R. L. Eubank

Testing for no effect in nonparametric regression

Abstract The large sample properties of three tests for no effect in nonparametric regression are investigated. The tests can all be represented as weighted sums of squared sample Fourier coefficients. The type of weighting employed by a test is shown to be crucial for its asymptotic and finite sample power properties.

Convergence rates for estimation in certain partially linear models

Abstract Rates of convergence are studied for estimation in certain partial linear models that include nonparametric regression models with discontinuous derivatives. The asymptotic behavior of two smoothing spline related estimators of the regression coefficient and regression function in these models are examined. Lower bounds are then derived for rates of convergence in estimating the size of jump discontinuities in a regression function or its derivative. The latter rates are nonparametric which indicates that parametric convergence rates are not possible in such instances.

Monotone Smoothing with Application to Dose-Response Curve

Motivated by problems that arise in dose-response curve estimation, we developed a new method to estimate a monotone curve. The resulting monotone estimator is obtained by combining techniques from smoothing splines with nonnegativity properties of cubic B-splines. Numerical experiments are given to exemplify the method.

A Simple Smoothing Spline

Abstract A simple, closed-form expression is derived for a linear smoothing spline. Examples are given of how this can be used as a pedagogical tool to help students understand the properties of smoothing splines and compare them with other nonparametric estimators.

Knot selection for least-squares and penalized splines

Two new stochastic search methods are proposed for optimizing the knot locations and/or smoothing parameters for least-squares or penalized splines. One of the methods is a golden-section-augmented blind search, while the other is a continuous genetic algorithm. Monte Carlo experiments indicate that the algorithms are very successful at producing knot locations and/or smoothing parameters that are near optimal in a squared error sense. Both algorithms are amenable to parallelization and have been implemented in OpenMP and MPI. An adjusted GCV criterion is also considered for selecting both the number and location of knots. The method performed well relative to MARS in a small empirical comparison.

On the Computation of Optimal Designs for Certain time Series Models with Applications to Optimal Quantile Selection for Location or Scale Parameter Estimation

Using the results of Chow (Ph.D. dissertation, Texas A & M Univ., 1978) on the optimal placement of knots in the approximation of functions by piecewise polynomials, we present an algorithm for the computation of optimal designs for certain time series models considered by Eubank, Smith and Smith (Ann. Statist., 9 (1981), pp. 486–493), (Tech. Rep. 150, Southern Methodist Univ., 1981). The ideas underlying this algorithm form a unified approach to the computation of optimal spacings for the sample quantiles used in the asymptotically best linear unbiased estimator of a location or scale parameter.

Hypothesis Testing for Fourier Based Edge Detection Methods

Edge detection is an essential task in image processing. In some applications, such as Magnetic Resonance Imaging, the information about an image is available only through its frequency (Fourier) data. In this case, edge detection is particularly challenging, as it requires extracting local information from global data. The problem is exacerbated when the data are noisy. This paper proposes a new edge detection algorithm which combines the concentration edge detection method (Gelb and Tadmor in Appl. Comput. Harmon. Anal. 7:101–135, 1999) with statistical hypothesis testing. The result is a method that achieves a high probability of detection while maintaining a low probability of false detection.

Adaptive Order Selection for Spline Smoothing

Computational methods are presented for spline smoothing that make it practical to compute smoothing splines of degrees other than just the standard cubic case. Specifically, an order n algorithm is developed that has conceptual and practical advantages relative to classical methods. From a conceptual standpoint, the algorithm uses only standard programming techniques that do not require specialized knowledge about spline functions, methods for solving sparse equation systems or Kalman filtering. This allows for the practical development of methods for adaptive selection of both the level of smoothing and degree of the estimator. Simulation experiments are presented that show adaptive degree selection can improve estimator efficiency over the use of cubic smoothing splines. Run-time comparisons are also conducted between the proposed algorithm and a classical, band-limited, computing method for the cubic case.

Unit canonical correlations and high-dimensional discriminant analysis

The problem of characterization of canonical vectors corresponding to unit sample canonical correlations in the non-full-rank case is considered. Classical work in this area is revisited and a new geometric characterization is developed. Applications are considered for classification problems that arise in the high-dimension low-sample-size setting. In that context, we show that Fishers linear discriminant analysis and canonical correlation do not, in general, coincide and conduct empirical comparisons between the two methods.

A simple smoothing spline, II

A simple derivation is given for a kernel-type approximation of the weight function for a linear smoothing spline that explicitly includes boundary corrections. As an application, asymptotic approximations are obtained for the estimators pointwise and integrated mean squared errors.