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Methods And Theory For Nonparametric Inference In High Dimensional Settings
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Book Synopsis Methods and Theory for Nonparametric Inference In High-dimensional Settings by : Yunhua Xiang
Download or read book Methods and Theory for Nonparametric Inference In High-dimensional Settings written by Yunhua Xiang and published by . This book was released on 2021 with total page 136 pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation addresses nonparametric estimation and inference problems of graphical modeling, linear association assessment, and matrix completion. First, we introduce a flexible framework for nonparametric graphical modeling. We propose three nonparametric measures of conditional dependence, which have theoretically optimal estimators that allow incorporation of flexible machine learning techniques and yield wald-type confidence intervals. In the second project, we propose a nonparametric parameter to measure the linear association between the outcome and explanatory variables. This parameter is always explicitly defined even when the true relationship is nonlinear and is equivalent with the regression coefficient under a linear model space. Thus, its estimator can be a more robust alternative to the standard model-based techniques to estimate the coefficients of a linear model. In the final project, we theoretically show that nuclear-norm penalization used for recovering low-rank matrices, remains effective even when the underlying matrices are generated by a low-dimensional non-linear manifold. The convergence rate can be expressed as a function of the size of the matrix, as well as the smoothness and dimension of the manifold, which is minimax optimal (up to a log term).
Book Synopsis Nonparametric Inference for High Dimensional Data by : Subhadeep Mukhopadhyay
Download or read book Nonparametric Inference for High Dimensional Data written by Subhadeep Mukhopadhyay and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Learning from data, especially 'Big Data', is becoming increasingly popular under names such as Data Mining, Data Science, Machine Learning, Statistical Learning and High Dimensional Data Analysis. In this dissertation we propose a new related field, which we call 'United Nonparametric Data Science' - applied statistics with "just in time" theory. It integrates the practice of traditional and novel statistical methods for nonparametric exploratory data modeling, and it is applicable to teaching introductory statistics courses that are closer to modern frontiers of scientific research. Our framework includes small data analysis (combining traditional and modern nonparametric statistical inference), big and high dimensional data analysis (by statistical modeling methods that extend our unified framework for small data analysis). The first part of the dissertation (Chapters 2 and 3) has been oriented by the goal of developing a new theoretical foundation to unify many cultures of statistical science and statistical learning methods using mid-distribution function, custom made orthonormal score function, comparison density, copula density, LP moments and comoments. It is also examined how this elegant theory yields solution to many important applied problems. In the second part (Chapter 4) we extend the traditional empirical likelihood (EL), a versatile tool for nonparametric inference, in the high dimensional context. We introduce a modified version of the EL method that is computationally simpler and applicable to a large class of "large p small n" problems, allowing p to grow faster than n. This is an important step in generalizing the EL in high dimensions beyond the p ≤ n threshold where the standard EL and its existing variants fail. We also present detailed theoretical study of the proposed method. The electronic version of this dissertation is accessible from http://hdl.handle.net/1969.1/149430
Book Synopsis Statistics for High-Dimensional Data by : Peter Bühlmann
Download or read book Statistics for High-Dimensional Data written by Peter Bühlmann and published by Springer Science & Business Media. This book was released on 2011-06-08 with total page 568 pages. Available in PDF, EPUB and Kindle. Book excerpt: Modern statistics deals with large and complex data sets, and consequently with models containing a large number of parameters. This book presents a detailed account of recently developed approaches, including the Lasso and versions of it for various models, boosting methods, undirected graphical modeling, and procedures controlling false positive selections. A special characteristic of the book is that it contains comprehensive mathematical theory on high-dimensional statistics combined with methodology, algorithms and illustrations with real data examples. This in-depth approach highlights the methods’ great potential and practical applicability in a variety of settings. As such, it is a valuable resource for researchers, graduate students and experts in statistics, applied mathematics and computer science.
Book Synopsis Nonparametric Learning in High Dimensions by : Han Liu
Download or read book Nonparametric Learning in High Dimensions written by Han Liu and published by . This book was released on 2010 with total page 279 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "This thesis develops flexible and principled nonparametric learning algorithms to explore, understand, and predict high dimensional and complex datasets. Such data appear frequently in modern scientific domains and lead to numerous important applications. For example, exploring high dimensional functional magnetic resonance imaging data helps us to better understand brain functionalities; inferring large-scale gene regulatory network is crucial for new drug design and development; detecting anomalies in high dimensional transaction databases is vital for corporate and government security. Our main results include a rigorous theoretical framework and efficient nonparametric learning algorithms that exploit hidden structures to overcome the curse of dimensionality when analyzing massive high dimensional datasets. These algorithms have strong theoretical guarantees and provide high dimensional nonparametric recipes for many important learning tasks, ranging from unsupervised exploratory data analysis to supervised predictive modeling. In this thesis, we address three aspects: 1 Understanding the statistical theories of high dimensional nonparametric inference, including risk, estimation, and model selection consistency; 2 Designing new methods for different data-analysis tasks, including regression classification, density estimation, graphical model learning, multi-task learning, spatial-temporal adaptive learning; 3 Demonstrating the usefulness of these methods in scientific applications, including functional genomics, cognitive neuroscience, and meteorology. In the last part of this thesis, we also present the future vision of high dimensional and large-scale nonparametric inference."
