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One-dimensional empirical measures, order statistics, and Kantorovich transport distances
This work is devoted to the study of rates of convergence of the empirical measures \\mu_{n} = \\frac {1}{n} \\sum_{k=1}^n \\delta_{X_k}, n \\geq 1, over a sample (X_{k})_{k \\geq 1} of independent identically distributed real-valued random variables towards the common distribution \\mu in Kantorovich transport distances W_p. The focus is on finite range bounds on the expected Kantorovich distances \\mathbb{E}(W_{p}(\\mu_{n},\\mu )) or \\big [ \\mathbb{E}(W_{p}^p(\\mu_{n},\\mu )) \\big ]^1/p in terms of moments and analytic conditions on the measure \\mu and its distribution function. The study describes a variety of rates, from the standard one \\frac {1}{\\sqrt n} to slower rates, and both lower and upper-bounds on \\mathbb{E}(W_{p}(\\mu_{n},\\mu )) for fixed n in various instances. Order statistics, reduction to uniform samples and analysis of beta distributions, inverse distribution functions, log-concavity are main tools in the investigation. Two detailed appendices collect classical and some new facts on inverse distribution functions and beta distributions and their densities necessary to the investigation.
eBook
Nonparametric inference of interaction laws in systems of agents from trajectory data
Inferring the laws of interaction in agent-based systems from observational data is a fundamental challenge in a wide variety of disciplines. We propose a nonparametric statistical learning approach for distance-based interactions, with no reference or assumption on their analytical form, given data consisting of sampled trajectories of interacting agents. We demonstrate the effectiveness of our estimators both by providing theoretical guarantees that avoid the curse of dimensionality and by testing them on a variety of prototypical systems used in various disciplines. These systems include homogeneous and heterogeneous agent systems, ranging from particle systems in fundamental physics to agent-based systems that model opinion dynamics under the social influence, prey–predator dynamics, flocking and swarming, and phototaxis in cell dynamics.
Journal Article
Exact p-values for pairwise comparison of Friedman rank sums, with application to comparing classifiers
Background
The Friedman rank sum test is a widely-used nonparametric method in computational biology. In addition to examining the overall null hypothesis of no significant difference among any of the rank sums, it is typically of interest to conduct pairwise comparison tests. Current approaches to such tests rely on large-sample approximations, due to the numerical complexity of computing the exact distribution. These approximate methods lead to inaccurate estimates in the tail of the distribution, which is most relevant for
p
-value calculation.
Results
We propose an efficient, combinatorial exact approach for calculating the probability mass distribution of the rank sum difference statistic for pairwise comparison of Friedman rank sums, and compare exact results with recommended asymptotic approximations. Whereas the chi-squared approximation performs inferiorly to exact computation overall, others, particularly the normal, perform well, except for the extreme tail. Hence exact calculation offers an improvement when small
p
-values occur following multiple testing correction. Exact inference also enhances the identification of significant differences whenever the observed values are close to the approximate critical value. We illustrate the proposed method in the context of biological machine learning, were Friedman rank sum difference tests are commonly used for the comparison of classifiers over multiple datasets.
Conclusions
We provide a computationally fast method to determine the exact
p
-value of the absolute rank sum difference of a pair of Friedman rank sums, making asymptotic tests obsolete. Calculation of exact
p
-values is easy to implement in statistical software and the implementation in R is provided in one of the Additional files and is also available at
http://www.ru.nl/publish/pages/726696/friedmanrsd.zip
.
Journal Article
A semi-nonparametric Poisson regression model for analyzing motor vehicle crash data
This paper develops a semi-nonparametric Poisson regression model to analyze motor vehicle crash frequency data collected from rural multilane highway segments in California, US. Motor vehicle crash frequency on rural highway is a topic of interest in the area of transportation safety due to higher driving speeds and the resultant severity level. Unlike the traditional Negative Binomial (NB) model, the semi-nonparametric Poisson regression model can accommodate an unobserved heterogeneity following a highly flexible semi-nonparametric (SNP) distribution. Simulation experiments are conducted to demonstrate that the SNP distribution can well mimic a large family of distributions, including normal distributions, log-gamma distributions, bimodal and trimodal distributions. Empirical estimation results show that such flexibility offered by the SNP distribution can greatly improve model precision and the overall goodness-of-fit. The semi-nonparametric distribution can provide a better understanding of crash data structure through its ability to capture potential multimodality in the distribution of unobserved heterogeneity. When estimated coefficients in empirical models are compared, SNP and NB models are found to have a substantially different coefficient for the dummy variable indicating the lane width. The SNP model with better statistical performance suggests that the NB model overestimates the effect of lane width on crash frequency reduction by 83.1%.
