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Classical statistical approaches for multiclass probability estimation are typically based on regression techniques such as multiple logistic regression, or density estimation approaches such as linear discriminant analysis (LDA) and quadratic discriminant analysis (QDA). These methods often make certain assumptions on the form of probability functions or on the underlying distributions of subclasses. In this article, we develop a model-free procedure to estimate multiclass probabilities based on large-margin classifiers. In particular, the new estimation scheme is employed by solving a series of weighted large-margin classifiers and then systematically extracting the probability information from these multiple classification rules. A main advantage of the proposed probability estimation technique is that it does not impose any strong parametric assumption on the underlying distribution and can be applied For a wide range of large-margin classification methods. A general computational algorithm is developed for class probability estimation. Furthermore, we establish asymptotic consistency of the probability estimates. Both simulated and real data examples are presented to illustrate competitive performance of the new approach and compare it with several other existing methods. KEY WORDS: Fisher consistency; Hard classification; Multicategory classification; Probability estimation: Soft classification; SVM.

While Distance Weighted Discrimination (DWD) is an appealing approach to classification in high dimensions, it was designed for balanced datasets. In the case of unequal costs, biased sampling, or unbalanced data, there are major improvements available, using appropriately weighted versions of DWD (wDWD). A major contribution of this paper is the development of optimal weighting schemes for various nonstandard classification problems. In addition, we discuss several alternative criteria and propose an adaptive weighting scheme (awDWD) and demonstrate its advantages over nonadaptive weighting schemes under some situations, the second major contribution is a theoretical study of weighted DWD. Both high-dimensional low sample-size asymptotics and Fisher consistency of DWD are studied. The performance of weighted DWD is evaluated using simulated examples and two real data examples. The theoretical results are also continued by simulations. KEYWORDS: Fisher consistency: High dimensional, low sample-size data; Linear discrimination: Nonstandard asymptolics; Unbalanced data.

A novel method for initiating discrimination training with nonverbal children combines a delayed S+ procedure that requires children to refrain from responding to either of 2 physically different choice stimuli until a prompt stimulus is added onto 1 of the choices, and a delayed prompting procedure that presents the same 2-choice stimulus display, but stimuli are initially added onto both choices. After a short delay, the added stimulus on the S- is removed, and the choice of the S+ is thus prompted. If the children learn to observe and respond to the defining features of the S+ choice stimulus, then they may respond to the S+ prior to the added-stimulus removal. Implementation was successful with 8 nonverbal children who had not previously exhibited simple simultaneous discrimination, suggesting a useful methodology for initiating discrimination training with populations for whom verbal instruction is ineffective., Among the most difficult challenges for behavioral scientists and educators is to communicate effectively with children who are nonverbal (Sidman & Stoddard, 1966). The challenge is similar to that faced [...]

This study analyzed psychophysical data from younger and elderly people on the influence of spacing between vertical lines, exposure time, and number of vertical lines in a stimulus for visual performance. A total of 50 elderly people and 31 graduate students participated in the experiment. Nine levels of spacing between lines (3, 6, 9, 12, 15, 18, 21, 24, and 29 points), three exposure times (0.2, 0.6, and 1.0 sec.), and five sets of numbers of lines (4, 5, 6, 7, and 8) were manipulated. Analysis showed enhanced spacing between lines and exposure time improved discrimination of separation. However, although performance on discrimination of separation increased as spacing between lines increased up to 21 points, it was degraded at higher values. A positive effect of number of lines on discrimination of separation was also observed, and performance increased as the number of lines decreased. The effect of age group, i.e., elderly versus younger, on performance was significant. The accuracy of the younger was greater than that for the elderly group. Moreover, three interactive two-way effects were found: group x spacing between lines, number of lines x spacing between lines, and number of lines x exposure time. The present findings could be used as a practical reference in the design of instrument displays in which the operator has to consider the scale and markings on a dial, especially if the display is operated in an emergency or is manipulated by elderly people.