Frontiers in Probability and the Statistical Sciences – serie

This book introduces advanced undergraduate, graduate students and practitioners to statistical methods for ranking data. An important aspect of nonparametric statistics is oriented towards the use of ranking data. Rank correlation is defined through the notion of distance functions and the notion of compatibility is introduced to deal with incomplete data. Ranking data are also modeled using a variety of modern tools such as CART, MCMC, EM algorithm and factor analysis.This book deals with statistical methods used for analyzing such data and provides a novel and unifying approach for hypotheses testing. The techniques described in the book are illustrated with examples and the statistical software is provided on the authors’ website.

This book introduces advanced undergraduate, graduate students and practitioners to statistical methods for ranking data. An important aspect of nonparametric statistics is oriented towards the use of ranking data. Rank correlation is defined through the notion of distance functions and the notion of compatibility is introduced to deal with incomplete data. Ranking data are also modeled using a variety of modern tools such as CART, MCMC, EM algorithm and factor analysis.This book deals with statistical methods used for analyzing such data and provides a novel and unifying approach for hypotheses testing. The techniques described in the book are illustrated with examples and the statistical software is provided on the authors’ website.

Microbiome research has focused on microorganisms that live within the human body and their effects on health. During the last few years, the quantification of microbiome composition in different environments has been facilitated by the advent of high throughput sequencing technologies. The statistical challenges include computational difficulties due to the high volume of data; normalization and quantification of metabolic abundances, relative taxa and bacterial genes; high-dimensionality; multivariate analysis; the inherently compositional nature of the data; and the proper utilization of complementary phylogenetic information. This has resulted in an explosion of statistical approaches aimed at tackling the unique opportunities and challenges presented by microbiome data.This book provides a comprehensive overview of the state of the art in statistical and informatics technologies for microbiome research. In addition to reviewing demonstrably successful cutting-edge methods, particular emphasis is placed on examples in R that rely on available statistical packages for microbiome data. With its wide-ranging approach, the book benefits not only trained statisticians in academia and industry involved in microbiome research, but also other scientists working in microbiomics and in related fields.

Microbiome research has focused on microorganisms that live within the human body and their effects on health. During the last few years, the quantification of microbiome composition in different environments has been facilitated by the advent of high throughput sequencing technologies. The statistical challenges include computational difficulties due to the high volume of data; normalization and quantification of metabolic abundances, relative taxa and bacterial genes; high-dimensionality; multivariate analysis; the inherently compositional nature of the data; and the proper utilization of complementary phylogenetic information. This has resulted in an explosion of statistical approaches aimed at tackling the unique opportunities and challenges presented by microbiome data.This book provides a comprehensive overview of the state of the art in statistical and informatics technologies for microbiome research. In addition to reviewing demonstrably successful cutting-edge methods, particular emphasis is placed on examples in R that rely on available statistical packages for microbiome data. With its wide-ranging approach, the book benefits not only trained statisticians in academia and industry involved in microbiome research, but also other scientists working in microbiomics and in related fields.

This book presents a general method for deriving higher-order statistics of multivariate distributions with simple algorithms that allow for actual calculations. Multivariate nonlinear statistical models require the study of higher-order moments and cumulants. The main tool used for the definitions is the tensor derivative, leading to several useful expressions concerning Hermite polynomials, moments, cumulants, skewness, and kurtosis. A general test of multivariate skewness and kurtosis is obtained from this treatment. Exercises are provided for each chapter to help the readers understand the methods. Lastly, the book includes a comprehensive list of references, equipping readers to explore further on their own.

This book presents a general method for deriving higher-order statistics of multivariate distributions with simple algorithms that allow for actual calculations. Multivariate nonlinear statistical models require the study of higher-order moments and cumulants. The main tool used for the definitions is the tensor derivative, leading to several useful expressions concerning Hermite polynomials, moments, cumulants, skewness, and kurtosis. A general test of multivariate skewness and kurtosis is obtained from this treatment. Exercises are provided for each chapter to help the readers understand the methods. Lastly, the book includes a comprehensive list of references, equipping readers to explore further on their own.

