Cook's fisher lectureship revisited for semi-supervised data reduction

R. D. Cook's Fisher lectureship (Cook, Stat Sci 22:1-26, 2007) opened a new and seminal paradigm in sufficient dimension reduction literature. It suggests a model-based approach, and it enables us to reduce the dimension of categorical and continuous predictors simultaneously. Here, the lectureship is extended for reducing the predictors in semi-supervised data under an isotonic error model, which has been common in many popular science fields such as speech recognition, spam email filtering, artificial intelligence, video surveillance, and so on. Under the isotonic error model, a combined dimension reduction model is proposed for semi-supervised data, and related theories are investigated. Numerical studies and real data example confirm its potential usefulness.