Eugenia Stoimenova

Rank Tests Based on Exceeding Observations

Rank tests based on the maximum number of exceeding observations for several standard nonparametric hypotheses are proposed. An approach to constructing nonparametric rank tests via metrics on the permutation group is used. The test statistics are based on a metric induced by Chebyshevs norm.

Big data techniques, systems, applications, and platforms: Case studies from academia

Big data is a broad term with numerous dimensions, most notably: big data characteristics, techniques, software systems, application domains, computing platforms, and big data milieu (industry, government, and academia). In this paper we briefly introduce fundamental big data characteristics and then present seven case studies of big data techniques, systems, applications, and platforms, as seen from academic perspective (industry and government perspectives are not subject of this publication). While we feel that it is difficult, if at all possible, to encapsulate all of the important big data dimensions in a strict and uniform, yet comprehensible language, we believe that a set of diverse case studies - like the one that is offered in this paper - a set that spreads over the principal big data dimensions can indeed be beneficial to the broad big data community by helping experts in one realm to better understand currents trends in the other realms.

Methodology in robust and nonparametric statistics

The original 1996 edition of this book, written by the first two authors, was a specialised treatment of the theoretical background of robust and nonparametric statistical methods. This new edition has been updated to reflect current research and publications on the topic. The authors are leading academics in the field and have written a fine text. The book is organised in nine chapters. Chapters 2–7 cover the basic theory of asymptotics and interrelations. They comprise a thorough update of Part I of the first edition. Chapter 1 is a short (historical) introduction to the topic and includes many up-to-date references. Chapter 2 provides the basic theory of robust statistical inference for classical linear models. The emphasis is on motivations rather than derivation. There are problem sections at the end of this and subsequent chapters which are helpful for graduate students. Chapters 3–7 deal with methodological tools of robust estimation of location, regression and scale parameters in univariate models. Chapter 3 provides a basic survey of robust and nonparametric estimators of location and regression for certain families of estimators. The asymptotic representations of L, M and R estimators are presented in an up-to-date and more general level in subsequent chapters. Chapter 7 deals with interrelations among these estimators. Chapter 8 covers various aspects of nonparametric estimation in multivariate models, with particular emphasis being placed on the most recent results. Nonparametric and robust statistical tests and confidence sets are discussed in the last chapter. There is an extensive updated bibliography at the end of the book. In summary, it is a modern book written for specialists and is a valuable contribution to theoretical statistics. It is not a book for practitioners and does not pretend to be. The level of the book is too high to use as a textbook except for those PhD students pursuing this or a closely related topic. Therefore, I recommend this book to someone working on the theory of robust and nonparametric methods from a mathematical point of view.

Nonparametric statistical inference

and a new chapter are particularly concerned with topics like rank procedures for nonlinear models, models with dependent error structure, rank methods for mixed models, and time series analysis. The book concentrates on a unified approach to rank-based analysis. The authors have enhanced the geometry of estimation and testing with rank-based procedures. In most of the cases the only difference in the geometry of the rank-based methods is that that the estimates and test statistics are based on L1 norm, while the traditional analysis is based on the Euclidean norm. Using the weighted L1 norm allows similar statistical interpretations to those of least squares. Additionally, some useful concepts such as distribution freeness and robustness have a simple formulation. Hettmansperger and McKean examine a wealth of interesting problems in connection with applying nonparametric robust methods. The first two chapters cover the oneand two-sample location models with many illustrative examples. Chapter 1 presents important results regarding robustness and L1 norm properties of the classical rank procedures. Chapter 3 is concerned with rank-based methods for the analysis of linear models. The theory of rank-based estimates and rank-based tests is presented for general rank scores. It includes material on properties of regression estimates, computational aspects of the estimates, and diagnostic procedures of the models. Special topics such as survival analysis, correlation models and high breakdown estimates are also discussed. The last section is new in this edition and treats the nonlinear estimation problem in the geometry of the L1 norm. In Chapter 4, the authors apply the estimation and testing theory of Chapter 3 to procedures for experimental designs. They give extensive development of methods for oneand two-way layouts but the ideas can be easily extended to rank-based analysis for any fixed effects design. The new chapter 5 is devoted to robust rank-based models with dependent error structure. It includes four sections on mixed linear models, and two short sections on generalised estimating equations and autoregressive time series models. Like other chapters, the new material is illustrated with examples using R libraries and functions. The coverage in the last chapter on multivariate models is highlighted by a treatment of multidimensional ranks, affine invariant and equivariant methods for multivariate location model, robustness properties of the estimates, and other advanced topics. The book also has a 45 page appendix containing theoretical results on asymptotics. In summary, this is a well-written and nicely presented book that is likely to appeal to a reader with a good mathematical background and an interest in robust and nonparametric statistical methods. In my opinion, the book could provide the basis for a seminar in robust non-parametric methods for graduate students in statistics or mathematics. However, the authors suggest selected topics from the first four chapters that could be used for a basic graduate course in rank-based methods.

