N. I. Fisher

Bump hunting in high-dimensional data

Many data analytic questions can be formulated as (noisy) optimization problems. They explicitly or implicitly involve finding simultaneous combinations of values for a set of (“input”) variables that imply unusually large (or small) values of another designated (“output”) variable. Specifically, one seeks a set of subregions of the input variable space within which the value of the output variable is considerably larger (or smaller) than its average value over the entire input domain. In addition it is usually desired that these regions be describable in an interpretable form involving simple statements (“rules”) concerning the input values. This paper presents a procedure directed towards this goal based on the notion of “patient” rule induction. This patient strategy is contrasted with the greedy ones used by most rule induction methods, and semi-greedy ones used by some partitioning tree techniques such as CART. Applications involving scientific and commercial data bases are presented.

Graphical Assessment of Dependence: Is a Picture Worth 100 Tests?

Nonparametric tests of association have been around for a long time, and new ones continue to be proposed to cope with specie c forms of association. However, graphs have the potential to assess a far richer class of bivariate dependence structures than any collection oftests. This article describes howchi-plots, used in conjunction with the usual scatterplot, provides a useful practical tool in this regard.Nonparametric tests of association have been around for a long time, and new ones continue to be proposed to cope with specific forms of association. However, graphs have the potential to assess a far richer class of bivariate dependence structures than any collection of tests. This article describes how chi-plots, used in conjunction with the usual scatterplot, provides a useful practical tool in this regard.

Some asymptotics for multimodality tests based on kernel density estimates

SummaryA test due to B.W. Silverman for modality of a probability density is based on counting modes of a kernel density estimator, and the idea of critical smoothing. An asymptotic formula is given for the expected number of modes. This, together with other methods, establishes the rate of convergence of the critically smoothed bandwidth. These ideas are extended to provide insight concerning the behaviour of the test based on bootstrap critical values.

Graphical Methods in Nonparametric Statistics: A Review and Annotated Bibliography

(ISI) are collaborating with JSTOR to digitize, preserve and extend access to International Statistical Review / Revue Internationale de Statistique. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Summary This paper reviews the range of graphical methods available for use with nonparametric procedures, and provides examples of the use of many of the methods under the broad groupings: two-sample procedures, one-sample procedures, association and regression procedures, and miscellaneous procedures. An annotated bibliography is also provided.

Nonparametric comparison of cumulative periodograms

Motivated by a problem in the analysis of hormonal time series data, this paper proposes a simple graphical method for comparing two periodograms and describes a new nonparametric approach to testing the hypothesis that the two underlying spectra are the same. Simulation studies show that the new test has power characteristics that are competitive with existing procedures. The relative merits of nonparametric and semiparametric tests are discussed.

Improved Pivotal Methods for Constructing Confidence Regions with Directional Data

Abstract The importance of pivoting is well established in the context of nonparametric confidence regions. It ensures enhanced coverage accuracy. However, pivoting for directional data cannot be achieved simply by rescaling. A somewhat cumbersome pivotal method, which involves passing first into a space of higher dimension, has been developed by Fisher and Hall for samples of unit vectors. Although that method has some advantages over nonpivotal techniques, it does suffer from certain drawbacks—in particular, the operation of passing to a higher dimension. Here we suggest alternative pivotal approaches, the implementation of which does not require us to increase the intrinsic dimension of the data and which in practice seem to achieve greater coverage accuracy. These methods are of two types: new pivotal bootstrap techniques and techniques that exploit the “implicit pivotalness” of the empirical likelihood algorithm. Unlike the method proposed by Fisher and Hall, these methods are also applicable to axia...

SPHERE: a contouring program for spherical data

Abstract This paper describes a method for displaying a sample of spherical data, by computing an “optimally” smoothed estimate of the underlying distribution and making a stereographic projection of the contours of this estimate. An interactive FORTRAN program which applies this method is supplied and described and examples given of its use.

Tests of discordancy for samples from Fisher's distribution on the sphere

Several tests of discordancy for an outlying value in a sample of data from Fishers distribution are investigated, and recommendations made about their use. One of the tests is modified for application to testing discordancy of several outliers in a single Fisher sample. The various tests are applied to some samples of artificial data and palaeomagnetic data.

Bootstrap Confidence Regions for Directional Data

Abstract Methods are proposed for constructing bootstrap confidence regions for the mean direction of a random p-dimensional unit vector X with an arbitrary unimodal distribution on the p sphere. The approach of this article differs from that of other authors in that it is based on pivotal statistics. A general pivotal method is introduced that produces a wide variety of confidence regions on general p-dimensional spheres; included are confidence cones and likelihood-based regions. It can readily be modified to incorporate extra assumptions about the underlying distribution, such as rotational symmetry. The general method leads to confidence pictures, which present information about the estimated posterior likelihood of mean orientation by shading spherical surfaces. An application is given to a sample of spherical cross-bed measurements. The methods extend to the case where X has random length, and to calculation of confidence regions for reference directions of axial bipolar or girdle distributions.

Bootstrap algorithms for small samples

Abstract We describe algorithms for exact computation of nonparametric bootstrap estimators, and show that they are practicable for small samples. It is argued that in this setting, enumeration and calculation of the entire bootstrap distribution is competitive with simulation. For example if the sample size is n = 6 then the entire bootstrap distribution has only 462 atoms, and exact calculation is competitive with simulation involving several hundred replications. We also describe the role of exact computation in the iterated bootstrap, and discuss a method of importance resampling appropriate to small samples.