Goodness-of-fit tests for semi-Markov and Markov survival models with one intermediate state (Q2771555)

A three-state illness-death model with duration dependence is considered where the intensity of the observed counting process \(N_j\) is of the form NEWLINE\[NEWLINE\lambda_j(t)=h_0(t) {\mathbf 1}_{(0,T_{j1}]}(t)+ h(T_{j1},t-T_{j1}){\mathbf 1}_{(T_{j1},T_{j2})}Y_j(t),NEWLINE\]NEWLINE where \(T_{j1}\) and \(T_{j2}\) are the times of transitions from the ``healthy'' to ``deceased'' and from ``deceased'' to ``death'' states, respectively, and \(Y_j\) is a \(\{0,1\}\) valued risk process of the \(j\)-th individual. The model is called semi-Markov if \(h(T_{j1}, t-T_{j1})\) depends only on \(t-T_{j1}\) for \(T_{j1}\leq t\), and is called Markov if \(h(T_{j1}, t-T_{j1})\) depends only on \(t\) for \(T_{j1}\leq t\). The observed data are \((N_j(t), Y_j(t), T_{j1}\wedge t,T_{j2}\wedge t)\).NEWLINENEWLINENEWLINEThe authors propose tests for the Markov and semi-Markov hypotheses based on the Nelson-Aalen estimator of the intensity. Asymptotic properties of the tests are investigated.