ABSTRACT
Cox, a seminal work, presented a model of breakages of thread as it is fed into looms. A Cox process model of precipitation occurrences is formulated in Smith and Karr. The unifying element in the applications is a physical interpretation for the directing random measure, that is, the structure of a Cox process mirrors the physical nature of the system being modeled. Even for statistical inference one ordinarily wishes to deal with probability laws of directing measures because it is they that are dictated by physical considerations, but must do so with observations solely of the Cox processes. Special structure of Cox processes is exploited for statistical inference through distributional relationships that make possible estimation and hypothesis testing for unobservable directing measures. The chapter discusses the problem of combined statistical inference and state estimation for mixed Poisson processes, with briefer analysis of more difficult techniques available for general Cox processes.