This book covers an extensive class of models involving inhomogeneous Poisson processes and deals with their identification, i.e. the solution of certain estimation or hypothesis testing problems based on the given dataset. These processes are mathematically easy-to-handle and appear in numerous disciplines, including astronomy, biology, ecology, geology, seismology, medicine, physics, statistical mechanics, economics, image processing, forestry, telecommunications, insurance and finance, reliability, queuing theory, wireless networks, and localisation of sources.Beginning with the definitions and properties of some fundamental notions (stochastic integral, likelihood ratio, limit theorems, etc.), the book goes on to analyse a wide class of estimators for regular and singular statistical models. Special attention is paid to problems of change-point type, and in particular cusp-type change-point models, then the focus turns to the asymptotically efficient nonparametric estimation of the mean function, the intensity function, and of some functionals. Traditional hypothesis testing, including some goodness-of-fit tests, is also discussed. The theory is then applied to three classes of problems: misspecification in regularity (MiR),corresponding to situations where the chosen change-point model and that of the real data have different regularity; optical communication with phase and frequency modulation of periodic intensity functions; and localization of a radioactive (Poisson) source on the plane using K detectors.Each chapter concludes with a series of problems, and state-of-the-art references are provided, making the book invaluable to researchers and students working in areas which actively use inhomogeneous Poisson processes.

This book covers an extensive class of models involving inhomogeneous Poisson processes and deals with their identification, i.e. the solution of certain estimation or hypothesis testing problems based on the given dataset. These processes are mathematically easy-to-handle and appear in numerous disciplines, including astronomy, biology, ecology, geology, seismology, medicine, physics, statistical mechanics, economics, image processing, forestry, telecommunications, insurance and finance, reliability, queuing theory, wireless networks, and localisation of sources.Beginning with the definitions and properties of some fundamental notions (stochastic integral, likelihood ratio, limit theorems, etc.), the book goes on to analyse a wide class of estimators for regular and singular statistical models. Special attention is paid to problems of change-point type, and in particular cusp-type change-point models, then the focus turns to the asymptotically efficient nonparametric estimation of the mean function, the intensity function, and of some functionals. Traditional hypothesis testing, including some goodness-of-fit tests, is also discussed. The theory is then applied to three classes of problems: misspecification in regularity (MiR),corresponding to situations where the chosen change-point model and that of the real data have different regularity; optical communication with phase and frequency modulation of periodic intensity functions; and localization of a radioactive (Poisson) source on the plane using K detectors.Each chapter concludes with a series of problems, and state-of-the-art references are provided, making the book invaluable to researchers and students working in areas which actively use inhomogeneous Poisson processes.

The book discusses various inequalities and sharp bounds for the usual order statistics as well as some functions of them.  In particular, deterministic bounds, bounds for the case of IID samples from general, symmetric and life distributions, IID samples from shape restricted family of distributions, and samples from finite populations are all discussed in detail.  An elaborate numerical evaluation and comparison of various bounds are also presented in order to illustrate their inherent differences as well as their precision.  Furthermore, their applications to inference, reliability theory and characterizations are also highlighted. The book provides an in-depth exposure to various mathematical inequalities and bounds established historically as well as in recent years and their applications to order statistics and some important functions of them.  It thus presents an up-to-date discussion of all results in this important area of mathematical and statistical research.  The results described here are general in nature and therefore could be useful in other areas of Probability and Statistics as well.

