Sufficient conditions for the uniqueness of parameter estimates from binary-response data

Abstract

The procedure of fitting parameterized models to experimental data is that of extremalizing a statistically meaningful scalar-valued vector function. The existence of multiple local extrema can greatly complicate the search for the global solution. Sufficient conditions for uniqueness of the parameter estimates are usually determined from the convexity of the criterion surface: the convexity properties are determined by the statistical criterion, the structure of the model, the underlying distribution, and the observations (data). In this paper, we seek the combinations of criteria, models, and distributions which yield sufficient conditions for unique parameter estimates regardless of the observed binary-response data values. Under mild sufficient conditions usually satisfied in practice, the Maximum Likelihood, Minimum Chi Square, and Minimum Transform Chi Square criteria are convex functions when the parameters appear linearly. These results are applied to equalvariance models of signal detection/recognition, sequential response, and additive learning models with implications on the experimental design. Unequal-variance models and models of discrete-sensory processing (rectilinear ROC curves) lead to nonconvex criteria for some observations (saddlepoints are demonstrated). Although convexity cannot be assured for these cases, the results suggest an efficient search procedure in a lower dimensional subspace to find global extrema. The extension of these results to more than two response levels is discussed.