Adaptive Regression for Dependent Data by Combining Different Procedures

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This proposal concerns research and education on adaptive regressionwhen the random errors are dependent. Many procedures have been (and will be) proposed for nonparametric regression based ondifferent assumptions. In applications, a difficulty a user oftenfaces is the choice of the best method for the data athand. This is specially the case for high-dimensional functionestimation, where to overcome the curse of dimensionality, various parsimonious models such as projection pursuit, CART, neural nets, additive models, MARS, etc. are proposed according to different characterizations of the target function. A main interest in this research is to construct adaptive estimators by combining acollection of candidate procedures. The goal for the combinedprocedure is to perform automatically as well as (or nearly as wellas) the best original procedure without knowing which one it is. The random errors will be assumed to be generally dependent,including both short- and long-range cases. The effects ofdependence on adaptation capability will be studied. It isanticipated that theoretically proven and computationally feasible algorithms will be proposed to combine regression procedurestargeted at various characteristics of the regression function and different dependence structures for the random errors.Function estimation is an important statistical tool that tries tounderstand accurately the functional relationships between variables based on data and it has applications in many disciplines for successfully addressing scientific questions. In reality, observations are alwayssubject to random noise (error) from different sources. When therandom errors are dependent on each other, the dependence maydisguise the functional relationship of interest. Long-rangedependence refers to a situation where the errors are still highlycorrelated even when they occur at times or locations that are faraway from each other. It is known that such a long-range dependence makesthe estimation of the target function much harder. In applications,the degree of dependence between the errors is usually unknown,which makes the function estimation problem even harder. In thisproposal, we intend to develop methods that adaptively handledifferent degrees of dependence among the errors so that thefunction of interest can be estimated optimally without knowing thedependence structure of the errors. The research results and relatedwork by others on long-range dependent data will be brought tostudents at various levels in several statisticscourses. Collaborations will be conducted with several professors atIowa State University and their students in atmospheric science, electrical engineering, agronomy and possibly other fields toappropriately address long-range dependence phenomena, which havebeen encountered often and known to cause problems in data analysis with the existing statistical methods.