After reading the work of Michael Nussbaum, Larry Brown, and Mark Low in 1996, I became interested in the idea of asymptotic equivalence and its application to nonparametric problems. My thesis work with David Pollard looked for a way to transform independent observations into a Gaussian process, a re-thinking of Nussbaum's important result.

This led me to work with Larry Brown, Mark Low, and Cun-Hui Zhang to refine the connection to approximate very small perturbations from uniformity in the parameter functions on a fine grid.

I have recently sought to extend the work of  Brown and Low on nonparametric regression problems. When there are nuisance parameters in the regression experiment like an unknown variance or design density, then the continuous Gaussian process with a single variance is not an appropriate approximation. There are however mixtures of Gaussian processes with different variance functions that can be used to approximate the nonregular regression problems.

My hope is that these results will lead to a new understanding of inference in density estimation and nonparametric regression problems. The structure of the constructions seems especially applicable to series or wavelet estimators. My future plans inlcude using these results in testing problems and in estimation for nonregular situations.

Here is a list of references in this area.

• Equivalent experiments in nonparametric regression problems with an unknown design density (presented at JSM '07). [Abstract]
  
• Approximating nonparametric regression experiments by continuous Gaussian processes when the variance is unknown. [Abstract]
• An asymptotic approximation of a nonparametric regression on the unit square by a Brownian sheet process with an unknown mean. [Abstract]
  
• Constructing an approximately normal set of observations from n independent observations. [Abstract]
• Establishing the equivalence of the density estimation experiment and a Gaussian process when the parameter space is a Besov space (with Larry Brown, Mark Low, and Cun-Hui Zhang). [Abstract]
• A bound on the error in coupling a symmetric binomial to a normal distribution that extends Tusnady’s lemma (with David Pollard). [Abstract]

• "Asymptotic equivalence of nonparametric experiments with nuisance parameters." NSF grant.
• Continuous Gaussian process approximations to nonparametric regression problems where nuisance parameters function as linear filters.
  
• Wavelet estimators of the mean of nonparametric regression observations with a random design.
  
• A simple example using the asymptotic equivalence of a density estimation and white noise to find  a goodness-of-fit test.We would like to compare the power of a Kolmogorov-Smirnov type test to a minimum Hellinger-distance (chi-squared) test.