Robust quantizers are designed for situations where there is only an incomplete statistical description of the quantizer input. The goal of the design is to closely approximate quantizer inputs by quantizer outputs without using more than a specified number of quantization levels. The exact probability distribution of the input is unknown, but this distribution is known to belong to some setC. The primary, setCconsidered is the set of all unimodal probability distributions which satisfy generalized moment constraint (e.g., mean-square value less than or equal to a constant). A quantizer is derived which minimizes over all quantizers the maximum distortion over all distributions inC. This robust quantizer guarantees a significanfiy lower worst case distortion than the classical Gaussian-optimal quantizer, while performing nearly as well as the Gaussian-optimal quantizer when the input is, in fact, Gaussian.

A problem in the sequential detection of weak signals in additive noise is solved under the assumption that the unknown noise density function is a member of some known class of symmetric densities. Two general approaches to the design of receivers which are asymptotically most robust in a minimax sense are established. A comparison between the two methods is provided for the special case when the noise density class is defined byF= \{ f: \int_{-a}^{a} f( x) dx = p, fsymmetric and contiauous at\pma \}. Furthermore the savings in the expected sample size over some nonsequential receivers are calculated.

The problem of minmax estimation of a location parameter introduced by Huber is considered. It is shown that under general conditions there exists a solution which is a form of the Robbins-Monro stochastic approximation algorithm. This generalizes earlier work by Martin and Masreliez who have given stochastic approximation (SA)-estimate solutions for two particular cases. As with theM-estimate solutions given by Huber, the SA solutions are completely determined by the probability distribution function with least Fisher information in the distribution set used to model the observation errors.

The problem of detection of known signals in additive noise is solved under the assumption that the unknown noise density is a member of some known family of symmetric densities. A general approach to the design of receivers that are asymptotically most robust is established. As an example, a detector is derived by applying the procedure to the special case obtained when the noise density family is defined byF=\left\{f \left| \int^{a}_{-a} f(x) dx = p, f \mbox{symmetric} \right}.Simulation results showing the detector's performance for small sample sizes are provided.

TheN-level uniform quantizer on[-c,c]plus the assignment ofy_{0}\deg = -(a _{s}+ c)/2andy_{N+1}\deg = (a_{s}+ c)/2to signal values falling in the saturation regions[-a_{s},- c) and (c,a_{s}], respectively, is shown to be the minimax(N + 2)-level quantizer with a nonsaturating input range[-c,c]. The performance criterion considered is the mean weighted quantization error and the input signals are only required to be amplitude bounded by\pm a_{s}wherea_{s} > c > 0. The worst case input signal marginal probability distributions are shown to be discrete. From the derivation of this result, the minimax error can be computed. An example is given which illustrates the performance of the minimax quantizer for several input ranges against different input signal probability distributions.

Simultaneous detection and estimation under multiple hypotheses when data from only one observation interval are available, are treated on the basis of statistical decision theory. Estimation is carried out under the assumption that the signal of interest is not present with probability 1, which is necessary if detection is to be a meaningful operation. Also, we consider the case where the operations of detection and estimation are coupled. Specific detector and estimator structures are determined for the case of strong coupling when the cost of estimation error is given by a quadratic function. The detector structures are in general complex nonlinear functions of the received data. However, a detailed analysis of the Gaussian case resulted in a type of correlation detector, which correlates the received data with the least square estimators of the possible signals in the absence of uncertainty. The associated optimum estimator structure is found to be a weighted sum of least square estimators in the absence of uncertainty. Also, joint detection and estimation under multiple hypotheses is discussed for the case of a simple cost function. The estimators that result can be interpreted as generalized maximum-likelihood estimators. Finally, optimum prediction and filtering are briefly considered.