Discontinuous decision processes and threshold autoregressive time series modelling

where p is a 'smooth' function. Suppose that we approximate 4u(x) by 0, a constant, for all x. In subsequent discussion we suppress the argument x whenever this may be done without obscuring the context. Clearly this approximation will incur some errors. On the other hand, as argued by Tong & Lim (1980), we could approximate / arbitrarily closely by a step function. At first sight, it seems that 'this could pose a horrendous computational problem. Indeed, this is the case from a purely deterministic point of view. However, we usually approximate the 'true' model with some purpose in mind, e.g. forecasting, control, filtering. Thus, we should really specify what we mean by approximating ,u arbitrarily closely. Bayesian decision theory seems a natural approach to this problem. In particular, the recent general results of Smith et al. (1981) provide the necessary framework.