The problem of state estimation with stochastic uncertainties in the initial state, model noise, and measurement noise is considered using the restricted risk Bayes approach. It is assumed that the a priori distributions of these quantities are not perfectly known, but that some information about them may be available. While offering robustness, the restricted risk Bayes approach incorporates the available a priori information to give less conservative state estimators than the Gamma-minimax approach popular in the literature. When attention is restricted to linear estimators based on a quadratic loss function, a systematic method to derive restricted risk Bayes estimators is proposed. Applying to the filtering problem, the restricted risk Bayes approach provides us with a robust method to calibrate the Kalman filter (KF), considering the presence of stochastic uncertainties. This method is illustrated with a target tracking example and a wireless channel tracking example for which the Bayes, minimax, and restricted risk Bayes estimators are derived and their performance is compared.

The problem of state estimation with initial state uncertainty is approached from a statistical decision theory point of view. The initial state is regarded as deterministic and unknown. It is only known that the initial state vector belongs to a specified parameter set. The (frequentist) risk is considered as the performance measure and the minimax approach is adopted. Minimax estimators are derived for some important cases of unbounded parameter sets. If the parameter set is bounded, a method of finding estimators whose maximum risk is arbitrarily close to that of a minimax estimator is provided. This method is illustrated with an example in which an estimator whose maximum risk is at most 3% larger than that of a minimax estimator is derived.

The problem of linear tracking of fast Rayleigh-faded MIMO channels is considered. Due to computational complexity limitations, Kalman filters based on low-order autoregressive (AR) models are often used although realistic Rayleigh-faded channels cannot be properly approximated by low-order AR models. By increasing the order of the AR model, the channel tracking performance is considerably improved, but the resulting computational complexity may become large. In this paper, the filters used to track the channel are restricted to have a recursive structure, based on the previous estimate and new observation. The knowledge of the autocorrelation function of the channel taps is used to derive filters that are significantly less complex than high-order AR model based Kalman filters at the price of only a minor degradation in performance.

The problem of estimating the location and time of origin of an instantaneous source of a particular gas using a simple sensor network is investigated in this paper. Here, the gas spreads by diffusion and the sensors make a binary decision on the existence of the gas by measuring its concentration in their immediate vicinity. The inability of the sensors to go beyond a binary resolution of the concentration justifies their classification as dumb. In this paper, it is restricted to the one-dimensional case where the sensors form a linear array. The analysis of the Cramer-Rao bound (CRB) to study the limits of estimation performance is numerically calculated. Based on the number of gas molecules the Poisson distribution by a Gaussian distribution with mean and variance is approximated.