Some characterizations of non-ergodic estimating functions for stochastic processes

We are concerned with the stochastic process for which the likelihood of the data, although it does exist, is possibly unknown to us. In order to estimate parameter, we are led to appropriate estimating functions (EF, for short) including the (quasi) maximum likelihood score. A formal definition of the general non-ergodic estimating function is made in this paper. This can be viewed as a generalization of the non-ergodic maximum likelihood score (due to Basawa and Koul (1979), and Basawa and Scott (1983)) toward the theory of EFs. In addition, some characterizations on the non-ergodic EFs are made. It is interesting to note non-standard cases where non-stationary process may yield an ergodic EF while stationary process can produce a non-ergodic EF. Various examples are presented to illustrate the main results.