The book discusses various inequalities and sharp bounds for the usual order statistics as well as some functions of them.  In particular, deterministic bounds, bounds for the case of IID samples from general, symmetric and life distributions, IID samples from shape restricted family of distributions, and samples from finite populations are all discussed in detail.  An elaborate numerical evaluation and comparison of various bounds are also presented in order to illustrate their inherent differences as well as their precision.  Furthermore, their applications to inference, reliability theory and characterizations are also highlighted. The book provides an in-depth exposure to various mathematical inequalities and bounds established historically as well as in recent years and their applications to order statistics and some important functions of them.  It thus presents an up-to-date discussion of all results in this important area of mathematical and statistical research.  The results described here are general in nature and therefore could be useful in other areas of Probability and Statistics as well.

This book is about copies-based nonparametric estimation of the drift function in stochastic differential equations (SDEs) driven by Brownian motion, a jump process, or fractional Brownian motion. While the estimators of the drift function in SDEs are classically computed from one long-time observation of the ergodic stationary solution, here the estimation framework – which is part of functional data analysis – involves multiple copies of the (non-stationary) solution observed over a short-time interval. Two kinds of nonparametric estimators are investigated for SDE models, first presented in the regression framework: the projection least squares estimator and the Nadaraya-Watson estimator. Adaptive procedures are provided for possible applications in statistical learning. Primarily intended for researchers in statistical inference for stochastic processes who are interested in the copies-based observation scheme, the book will also be useful for graduate and PhD students in probability and statistics, thanks to its multiple reminders of the requisite theory, especially the chapter on nonparametric regression.

This book presents recent findings on central and non-central limit theorems for Toeplitz and tapered Toeplitz random quadratic functionals of stationary processes, with applications in spectral-based statistical inference. It focuses on Gaussian, orthogonal increment-driven, and Lévy-driven linear stationary processes with memory, in both discrete and continuous time.Toeplitz matrices and operators are central to the study of stationary processes. The covariance matrix of a discrete-time stationary process is a truncated Toeplitz matrix generated by the process's spectral density; in continuous-time, this becomes a Toeplitz operator. The foundations of the trace approximation problem were laid by Grenander and Szegö in their classical monograph “Toeplitz Forms and Their Applications” (1958), and the subject has recently seen renewed interest due to developments in long-range dependence and tapered data analysis.The book addresses topics that are often overlooked in other texts, including the trace approximation problem, central limit theorems in continuous time, functional central and non-central limit theorems for Toeplitz processes, and central limit theorems for tapered functionals. It also covers approaches to estimating linear and nonlinear spectral functionals, Whittle estimators, and goodness-of-fit tests using tapered data – each enriched by new advances in the field.Comprising ten chapters and two appendices, the book begins with an overview of the main problems and a review of foundational concepts from real analysis, functional analysis, and matrix analysis. It then introduces a model that is a second-order stationary process and discusses key concepts and results from the general theory of stationary processes, before delving into the trace approximation problem. Subsequent chapters cover central and non-central limit theorems for Toeplitz and tapered Toeplitz random quadratic functionals and explore statistical inference problems. The appendices discuss the motivations and benefits of data tapering, and outline several important problems closely related to the main themes of the book.The text will be a valuable resource for researchers in time series analysis, econometrics, finance, and applied statistics. It is suitable for graduate-level courses in time series analysis or the statistics of stochastic processes, and as a supplementary reference for students of advanced statistics, probability, econometrics, or finance.

Next Generation Sequencing (NGS) is the latest high throughput technology to revolutionize genomic research. NGS generates massive genomic datasets that play a key role in the big data phenomenon that surrounds us today. To extract signals from high-dimensional NGS data and make valid statistical inferences and predictions, novel data analytic and statistical techniques are needed. This book contains 20 chapters written by prominent statisticians working with NGS data. The topics range from basic preprocessing and analysis with NGS data to more complex genomic applications such as copy number variation and isoform expression detection. Research statisticians who want to learn about this growing and exciting area will find this book useful. In addition, many chapters from this book could be included in graduate-level classes in statistical bioinformatics for training future biostatisticians who will be expected to deal with genomic data in basic biomedical research, genomic clinical trials and personalized medicine.About the editors:Somnath Datta is Professor and Vice Chair of Bioinformatics and Biostatistics at the University of Louisville. He is Fellow of the American Statistical Association, Fellow of the Institute of Mathematical Statistics and Elected Member of the International Statistical Institute. He has contributed to numerous research areas in Statistics, Biostatistics and Bioinformatics.Dan Nettleton is Professor and Laurence H. Baker Endowed Chair of Biological Statistics in the Department of Statistics at Iowa State University.  He is Fellow of the American Statistical Association and has published research on a variety of topics in statistics, biology and bioinformatics.

