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STS535 Edsel A. P. et al.
• The regression coefficients , , and in the sub-model for the
1
1
1
evolution of the -process.
• The regression coefficients , , and in the sub-model for the
2
2
2
evolution of the -process.
• The regression coefficients , , , and in the sub-model for the
3
3
3
evolution of the -process.
• The parameters in the functions (·; ) in the -process sub-

model.
In the CTMCs for the - and -processes, ordinarily the state holding
times and the transition probabilities are completely determined by the
infinitesimal generators, but in our model these are affected by the other
components through the exponential link functions and the relevant
regression coefficients. Since the values of the processes may change at each
event or transition time, then the state holding times are governed by
piecewise exponential distributions, but where the changes occur are
determined by where the transitions or events occur, hence are dynamic in
some sense.
In each of the models for the three components, there exists a ‘competing
risks’ aspect. In the -process, the states are in some sense competing with
each other.

This is also the case with the -process; and also with the -process. Thus,
when the likelihood function is constructed, this competing aspect needs to
be incorporated, but this is immediately taken cared of by the likelihood
construction using Jacod’s [2] (see also [1]) approach.

4. Statistical Inference Issues
Of critical importance is to be able to infer about the model parameters of
this class of joint models in order that the model could be used in practice.
Such statistical inference will be based on independent observations of n
subjects or units that are monitored over their respective monitoring periods.
For the th unit the random observables , , , , and are observed

over [0, ]. The likelihood process is then constructed from their realizations.

However, due to space limitations, we do not present the statistical inference
approach in this paper, but defer its discussion for the talk during the WSC.
Suffice it to say that the first step in performing the statistical inference is the
construction of the appropriate likelihood process. This is constructed by
exploiting the Markovian structure and also the conditional independence
among the three components given the present. Inference for the parameters

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