The method of maximum likelihood is used to estimate parameterized transition probabilities of a non-homogeneous Markov chain model of movements between the health states disability-free, disabled, and death. A complication is that individuals are observed at irregular intervals, implying that particular attention must be paid to computational issues. Numerical results including estimated health expectancies and their standard errors are given for data from the Longitudinal Study on Aging (LSOA) of 1984-86-88-90 (Kovar et al. 1992). The weak ergodicity of prevalence on the non-absorbing states is established and supports the hypothesis of the compression of morbidity.
Summary
It is well known that in the absence of immigration, a population of like particles developing under the usual laws for branching processes either increases unboundedly with time or becomes extinct (Feller, 1957, Chapter 12; Harris, 1963, Chapter 1). If migration into the system is permitted, then it is clear that under certain conditions a proper stationary distribution for population size will exist. That this is the case has been shown by Bartlett (1956, Section 3.41) for processes in continuous time, and the present note is concerned with the same problem but considering discrete generations. Our result for the generating function of the stationary distribution of population size reduces to a form analogous to Bartlett's result when the immigration distribution is Poisson.
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