Cornelis J. Potgieter

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.

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.

Density estimation in the presence of heteroscedastic measurement error of unknown type using phase function deconvolution: Density estimation in the presence of heteroscedastic measurement error of unknown type using phase function deconvolution

It is important to properly correct for measurement error when estimating density functions associated with biomedical variables. These estimators that adjust for measurement error are broadly referred to as density deconvolution estimators. While most methods in the literature assume the distribution of the measurement error to be fully known, a recently proposed method based on the empirical phase function (EPF) can deal with the situation when the measurement error distribution is unknown. The EPF density estimator has only been considered in the context of additive and homoscedastic measurement error; however, the measurement error of many biomedical variables is heteroscedastic in nature. In this paper, we developed a phase function approach for density deconvolution when the measurement error has unknown distribution and is heteroscedastic. A weighted EPF (WEPF) is proposed where the weights are used to adjust for heteroscedasticity of measurement error. The asymptotic properties of the WEPF estimator are evaluated. Simulation results show that the weighting can result in large decreases in mean integrated squared error when estimating the phase function. The estimation of the weights from replicate observations is also discussed. Finally, the construction of a deconvolution density estimator using the WEPF is compared with an existing deconvolution estimator that adjusts for heteroscedasticity but assumes the measurement error distribution to be fully known. The WEPF estimator proves to be competitive, especially when considering that it relies on minimal assumption of the distribution of measurement error.

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.