As chapters in this book demonstrate, BNP has important uses in clinical sciences and inference for issues like unknown partitions in genomics. Nonparametric Bayesian approaches (BNP) play an ever expanding role in biostatistical inference from use in proteomics to clinical trials. Many research problems involve an abundance of data and require flexible and complex probability models beyond the traditional parametric approaches. As this book's expert contributors show, BNP approaches can be the answer. Survival Analysis, in particular survival regression, has traditionally used BNP, but BNP's potential is now very broad. This applies to important tasks like arrangement of patients into clinically meaningful subpopulations and segmenting the genome into functionally distinct regions. This book is designed to both review and introduce application areas for BNP. While existing books provide theoretical foundations, this book connects theory to practice through engaging examples and research questions. Chapters cover: clinical trials, spatial inference, proteomics, genomics, clustering, survival analysis and ROC curve.

The Model-Free Prediction Principle expounded upon in this monograph is based on the simple notion of transforming a complex dataset to one that is easier to work with, e.g., i.i.d. or Gaussian. As such, it restores the emphasis on observable quantities, i.e., current and future data, as opposed to unobservable model parameters and estimates thereof, and yields optimal predictors in diverse settings such as regression and time series. Furthermore, the Model-Free Bootstrap takes us beyond point prediction in order to construct frequentist prediction intervals without resort to unrealistic assumptions such as normality.Prediction has been traditionally approached via a model-based paradigm, i.e., (a) fit a model to the data at hand, and (b) use the fitted model to extrapolate/predict future data. Due to both mathematical and computational constraints, 20th century statistical practice focused mostly on parametric models. Fortunately, with the advent of widely accessible powerful computing in the late 1970s, computer-intensive methods such as the bootstrap and cross-validation freed practitioners from the limitations of parametric models, and paved the way towards the `big data' era of the 21st century. Nonetheless, there is a further step one may take, i.e., going beyond even nonparametric models; this is where the Model-Free Prediction Principle is useful.Interestingly, being able to predict a response variable Y associated with a regressor variable X taking on any possible value seems to inadvertently also achieve the main goal of modeling, i.e., trying to describe how Y depends on X. Hence, as prediction can be treated as a by-product of model-fitting, key estimation problems can be addressed as a by-product of being able to perform prediction. In other words, a practitioner can use Model-Free Prediction ideas in order to additionally obtain point estimates and confidence intervals for relevant parameters leading to an alternative, transformation-based approach to statistical inference.

The Model-Free Prediction Principle expounded upon in this monograph is based on the simple notion of transforming a complex dataset to one that is easier to work with, e.g., i.i.d. or Gaussian. As such, it restores the emphasis on observable quantities, i.e., current and future data, as opposed to unobservable model parameters and estimates thereof, and yields optimal predictors in diverse settings such as regression and time series. Furthermore, the Model-Free Bootstrap takes us beyond point prediction in order to construct frequentist prediction intervals without resort to unrealistic assumptions such as normality.Prediction has been traditionally approached via a model-based paradigm, i.e., (a) fit a model to the data at hand, and (b) use the fitted model to extrapolate/predict future data. Due to both mathematical and computational constraints, 20th century statistical practice focused mostly on parametric models. Fortunately, with the advent of widely accessible powerful computing in the late 1970s, computer-intensive methods such as the bootstrap and cross-validation freed practitioners from the limitations of parametric models, and paved the way towards the `big data' era of the 21st century. Nonetheless, there is a further step one may take, i.e., going beyond even nonparametric models; this is where the Model-Free Prediction Principle is useful.Interestingly, being able to predict a response variable Y associated with a regressor variable X taking on any possible value seems to inadvertently also achieve the main goal of modeling, i.e., trying to describe how Y depends on X. Hence, as prediction can be treated as a by-product of model-fitting, key estimation problems can be addressed as a by-product of being able to perform prediction. In other words, a practitioner can use Model-Free Prediction ideas in order to additionally obtain point estimates and confidence intervals for relevant parameters leading to an alternative, transformation-based approach to statistical inference.

