Ph.D. Recipients and their Thesis Abstracts

Statistics

Algebra, Analysis, Combinatorics, Differential Equations / Dynamical Systems, Differential Geometry, Geometry, Interdisciplinary, Lie Groups, Logic, Mathematical Physics, PDE / Numerical Analysis, Probability, Statistics, Topology

Nonparametric Function Estimation Via Wavelets

Abstract: The nonparametric function estimation has very important applications in statistics as well as many other fields such as signal processing. The recent development of wavelet bases based on multiresolution analyses suggests new techniques for statistical estimation. In this dissertation, we consider two problems in nonparametric regression model: estimation of regression functions for unequispaced designs and estimation of derivative of regression functions.

For equispaced designs, the Donoho-Johnstone wavelet shrinkage procedures for estimating regression functions have been shown to be near optimal in theory and very useful in practice. But in many applications, the samples are unequispaced. We first show that direct application of the Donoho-Johnstone procedures to unequispaced samples are in many cases suboptimal. Then we propose a new adaptive wavelet shrinkage estimate for unequispaced designs. We show that the estimate is near-minimax in global estimation and attains exact adaptive minimax rate for pointwise estimation. The procedure is implemented in SPlus. Simulations are conducted and confirm the theoretical results.

We are interested in one type of homogeneous linear inverse problems — estimating derivative of regression functions. By using wavelets as a tool and applying the so-called Wavelet-Vaguelette decomposition and Vaguelette-Wavelet decomposition, we construct two estimates of derivative of regression functions for equispaced samples. Both estimates are shown to be adaptive and near minimax in global estimation. They attain the exact adaptive minimax rate for estimation at a point.

The techniques developed in this dissertation can also be applied to other statistical problems such as nonparametric hypothesis testing which includes signal detection problem as a special case.

On Assessment of Bioequivalence

Abstract: When we consider bioequivalence in distribution, the equivalence in mean between formulations may not be sufficient for assessment of bioequivalence in distribution. The difference in variability between formulations should also be considered. Under the normality assumption, the equivalences in both mean and variability can result in bioequivalence in distribution. In this thesis an unbiased test procedure for bioequivalence in variability is provided. The test is uniformly more powerful than the two one-sided tests procedure proposed by Liu and Chow. Under a stronger condition, a UMPI (uniformly most powerful invariant) test for bioequivalence in variability is proposed.

One may want to consider, especially when the logarithmic transformation was not applied to the data, the ratio of two population means instead of the difference. An unbiased test is constructed for assessment of bioequivalence in mean in terms of ratio.

When we consider the individual bioequivalence of Anderson and Hauck, it is generally believed that Anderson and Hauck's nonparametric test procedure (TIER) is valid. Indeed, their procedure is usually valid. A sufficient condition that implies the validity of Anderson and Hauck's test is established. However, their procedure is not always valid. A counterexample is constructed in Chapter 6.

Under the normality assumption, the existence of nonrandomized unbiased test for an arbitrary alternative region is investigated and a necessary condition for the existence is given. The region determined by individual bioequivalence is derived, which greatly improves in power upon Anderson and Hauck's test and Liu and Chow's two one-sided tests procedure.

Frequentist and Bayesian Aspects of Some Nonparametric Estimation Problems

Abstract: We first study estimation for nonparametric regression problems. The class of linear estimators is introduced. Viewing the nonparametric problem in this fashion enables us to apply some classical techniques from the multivariate normal mean estimation problem. Using these, we show how to improve certain broad classes of linear estimators. The results also yield generalizations of some well known theorems about admissibility of linear estimators in the multivariate normal mean estimation problem. It is shown that the commonly used nonparametric estimators can be improved. An asymptotic result is described which gives a quantitative measure of the maximum improvement to be gained in certain situations.

We then study the white noise model, which is asymptotically equivalent to the nonparametric regression and density estimation problems. Estimating the entire expectation function is considered. A class of conjugate priors is introduced. It is shown that these priors yield consistent Bayes estimators. An inconsistent Bayes estimator is also found. We then study the situation where the parameter family is restricted to be an ellipsoidal subspace (i.e., a Sobolev space). We first produce a conjugate prior whose Bayes solution attains the optimal minimax rate. We also show that no conjugate prior whose support is on the space will have Bayes solutions which attain the optimal minimax rate. Similar results for estimating the function value at a point are also established.

Finally, linear estimation of the expectation function, f(x), at a point in the white noise model is considered. The exact linear minimax estimator of f(0) is found for the family of f(x) in which f '(x) is Lip(M). The resulting estimator is then used to verify a conjecture of Sacks and Ylvisaker concerning the near optimality of the Epanechnikov kernel.