Consistency and Limiting Distribution of the Maximum Likelihood Estimator of a Generalized Threshold Model

The open-loop Threshold Model, proposed by Tong [30], is a piecewise-linear stochastic regression model useful for modeling conditionally normal response time-series data. However, in many applications, the response variable is conditionally non-normal, e.g. Poisson or binomially distributed. We generalize the open-loop Threshold Model by introducing the Generalized Threshold Model (GTM). 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. We introduce a likelihood-based estimation scheme for the GTM, and the consistency and limiting distribution of the maximum likelihood estimator are derived. A simulation study is conducted to illustrate the asymptotic results.