Journal Article
Nonparametric Inference on Manifolds
This book introduces in a systematic manner a general nonparametric theory of statistics on manifolds, with emphasis on manifolds of shapes. The theory has important and varied applications in medical diagnostics, image analysis, and machine vision. An early chapter of examples establishes the effectiveness of the new methods and demonstrates how they outperform their parametric counterparts. Inference is developed for both intrinsic and extrinsic Fréchet means of probability distributions on manifolds, then applied to shape spaces defined as orbits of landmarks under a Lie group of transformations - in particular, similarity, reflection similarity, affine and projective transformations. In addition, nonparametric Bayesian theory is adapted and extended to manifolds for the purposes of density estimation, regression and classification. Ideal for statisticians who analyze manifold data and wish to develop their own methodology, this book is also of interest to probabilists, mathematicians, computer scientists, and morphometricians with mathematical training.
eBook
Nonparametric analysis of quality-of-life measures for randomized clustered design: the R package npclust
Pre-Post intervention factorial design with multiple observations for each subject before and after the intervention frequently arises in Quality-of-Life (QoL) research. Usually not only a global hypothesis on intervention is of interest, but also the hypotheses of pre-post change and also the interaction between intervention and pre-post change. In most practical situations, the distribution of the observed data is unknown and there may exist a number of atypical measurements and outliers. More importantly, QoL outcomes are measured in rating scales and the multiple Quality-of-Life measurements before and after the intervention present complex within-subject correlations. Hence, use of parametric and semi-parametric procedures that impose restrictive distributional and correlation assumptions becomes questionable. This emphasizes the demand for statistical procedures that enable us to accurately and reliably analyze QoL outcomes with minimal conditions. Nonparametric methods offer such a possibility and thus become of particular practical importance in the field of QoL. In this article, we aim to expose researchers and practitioners in the biomedical and behavioral science to nonparametric methods for the analysis of data collected in clustered randomized design. We also illustrate the use of the
R
package
npclust
we developed for an easy and user-friendly access to nonparametric methods for the analysis of Quality-of-Life data. It provides procedures for pre-processing the data, performing various hypothesis tests and computing confidence intervals for the estimated effects. The procedures also contain methods for visual display of the results. We illustrate the implemented procedures with Pediatric Asthma Quality-of-Life data collected from the Asthma Randomized Trial of Indoor wood Smoke (ARTIS).
Journal Article
Nonparametric statistics with applications to science and engineering with R
NONPARAMETRIC STATISTICS WITH APPLICATIONS TO SCIENCE AND ENGINEERING WITH R
Introduction to the methods and techniques of traditional and modern nonparametric statistics, incorporating R code
Nonparametric Statistics with Applications to Science and Engineering with R presents modern nonparametric statistics from a practical point of view, with the newly revised edition including custom R functions implementing nonparametric methods to explain how to compute them and make them more comprehensible.
Relevant built-in functions and packages on CRAN are also provided with a sample code. R codes in the new edition not only enable readers to perform nonparametric analysis easily, but also to visualize and explore data using R's powerful graphic systems, such as ggplot2 package and R base graphic system.
The new edition includes useful tables at the end of each chapter that help the reader find data sets, files, functions, and packages that are used and relevant to the respective chapter. New examples and exercises that enable readers to gain a deeper insight into nonparametric statistics and increase their comprehension are also included.
Some of the sample topics discussed in Nonparametric Statistics with Applications to Science and Engineering with R include:
* Basics of probability, statistics, Bayesian statistics, order statistics, Kolmogorov–Smirnov test statistics, rank tests, and designed experiments
* Categorical data, estimating distribution functions, density estimation, least squares regression, curve fitting techniques, wavelets, and bootstrap sampling
* EM algorithms, statistical learning, nonparametric Bayes, WinBUGS, properties of ranks, and Spearman coefficient of rank correlation
* Chi-square and goodness-of-fit, contingency tables, Fisher exact test, MC Nemar test, Cochran's test, Mantel–Haenszel test, and Empirical Likelihood
Nonparametric Statistics with Applications to Science and Engineering with R is a highly valuable resource for graduate students in engineering and the physical and mathematical sciences, as well as researchers who need a more comprehensive, but succinct understanding of modern nonparametric statistical methods.
eBook
Robust nonparametric tests of general linear model coefficients: A comparison of permutation methods and test statistics
Statistical inference in neuroimaging research often involves testing the significance of regression coefficients in a general linear model. In many applications, the researcher assumes a model of the form Y=α+Xβ+Zγ+ε, where Y is the observed brain signal, and X and Z contain explanatory variables that are thought to be related to the brain signal. The goal is to test the null hypothesis H0:β=0 with the nuisance parameters γ included in the model. Several nonparametric (permutation) methods have been proposed for this problem, and each method uses some variant of the F ratio as the test statistic. However, recent research suggests that the F ratio can produce invalid permutation tests of H0:β=0 when the ε terms are heteroscedastic (i.e., have non-constant variance), which can occur for a variety of reasons. This study compares the classic F test statistic to the robust W (Wald) test statistic using eight different permutation methods. The results reveal that permutation tests using the F ratio can produce accurate results when the errors are homoscedastic, but high false positive rates when the errors are heteroscedastic. In contrast, permutation tests using the W test statistic produced valid results when the errors were homoscedastic, and asymptotically valid results when the errors were heteroscedastic. In the situation with homoscedastic errors, permutation tests using the W statistic showed slightly reduced power compared to the F statistic, but the difference disappeared as the sample size n increased. Consequently, the W test statistic is recommended for robust nonparametric hypothesis tests of regression coefficients in neuroimaging research.
•Permutation tests of regression coefficients are popular in neuroimaging research.•Most applications assume exchangeable errors and use the F statistic for inference.•Such tests can produce high false positive rates if the errors are heteroscedastic.•The W (Wald) statistic does not require exchangeable errors for valid results.•For robust permutation tests, the W statistic is preferred over the F statistic.
Journal Article