As chapters in this book demonstrate, BNP has important uses in clinical sciences and inference for issues like unknown partitions in genomics. Nonparametric Bayesian approaches (BNP) play an ever expanding role in biostatistical inference from use in proteomics to clinical trials. Many research problems involve an abundance of data and require flexible and complex probability models beyond the traditional parametric approaches. As this book's expert contributors show, BNP approaches can be the answer. Survival Analysis, in particular survival regression, has traditionally used BNP, but BNP's potential is now very broad. This applies to important tasks like arrangement of patients into clinically meaningful subpopulations and segmenting the genome into functionally distinct regions. This book is designed to both review and introduce application areas for BNP. While existing books provide theoretical foundations, this book connects theory to practice through engaging examples and research questions. Chapters cover: clinical trials, spatial inference, proteomics, genomics, clustering, survival analysis and ROC curve.

Next Generation Sequencing (NGS) is the latest high throughput technology to revolutionize genomic research. NGS generates massive genomic datasets that play a key role in the big data phenomenon that surrounds us today. To extract signals from high-dimensional NGS data and make valid statistical inferences and predictions, novel data analytic and statistical techniques are needed. This book contains 20 chapters written by prominent statisticians working with NGS data. The topics range from basic preprocessing and analysis with NGS data to more complex genomic applications such as copy number variation and isoform expression detection. Research statisticians who want to learn about this growing and exciting area will find this book useful. In addition, many chapters from this book could be included in graduate-level classes in statistical bioinformatics for training future biostatisticians who will be expected to deal with genomic data in basic biomedical research, genomic clinical trials and personalized medicine.About the editors:Somnath Datta is Professor and Vice Chair of Bioinformatics and Biostatistics at the University of Louisville. He is Fellow of the American Statistical Association, Fellow of the Institute of Mathematical Statistics and Elected Member of the International Statistical Institute. He has contributed to numerous research areas in Statistics, Biostatistics and Bioinformatics.Dan Nettleton is Professor and Laurence H. Baker Endowed Chair of Biological Statistics in the Department of Statistics at Iowa State University.  He is Fellow of the American Statistical Association and has published research on a variety of topics in statistics, biology and bioinformatics.

This book presents an overview of computational and statistical design and analysis of mass spectrometry-based proteomics, metabolomics, and lipidomics data. This contributed volume provides an introduction to the special aspects of statistical design and analysis with mass spectrometry data for the new omic sciences. The text discusses common aspects of design and analysis between and across all (or most) forms of mass spectrometry, while also providing special examples of application with the most common forms of mass spectrometry. Also covered are applications of computational mass spectrometry not only in clinical study but also in the interpretation of omics data in plant biology studies.Omics research fields are expected to revolutionize biomolecular research by the ability to simultaneously profile many compounds within either patient blood, urine, tissue, or other biological samples. Mass spectrometry is one of the key analytical techniques used in these new omic sciences. Liquid chromatography mass spectrometry, time-of-flight data, and Fourier transform mass spectrometry are but a selection of the measurement platforms available to the modern analyst. Thus in practical proteomics or metabolomics,  researchers will not only be confronted with new high dimensional data types—as opposed to the familiar data structures in more classical genomics—but also with great variation between distinct types of mass spectral measurements derived from different platforms, which may complicate analyses, comparison, and interpretation of results.