Search

We study a discriminant direction vector that generally exists only in high-dimension, low sample size settings. Projections of data onto this direction vector take on only two distinct values, one for each class. There exist infinitely many such directions in the subspace generated by the data; but the maximal data piling vector has the longest distance between the projections. This paper investigates mathematical properties and classification performance of this discrimination method. Copyright 2010, Oxford University Press.

High-dimension, low-small-sample size datasets have different geometrical properties from those of traditional low-dimensional data. In their asymptotic study regarding increasing dimensionality with a fixed sample size, Hall et al. (2005) showed that each data vector is approximately located on the vertices of a regular simplex in a high-dimensional space. A perhaps unappealing aspect of their result is the underlying assumption which requires the variables, viewed as a time series, to be almost independent. We establish an equivalent geometric representation under much milder conditions using asymptotic properties of sample covariance matrices. We discuss implications of the results, such as the use of principal component analysis in a high-dimensional space, extension to the case of nonindependent samples and also the binary classification problem. Copyright 2007, Oxford University Press.

SUMMARY The problem of identifying the number of modal groups manifested in a sample of data is of continuing interest and importance in statistical practice. In this paper, we combine features of two current approaches to the mode-testing problem with some new ideas, to produce a method that provides the data analyst with a powerful and flexible way of exploring modality in univariate data. This approach also provides what appears to be the first satisfactory test of modality for samples of circular data.

The local linear smoother usually gives better performance than the Nadaraya-Watson smoother. An exception is the case of data sparsity. Here we discuss a modification of the Nadaraya-Watson smoother by Muller & Song (1993), based on a horizontal shift of the kernel weights towards the local centre of mass of the design points. This gives performance similar to the local linear when that works well and better performance when it does not. The new smoother also preserves monotonicity. Shifting towards the centre of mass is also used to develop a modified kernel density estimate which cancels the well-known peak spreading effect.

SUMMARY We compare the performances of local and global rules for smoothing parameter choice, in terms of asymptotic mean squared errors of the resulting estimators. In some instances there is surprisingly little to choose between local and global approaches; our analysis identifies contexts where the differences are small or large. This work motivates development of smoothing rules that form a 'half-way house' between local and global smoothing. There, interpolation provides a basis for partial local smoothing. A key result shows that interpolation on even a coarse grid can produce a very good approximation to full local smoothing. Our theoretical and numerical results lead us to suggest linear interpolation of a bandwidth obtained by integral approximations on discrete intervals.

A bandwidth selection method is proposed for kernel density estimation. This is based on the straightforward idea of plugging estimates into the usual asymptotic representation for the optimal bandwidth, but with two important modifications. The result is a bandwidth selector with the, by nonparametric standards, extremely fast asymptotic rate of convergence of n−½ where n → ∞ denotes sample size. Comparison is given to other bandwidth selection methods, and small sample impact is investigated.

A simple explicit estimator of variance in nonparametric regression is suggested, based on the mean square of a sequence of residuals. The estimator is valid for general error distributions, has elementary bias and variance properties, and is optimal to both first and second orders

SUMMARY The bandwidth selection problem in nonparametric kernel regression is considered. Bandwidth selectors based on cross-validation and on Akaike's information criterion, AIC, and his finite prediction error, FPE, are among those compared. It is seen that they are not necessarily asymptotically equivalent. Conditions are given under which the equivalence holds and modifications are suggested which make the selectors equivalent.