Asymptotic equivalence of nonparametric experiments with nuisance parameters.

NSF grant DMS-05-04233

2005 - 2008

Abstract

Asymptotic equivalence theory has emerged as a growing new area of mathematical statistics research. The purpose of this project is to describe the asymptoticbehavior of some nonparametric curve estimation models, which have distributions that depend on a smooth function f(x). For each of these nonparametric models, a limiting model will be found that can be used as an asymptotic approximation, such that inference in the original experiment can be performed under the limiting experiment without loss of information. Two important nonparametric models are nonparametric regression, which observes f(x) at n sample points plus some normally distributed noise, and density estimation, which has n independent observations from a distribution with unknown density f(x). It has been shown that both of these experiments can be approximated in the limit by a white-noise-with-drift experiment which observes a Brownian motion plus a mean that depends on the mean function. This project considers extensions to the regular nonparametric regression and density estimation experiments. In the regression problem, the presence of nuisance parameters such as an unknown variance and an unknown distribution of sample points is explored. The limiting experiment in this case is the white-noise-with-drift experiment plus a secondary component that describes the nuisance parameter. The nonparametric regression with a random design on a two-dimensional sample space is also investigated. In the density estimation experiment, an important extension is to two-dimensional observations on the unit square. This experiment has a limiting model that observes a Brownian sheet process plus a mean.


This project provides a way of taking estimation techniques in certain difficult problems and transforming them into situations where the estimation may be easier. In particular,  the problem of estimating a smooth function from a set of noisy data is studied when either the magnitude of the noise is unknown or the locations where the function is observed are random with an unknown distribution. Both of these problems include additional unknown components that make estimation more difficult. This project proves that for a large number of observations the additional unknown components can be separated as nearly independent estimation problems. These asymptotic approximations allow standard techniques to be applied to a wider range of problems. In particular, standard wavelet estimators can be adapted to problems that do not fit the normal regularity assumptions, thus providing new estimators and added rationale for some estimation techniques already proposed. Non-regular estimation problems like this might come up in problems such as filtering noisy medical images or signals, estimating distributions of plant species, or analyzing financial series. Furthermore, this project will use asymptotic approximations to help calculate the power of some goodness-of-fit tests. Applications of these results demonstrate the usefulness of the asymptotic techniques, and help to motivate more people to learn about them. The results are disseminated through published articles, conferences, and web sites.