The Power of Exceedance-Type Tests Under Lehmann Alternatives

The distribution of a rank test statistic for the two-sample problem is derived under Lehmann alternative. The test statistic is based on extreme ranks of both samples in the combined sample. The exact power function of the test is studied and compared to the power functions of several two-sample rank tests.

Statistical approach in soil-water characteristic curve modelling

In this paper we evaluate a number of model equations for the soil-water characteristic curve (SWCC), provided a relationship exists for each relevant soil. The models are with two parameters and relate suction to volumetric water content for values of suction higher then air entry value. Logarithmic tranformations are applied to the variables in order to find linear patern. Most of generally used models are also discussed and compared to the proposed. An example is given how to assess the experimental data to define the variable transformation and how to choose the most proper model. The procedure for estimating the air entry value is also presented.

On statistical properties of Chebyshev's norm

A measure of association between two rankings is proposed. This measure -- the maximum of the absolute values of the difference between the ranks -- is treated as a metric on the set of permutations. We calculate the mean and the variance of the metric under uniformity assumption. The limiting distribution is established.

Šidák-type tests for the two-sample problem based on precedence and exceedance statistics

ABSTRACT This paper deals with a class of nonparametric two-sample tests for ordered alternatives. The test statistics proposed are based on the number of observations from one sample that precede or exceed a threshold specified by the other sample, and they are extensions of Šidáks test. We derive their exact null distributions and also discuss a large-sample approximation. We then study their power properties exactly against the Lehmann alternative and make some comparative comments. Finally, we present an example to illustrate the proposed tests.

EM Estimation of the Parameters in Latent Mallows’ Models

Mallows’ models are often convenient initial tool for analyzing a set of rank data. They capture the main structure of the data with only one parameter and could be the basis for further research. However, it is usually unrealistic to expect a one-parameter model to reveal all features of the data. One possible generalization of these models could be made by assuming that there are several latent groups in the population. In this paper, we propose an algorithm to find maximum likelihood estimates of the unknown parameters of Latent Mallows’ models by making use of the EM algorithm. As an application of the considered estimation algorithm, a comparison between models based on different metrics is made via simulation study.

Robust nonparametric statistical methods

As elsewhere in the book, the commands shown load any necessary add-on packages. The HELP examples section for this chapter is the longest in the book. Chapter 6 is devoted to R graphics and is split into aptly named sections: A compendium of useful plots, Adding elements, Options and parameters, Saving graphs. This chapter provides an easy to navigate reference resource for finding out or recalling how to do many tasks related to graphics. A brief overview of the graphics capabilities of R is provided and the example commands use the ‘classic’ graphics commands, lattice, ggplot2 and others, as appropriate. There is no confusion since commands that use functions from these packages are preceded by a library command that loads them. The last chapter presents some more advanced topics – power and size calculations, reading geocoded data and drawing maps, data scraping, some more elaborated data management and simulation tasks. The entries in this chapter are generally longer and involve longer scripts and functions. The HELP study data set used in the end-of-chapter examples is described in the appendix (‘HELP’ is an acronym here). The content is reliable and the material is well structured, though a persistent reader may find a typo (advice on p. 55 to prepend ‘d’ to a probability distribution name to obtain quantiles). The reader may also be surprised to find sections on control flow and probability distributions in the chapter on data management. This book is an excellent reference resource. Used this way, it can be helpful for years to come for both experienced and novice users. The organization of the material makes it easy to find the relevant piece of information either by topic (from the table of contents) or using one of the indexes. The task entries are self-contained. Users with experience in technical computing may use it as a quick starter in R, as well. Those who like the tangible feeling of reading from a printed book will definitely use it actively and not put it in a few weeks on a bookcase to gather dust among other nice books. It can offer much to the elusive ‘experienced’ user, as well, in addition to the online help system and other freely available documentation. Such users may prefer the electronic edition of this book.