,\dots, T_n)\leq (V_1,\dots, V_n)\) if and only if \(T_1\leq V_1\) and then \(T_n\leq V_n\), \(n\geq 1\), all in the hazard rate order.NEWLINENEWLINENEWLINEAnalogous results are derived for two nonhomogeneous birth processes where NEWLINE\[NEWLINEP(N(t+ dt)= n+ 1\mid N(t)= n)+ r_n(t) dt\text{ and }s_n(t) dt,NEWLINE\]NEWLINE respectively. Orderings now are derived by a representation in distribution of the \(T_i\) in terms of a sequence of independent random variables with hazard rate \(r_i\) and similarly for the \(V_i\).NEWLINENEWLINENEWLINEApplications: The nonhomogeneous Yule process in epidemiology. Load sharing models and comparison of items that are continuously minimally repaired.