B. M. Brown

Beta-Bernstein Smoothing for Regression Curves with Compact Support

ABSTRACT. The problem of boundary bias is associated with kernel estimation for regression curves with compact support. This paper proposes a simple and uni(r)ed approach for remedying boundary bias in non‐parametric regression, without dividing the compact support into interior and boundary areas and without applying explicitly different smoothing treatments separately. The approach uses the beta family of density functions as kernels. The shapes of the kernels vary according to the position where the curve estimate is made. Theyare symmetric at the middle of the support interval, and become more and more asymmetric nearer the boundary points. The kernels never put any weight outside the data support interval, and thus avoid boundary bias. The method is a generalization of classical Bernstein polynomials, one of the earliest methods of statistical smoothing. The proposed estimator has optimal mean integrated squared error at an order of magnitude n−4/5, equivalent to that of standard kernel estimators when the curve has an unbounded support.

Normal Scores, Normal Plots Tests for Normality

Abstract In this article we develop new plotting positions for normal plots. The use of the plots usually centers on detecting irregular tail behavior or outliers. Along with the normal plot, we develop tests for various departures from normality, especially for skewness and heavy tails. The tests can be considered as components of a Shapiro-Wilk type test that has been decomposed into different sources of nonnormality. Convergence to the limiting distributions is slow, so finite sample corrections are included to make the tests useful for small sample sizes.

On Certain Bivariate Sign Tests and Medians

Abstract Brown and Hettmansperger proposed an affine invariant bivariate analogue of the sign test, the OS test, based on the generalized median of Oja. On the other hand, Oja and Nyblom introduced a family of locally most powerful affine invariant sign tests. In the case of elliptic distributions, the locally most powerful Blumens test and the OS test are shown to be asymptotically equivalent. Formulas for calculating asymptotic relative efficiencies of the OS test and the Oja bivariate median are given. It is shown that if the contours of a distribution are of a similar shape, the relative efficiencies of the OS test and Blumens test depend on the distribution only through the shape of the contours. For the power family of contours |x 1| p + |x 2| p = c, p > 0, numerical calculations show that the efficiency of the OS test relative to Blumens test attains its minimum 1 as p = 2 (spherical/elliptic case) and increases to infinity as p → 0. In the bivariate Laplace case with independent marginals (p = ...

Combined and Least Squares Empirical Likelihood

In conventional empirical likelihood, there is exactly one structural constraint for every parameter. In some circumstances, additional constraints are imposed to reflect additional and sought-after features of statistical analysis. Such an augmented scheme uses the implicit power of empirical likelihood to produce very natural adaptive statistical methods, free of arbitrary tuning parameter choices, and does have good asymptotic properties. The price to be paid for such good properties is in extra computational difficulty. To overcome the computational difficulty, we propose a ‘least-squares’ version of the empirical likelihood. The method is illustrated by application to the case of combined empirical likelihood for the mean and the median in one sample location inference.

Regular Redescending Rank Estimates

Abstract A study is made of some unusual location estimates first proposed in 1977 by J. S. Maritz, M. Wu, and R. G. Staudte, Jr., who established some strong robustness properties of these estimates, such as redescending influence functions and, in some cases, full efficiency at the Cauchy model. It is further shown here that the estimates, called M-W-S estimates, are derivable from a rank-based scheme founded on convex functions and hence are regular, are unique, and have associated tests and confidence intervals. The tests belong to a family of signed-rank-type tests, and their distributions have nice combinatorial properties. It is not the case, therefore, that a redescending influence function need imply all the computational irregularities possessed by redescending M estimates. In addition, M-W-S estimates are shown to have breakdown point of .5, a very strong robustness property.