Abstract
The work described in this thesis resulted from the author's attempts to analyse some data collected by about seventy general practitioners (see Chapter III). There was no intention, initially, of developing statistical methods not previously described, but it gradually became apparent that there were many gaps, both in theory and practice, to be filled. I hope that the content here may be seen as a contribution, along with the work of many others, towards statistical analyses for data which are not "i.i.d." In the first section a condition for consistency of maximum likelihood estimation is derived. The situation treated is general, where the observations are dependent, non-identically distributed, except that the usual regularity conditions are assumed to be satisfied by the conditional probability density functions, Asymptotic Normality is also discussed, using a martingale central limit theorem. It is then shown that a wide class of stochastic processes (so called mixing processes) satisfy our requirements for c.a.n. estimation. Next we apply the above results to the Logit analysis of nonhomogeneous Markov chains. The binary case is treated first, then the case of m states (2<m<infinity). Analogous models are suggested for the case of a denumerable state-space, with a Poisson example worked in detail. Also more general models ("quasi-linear") are discussed which have similar asymptotic properties to the basic logit model. An application follows. The data, on certain common infectious diseases, has been collected for some years by the Royal College of General Practitioners. We use a logit model of the type examined in section 2 for its analysis, and briefly discuss points arising out of fitting and testing the goodness-of-fit of the model, and in assessing various hypotheses of medical interest. In the last section some further questions concerning m.l.e. are discussed, with applications.