Fred Lombard

Nonparametric Confidence Bands for a Quantile Comparison Function

This article develops statistical methods useful for comparing the marginals F and G of a bivariate distribution. Such comparisons are appropriate in the analysis of matched-pair data involving a treatment and a control (or two treatments). The quantile comparison functionq = G−1(F), proposed by Lehmann as a generalized measure of treatment effect in fully randomized experiments, is shown to be applicable and also useful in the analysis of matched-pair data. Nonparametric simultaneous confidence bands for q are constructed using asymptotic and permutation methods. The copula of the bivariate distribution plays a significant role in the methodology. Application of the results is illustrated on a dataset from an industrial experiment.

A cusum procedure to detect deviations from uniformity in angular data

We propose a sequential cumulative sum procedure to detect deviations from uniformity in angular data. The method is motivated by a problem in high-energy astrophysics and is illustrated by an application to data.

A New Approach to Function-Based Hypothesis Testing in Location-Scale Families

Motivated by two applications in the mining industry, we introduce a new approach to testing the hypothesis that two-sampled distributions are simply location and scale changes of one another. The test, applicable to both paired data and two-sample data, is based on the empirical characteristic function. More conventional techniques founded on the empirical distribution function suffer from serious drawbacks when used to test for location-scale families. In the motivating applications, knowing that the distributions differ only in location and scale has significant operational and economic advantages, enabling protocols for one type of data to be applied directly to another. Supplementary material in the form of Matlab code is available online.

Segmentation of circular data

Circular data – data whose values lie in the interval [0,2π) – are important in a number of application areas. In some, there is a suspicion that a sequence of circular readings may contain two or more segments following different models. An analysis may then seek to decide whether there are multiple segments, and if so, to estimate the changepoints separating them. This paper presents an optimal method for segmenting sequences of data following the von Mises distribution. It is shown by example that the method is also successful in data following a distribution with much heavier tails.

A Note on Two-Sided Cusums for a Normal Mean

A two-sided cumulative sum (cusum) scheme for a normal mean involves running two one-sided schemes, one to control upward and the other to control downward changes in the mean. We show that the two-sided scheme can be replaced by a single upward cusum scheme that has the same in-control average run length (ARL) and a smaller out-of-control ARL. The benefits of the new scheme are especially apparent at small mean shifts.

Cusum control for data following the von Mises distribution

ABSTRACT The von Mises distribution is widely used for modeling angular data. When such data are seen in a quality control setting, there may be interest in checking whether the values are in statistical control or have gone out of control. A cumulative sum (cusum) control chart has desirable properties for checking whether the distribution has changed from an in-control to an out-of-control setting. This paper develops cusums for a change in the mean direction and concentration of angular data and illustrates some of their properties.

Sequential rank CUSUM charts for angular data

A cumulative sum (CUSUM) control chart has desirable properties for checking whether a distribution has changed from an in-control to an out-of-control setting. Distribution-free CUSUMs based on sequential ranks to detect changes in the mean direction and dispersion of angular data are developed and some of their properties are illustrated by theoretical calculations and Monte Carlo simulation. Three applications to sequentially observed angular data from health science, industrial quality control and astrophysics are discussed.

A multivariate rank test for comparing mass size distributions

Particle size analyses of a raw material are commonplace in the mineral processing industry. Knowledge of particle size distributions is crucial in planning milling operations to enable an optimum degree of liberation of valuable mineral phases, to minimize plant losses due to an excess of oversize or undersize material or to attain a size distribution that fits a contractual specification. The problem addressed in the present paper is how to test the equality of two or more underlying size distributions. A distinguishing feature of these size distributions is that they are not based on counts of individual particles. Rather, they are mass size distributions giving the fractions of the total mass of a sampled material lying in each of a number of size intervals. As such, the data are compositional in nature, using the terminology of Aitchison [1] that is, multivariate vectors the components of which add to 100%. In the literature, various versions of Hotellings T 2 have been used to compare matched pairs of such compositional data. In this paper, we propose a robust test procedure based on ranks as a competitor to Hotellings T 2. In contrast to the latter statistic, the power of the rank test is not unduly affected by the presence of outliers or of zeros among the data.

Signed sequential rank CUSUMs

CUSUMs based on the signed sequential ranks of observations are developed for detecting location and scale changes in symmetric distributions. The CUSUMs are distribution-free and fully self-starting: given a specified in-control median and nominal in-control average run length, no parametric specification of the underlying distribution is required in order to find the correct control limits. If the underlying distribution is normal with unknown variance, a CUSUM based on the Van der Waerden signed rank score produces out-of-control average run lengths that are commensurate with those produced by the standard CUSUM for a normal distribution with known variance. For heavier tailed distributions, use of a CUSUM based on the Wilcoxon signed rank score is indicated. The methodology is illustrated by application to real data from an industrial environment.

Nonparametric two-sample estimation of location and scale parameters from empirical characteristic functions

ABSTRACT Two random variables X and Y are said to belong to the same location-scale family when with unknown constants and . Given iid observations , and , satisfying this location-scale assumption, we wish to estimate μ and σ with high efficiency in the absence of knowledge of the functional form of the underlying common family of distributions of X and Y. Here, ‘high efficiency’ means that the estimator is asymptotically unbiased and that its asymptotic variance is close to the asymptotic variance of the maximum likelihood estimator that would be used had the form of the underlying location-scale family of distributions been known. We propose in the present paper two methods for estimating these parameters based on the empirical characteristic function (ECF). The first approach considered minimizes a weighted distance between the ECFs of the X and Y data. The second approach constructs a quadratic form comparing the real and imaginary parts of the X- and Y-sample ECFs at a preselected number of points. In both approaches, the constructed distance metric is minimized to estimate μ and σ. The asymptotic distributions of the estimators are found, and small sample performance is investigated via a Monte Carlo simulation study.