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Article Excerpt
1. INTRODUCTION
Markov chains in both discrete-time and continuous-time have been very popular modeling tools in applications, and the estimation methods for these models have been well established. Modeling the evolution of a population of individual entities with a Markov chain assumes that the population is homogeneous with respect to the movement behavior among the states. However, in many applications the studied population is heterogeneous with respect to the dynamics of transitions among states, and a Markov chain provides an inaccurate description of the evolution of the population.
The purpose of this article is to define and estimate the mixture of time-homogeneous Markov chains that captures the population heterogeneity in the rate of movement among states. Identification of different regimes with respect to the rate of movement, or with respect to some other aspect of movement behavior, is of obvious interest for describing the dynamics of the observed process. It also has an exploratory value for identifying and characterizing possible nonhomogeneous subpopulations. In addition, the non-Markovian property of a mixture process offers a possibility of taking into account not only a present state, but also the past history of the process when making forecasts of the future states.
Both continuous-time and discrete-time versions of the mixture process are considered herein. To define a continuous-time version, let X = {X(t), t > 0} be a stochastic process with state space D = {1, 2,...,w}, which, conditionally on the initial state, is a mixture of homogeneous Markov chains, [X.sub.m] = {[X.sub.m](t), t > 0}, with generators [A.sub.m], 1 [less than or equal to] m [less than or equal to] N. It is assumed that the generators are related by
[A.sub.m] = [[GAMMA].sub.m]Q, 1 [less than or equal to] m [less than or equal to] N, (1)
where [[GAMMA].sub.m] = diag([[gamma].sub.1,m], [[gamma].sub.2,m],...,[[gamma].sub.w,m]), 1 [less than or equal to] m [less than or equal to] N - 1, [[GAMMA].sub.N] = I. The mth generator, [A.sub.m], is a matrix with entries [a.sub.ij] satisfying
[a.sub.ii,m] [less than or equal to] 0, [a.sub.ij,m] [greater than or equal to] 0,
[summation over (j[not equal to]i)] [a.sub.ij,m] = -[a.sub.ii,m] [equivalent to] [a.sub.i,m], i [member of] D,
where [a.sub.ij,m] = [[gamma].sub.i,m][q.sub.i,j], i [not equal to] j, and [a.sub.i,m] = [[gamma].sub.i,m][q.sub.i]. In particular, 1/(-[q.sub.ii]) is the expected length of time that [X.sub.N](t) remains in state i, i [member of] D. The discrete mixing distribution on these Markov chains is defined conditionally on X(0). The mixture probabilities, [s.sub.i,m], depend on the initial state i,
[s.sub.i,m] = P(X = [X.sub.m]|X(0) = i),
and for each i satisfy [s.sub.i,m] [greater than or equal to] 0, m = 1, 2,..., N, and [[summation].sub.m=1.sup.N] [s.sub.i,m] = 1. The transition probability matrix of the mixture process is
P(t) = [N.summation over (m=1)][S.sub.m][P.sub.m](t), t [greater than or equal to] 0, (2)
where [P.sub.m](t) = exp(t[A.sub.m]), and [S.sub.m] = diag([s.sub.1,m], [s.sub.2,m],..., [s.sub.w,m]). The Markov chains that compose the mixture may differ in the rates ([a.sub.i,m]) at which they move among the states, but all chains when leaving a state i have the same probability of entering state j. Depending on whether [[gamma].sub.i,m] = 0, < [[gamma].sub.i,m] < 1, or [[gamma].sub.i,m] > 1, the realizations generated by [A.sub.m] do not move out of state i or move out of state i at a lower or a higher rate than those generated by Q.
The particular case of the proposed mixture process, known as a mover-stayer model, arises when N = 2 and [[gamma].sub.1,1] = [[gamma].sub.2,1] = ... = [[gamma].sub.w,1] = 0. It postulates that population consists of "movers," who evolve according to a Markov chain and of "stayers," who never leave their initial states. The motivation for the new mixture was to relax the "mover-stayer" dichotomy and allow for subpopulations to have different rates of movement from states. Despite its rather restrictive assumptions, the mover-stayer mixture, particularly its discrete-time version, has gained considerable popularity since its introduction by Blumen, Kogan, and McCarthy (1955). It has been applied to modeling of occupational mobility (Sampson 1990), income dynamics (Dutta, Sefton, and Weale 2001), consumer brand preferences (Chatterjee and Ramaswamy 1996; Colombo and Morrison 1989), bond ratings migrations (Altman and Kao 1991), credit behavior (Frydman, Kallberg, and Kao 1985), and tumor progression (Tabar et al. 1996; Chen, Duffy, and Tabar 1997).
A discrete-time analog of the mixture process in (2) is defined in Section 4. For both continuous-time and discrete-time mixtures, this article develops the maximum likelihood estimation of their parameters from a sample of independent, continuously observed realizations of the mixture process. For each mixture, the maximum likelihood estimators (MLEs) of their parameters are first obtained under the complete information, that is, when it is known which Markov chain generated each realization. These are then used in the expectation step of the EM algorithm for estimating the mixtures under incomplete information.
This development shows that in both the continuous-time and discrete-time cases, all mixture parameters can be identified. In the continuous-time case, this is a consequence of restriction (1) on the form of generators, and in the discrete-time case, it is a consequence of restriction (10). As suggested by a referee, it is of interest to consider more general identifiable mixtures than the one defined in (2). One possible generalization of (2) is to assume that [A.sub.m] is obtained from the basic generator Q by introducing two gamma parameters for each row (in the present model there is one gamma parameter, [[gamma].sub.i,m], for the ith row), a gamma parameter for the upward transitions (above diagonal) and one for the downward transitions (below diagonal). The estimation in this extension of (2) and in more general ones is left to the future investigation.
Maximum likelihood estimation in the discrete-time mover-stayer model was developed by Frydman (1984) and Fuchs and Greenhouse (1988). Some additional aspects of this estimation were discussed by Swensen (1996). Estimation in the continuous-time mover-stayer model from continuous data was considered by Frydman and Kadam (2004) and from panel data by Fougere and Kamionka (2003). Cook, Kalbfleisch, and Yi (2002) considered estimation of a generalized mover-stayer model from panel data. Because the mover-stayer model is a very special case of the mixture of Markov chains proposed...
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