In many parameteric statistical decision problems, one has to make a decision in the presence of some uncertainties about the parameters in the statistical model of the observed random variables. Examples include estimation of signal parameters in additive noise of unknown power and detection of a signal of unknown amplitude. We present a methodology to test multiple hypotheses on the distribution of a random variable when the hypotheses are parameterized by fuzzy variables. The proposed approach has a Bayesian favor in the sense that the objective is to minimize a fuzzy average decision error probability by a proper choice of decision regions. We use a scalar index, called the total distance criterion (TDC) ranking index, in order to rank the fuzzy average decision error probabilities of different decision rules. We derive the optimal decision rule which minimizes the TDC index of the fuzzy average decision error probability. As an example we apply the general approach proposed here to the classification of the fuzzy mean of a Gaussian random variable.

This paper addresses the problem of finite sample simultaneous detection and estimation which arises when estimation of signal parameters is desired but signal presence is uncertain. In general, a joint detection and estimation algorithm cannot simultaneously achieve optimal detection and optimal estimation performance. We develop a multihypothesis testing framework for studying the tradeoffs between detection and parameter estimation (classification) for a finite discrete parameter set. Our multihypothesis testing problem is based on the worst case detection and worst case classification error probabilities of the class of joint detection and classification algorithms which are subject to a false alarm constraint. This framework leads to the evaluation of greatest lower bounds on the worst case decision error probabilities and a construction of decision rules which achieve these lower bounds. For illustration, we apply these methods to signal detection, order selection, and signal classification for a multicomponent signal in noise model. For two or fewer signals, an SNR of 3 dB and signal space dimension of N=10 numerical results are obtained which establish the existence of fundamental tradeoffs between three performance criteria: probability of signal detection, probability of correct order selection, and probability of correct classification. Furthermore, based on numerical performance comparisons between our optimal decision rule and other suboptimal penalty function methods, we observe that Rissanen's (1978) order selection penalty method is nearly min-max optimal in some nonasymptotic regimes.>

A constrained optimization method is used to reconstruct signals from their noisy linear transforms. In this approach the fit error energy is minimized over a set of signals whose first few normalized autocorrelation coefficients are larger than specified constraint values. This provides another method of introducing regularization to the least squares signal reconstruction. By choosing the constraint values larger more regularization can be obtained. Significant computational saving is gained by solving the corresponding dual optimization problem in the actual computation of the reconstruction. Comparison of the results of this and other approaches leads us to the conclusion that although the proposed approach requires more computation, it provides better reconstructions.>

For a general discrete parameter signal model and a false alarm constraint, the authors give a general theorem which specifies lower bounds on worst-case probability of miss, worst-case probability of incorrectly specifying the number of parameters (order selection), and worst-case probability of parameter estimation error. These bounds are achievable by constrained min-max decision rules: the constrained min-max detector, the constrained min-max order selector, and the constrained min-max parameter estimator, respectively. As in previous work, the authors use the theory to study fundamental tradeoffs between achievable estimation, order selection, and detection performance for a multiple component signal model. Numerical results indicate that for the example studied, parameter estimation optimality entails very little loss in detection performance, while detection optimality severely sacrifices parameter estimation performance.>

A method for gauging the tradeoff between parameter estimation and order selection estimation performance for discrete parameters is developed. The method is based on evaluation of lower bounds on worst-case probability of estimation error and worst-case probability of order selection error subject to a constraint on maximum false alarm probability. Since each of the two bounds is derived by evaluating the performance of an optimal order estimator and an optimal parameter estimator, respectively, the bounds are achievable. It is shown that it is generally impossible to achieve simultaneously the order selection and parameter estimation lower bounds: there is a necessary compromise between estimation and order selection. The authors present numerical studies of these bounds for a common multiple signal model.>

The performance of the MUSIC (Multiple Signal Classification) algorithm in determining the bearing of a single emitter from the signals of an array of sensors is investigated. The causes and cures of ambiguities in bearing estimation, due to array response duplications, are addressed first. The statistical properties of unambiguous bearing estimates are then considered. In particular, expressions for the first two moments of the error in these estimates are presented. These expressions, being more general than those that have appeared in the literature, are not limited to a line array of isotropic elements; hence they include the effects of antenna pattern and array geometry on estimation accuracy. The numerical examples verify the analytical moment expressions.>