Partial likelihood for online order selection

Partial likelihood (PL) is a flexible framework for adaptive nonlinear signal processing allowing the use of a wide class of nonlinear structures-probability models-as filters. PL maximization has been shown to be equivalent to relative entropy minimization for the general case of time-dependent observations and its large sample properties have been established. In this paper, we use these properties to derive an information-theoretic criterion for order selection-the penalized partial likelihood (PPL) criterion,-for the general case of dependent observations. We then consider nonlinear signal processing by conditional finite normal mixtures as an example, a problem for which true order selection is particularly important. For this case, in which the PL coincides with the usual likelihood formulation, we present a formulation for online order selection by eliminating the need to store all data samples up to the current time. We demonstrate the successful application of the PPL criterion and its online implementation for the equalization problem by simulation examples.