Joint detection, estimation and system identification

Recent results of Middleton and Esposito (1968) and Lainiotis (1969) on single-shot joing detection-estimation for discrete data are extended to the single-shot continuous data case and generalized to joint Bayesian detection-estimation-system identification. Moreover, previous results were generalized to the case of causal estimator. Specifically, it is shown that the above problem constitutes a class of nonlinear mse estimation problems, with the attendant difficulties in realizing the optimal nonlinear estimators. However, by utilizing the adaptive approach, closed form integral expressions are given. These are given in terms of the generalized likelihood ratio gλ(t), which is a sufficient statistic for Bayes-optimal compound detection. The latter in turn is specified by a continuum (for continuous θ) θ-conditional likelihood ratious λ(t/θ) each of which is the LR for testing for the model specified by the parameter value θ. The latter LR's are, moreover, given in terms of optimal mse causal estimators. In essence then, it has been shown that system identification is equivalent to multihypothesis testing, with a continuum or finite sequence of hypotheses, respectively, for continuous or finite discrete range of θ