Adaptive Regression for Dependent Data by Combining Different Procedures

This proposal concerns research and education on adaptive regression

when the random errors are dependent. Many procedures have been

(and will be) proposed for nonparametric regression based on

different assumptions. In applications, a difficulty a user often

faces is the choice of the best method for the data at

hand. This is specially the case for high-dimensional function

estimation, 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 a

collection of candidate procedures. The goal for the combined

procedure is to perform automatically as well as (or nearly as well

as) 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 of

dependence on adaptation capability will be studied. It is

anticipated that theoretically proven and computationally feasible

algorithms will be proposed to combine regression procedures

targeted at various characteristics of the regression function and

different dependence structures for the random errors.

Function estimation is an important statistical tool that tries to

understand 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 always

subject to random noise (error) from different sources. When the

random errors are dependent on each other, the dependence may

disguise the functional relationship of interest. Long-range

dependence refers to a situation where the errors are still highly

correlated even when they occur at times or locations that are far

away from each other. It is known that such a long-range dependence makes

the 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 this

proposal, we intend to develop methods that adaptively handle

different degrees of dependence among the errors so that the

function of interest can be estimated optimally without knowing the

dependence structure of the errors. The research results and related

work by others on long-range dependent data will be brought to

students at various levels in several statistics

courses. Collaborations will be conducted with several professors at

Iowa State University and their students in atmospheric science,

electrical engineering, agronomy and possibly other fields to

appropriately address long-range dependence phenomena, which have

been encountered often and known to cause problems in data analysis

with the existing statistical methods.