A generalized threshold mixed model for analyzing non-normal nonlinear time series

We introduce the Generalized Threshold Mixed Model (GTMM) for piecewise-linear stochastic regression with (possibly) non-normal time-series data. Specifically, it is assumed that the conditional probability distribution of the response variable belongs to the exponential family, and the conditional mean response is linked to some piecewise-linear stochastic regression function; the domain of each linear submodel is referred to as a regime.; We first study the specific case where the response variable equals zero in the lower regime. Some large-sample properties of a likelihood-based estimation scheme are derived. Our approach is motivated by the need for modeling nonlinearity in serially correlated epizootic events. Data coming from monitoring conducted in a natural plague focus in Kazakhstan are used to illustrate this model by obtaining biologically meaningful conclusions regarding the threshold relationship between the prevalence of plague and some covariates including past abundance of great gerbils and other climatic variables.; The real application illustrates the potential usefulness of the GTMM in analyzing epidemiological time series subject to a threshold condition. While the specific case of zero in the lower regime has a sound epidemiological justification, it is of interest to study the more general model that the non-normal response follows an unrestricted generalized piecewise-linear model. We investigate this problem by focusing on the Generalized Threshold Model (GTM) which is a GTMM without the random effect. We study maximum likelihood estimation of the GTM, and derive its large-sample properties.