## Friday, 26 October 2012

### Estimating parametric semi-Markov models from panel data using phase-type approximations

Andrew Titman has a new paper in Statistics and Computing. This extends previous work on fitting semi-Markov models to panel data using phase-type distributions. Here, rather than assume a model in which each state is assumed to have a Coxian phase-type sojourn distribution, the model assumes a standard parametric sojourn distribution (e.g. Weibull or Gamma). The computational tractability of phase-type distributions is exploited by approximating the parametric semi-Markov model by a model in which each state has a 5-phase phase-type distribution. In order to achieve this, a family of approximations to the parametric distribution, with scale parameter 1, is developed by solving a relatively large one-off optimization assuming the optimal phase-type parameters for given shape parameters evolve as B-spline functions. The resulting phase-type approximations can then be scaled to give an approximation for any combination of scale and shape parameter and then embedded into the overall semi-Markov process. The resulting approximate likelihood appears to be very close to the exact likelihood, both in terms of shape and magnitude.

In general a 2-phase Coxian phase-type model is likely to give similar results to a Weibull or Gamma model. The only advantages of the Weibull or Gamma model are that the parameters are identifiable under the null (Markov) model so standard likelihood ratio tests can be used to compare models (unlike for the 2-phase model). Also the Weibull or Gamma model requires fewer parameters so may be useful for smaller sample sizes.

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