Book Synopsis Nonparametric Statistical Inference with an Emphasis on Information-Theoretic Methods by : Jan Mielniczuk
Download or read book Nonparametric Statistical Inference with an Emphasis on Information-Theoretic Methods written by Jan Mielniczuk and published by Mdpi AG. This book was released on 2022-06-02 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book addresses contemporary statistical inference issues when no or minimal assumptions on the nature of studied phenomenon are imposed. Information theory methods play an important role in such scenarios. The approaches discussed include various high-dimensional regression problems, time series and dependence analyses.
Book Synopsis High-Dimensional Statistics by : Martin J. Wainwright
Download or read book High-Dimensional Statistics written by Martin J. Wainwright and published by Cambridge University Press. This book was released on 2019-02-21 with total page 571 pages. Available in PDF, EPUB and Kindle. Book excerpt: A coherent introductory text from a groundbreaking researcher, focusing on clarity and motivation to build intuition and understanding.
Book Synopsis Partially Linear Models by : Wolfgang Härdle
Download or read book Partially Linear Models written by Wolfgang Härdle and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 210 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the last ten years, there has been increasing interest and activity in the general area of partially linear regression smoothing in statistics. Many methods and techniques have been proposed and studied. This monograph hopes to bring an up-to-date presentation of the state of the art of partially linear regression techniques. The emphasis is on methodologies rather than on the theory, with a particular focus on applications of partially linear regression techniques to various statistical problems. These problems include least squares regression, asymptotically efficient estimation, bootstrap resampling, censored data analysis, linear measurement error models, nonlinear measurement models, nonlinear and nonparametric time series models.
Book Synopsis An Introduction to the Advanced Theory of Nonparametric Econometrics by : Jeffrey S. Racine
Download or read book An Introduction to the Advanced Theory of Nonparametric Econometrics written by Jeffrey S. Racine and published by Cambridge University Press. This book was released on 2019-06-27 with total page 435 pages. Available in PDF, EPUB and Kindle. Book excerpt: Provides theory, open source R implementations, and the latest tools for reproducible nonparametric econometric research.
Book Synopsis Inference of Nonparametric Hypothesis Testing on High Dimensional Longitudinal Data and Its Application in DNA Copy Number Variation and Micro Array Data Analysis by : Ke Zhang
Download or read book Inference of Nonparametric Hypothesis Testing on High Dimensional Longitudinal Data and Its Application in DNA Copy Number Variation and Micro Array Data Analysis written by Ke Zhang and published by . This book was released on 2008 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: High throughput screening technologies have generated a huge amount of biological data in the last ten years. With the easy availability of array technology, researchers started to investigate biological mechanisms using experiments with more sophisticated designs that pose novel challenges to statistical analysis. We provide theory for robust statistical tests in three flexible models. In the first model, we consider the hypothesis testing problems when there are a large number of variables observed repeatedly over time. A potential application is in tumor genomics where an array comparative genome hybridization (aCGH) study will be used to detect progressive DNA copy number changes in tumor development. In the second model, we consider hypothesis testing theory in a longitudinal microarray study when there are multiple treatments or experimental conditions. The tests developed can be used to detect treatment effects for a large group of genes and discover genes that respond to treatment over time. In the third model, we address a hypothesis testing problem that could arise when array data from different sources are to be integrated. We perform statistical tests by assuming a nested design. In all models, robust test statistics were constructed based on moment methods allowing unbalanced design and arbitrary heteroscedasticity. The limiting distributions were derived under the nonclassical setting when the number of probes is large. The test statistics are not targeted at a single probe. Instead, we are interested in testing for a selected set of probes simultaneously. Simulation studies were carried out to compare the proposed methods with some traditional tests using linear mixed-effects models and generalized estimating equations. Interesting results obtained with the proposed theory in two cancer genomic studies suggest that the new methods are promising for a wide range of biological applications with longitudinal arrays.
Book Synopsis Nonparametric Statistical Inference with an Emphasis on Information-Theoretic Methods by : Jan Mielniczuk
Download or read book Nonparametric Statistical Inference with an Emphasis on Information-Theoretic Methods written by Jan Mielniczuk and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book addresses contemporary statistical inference issues when no or minimal assumptions on the nature of studied phenomenon are imposed. Information theory methods play an important role in such scenarios. The approaches discussed include various high-dimensional regression problems, time series and dependence analyses.
Book Synopsis Introduction to High-Dimensional Statistics by : Christophe Giraud
Download or read book Introduction to High-Dimensional Statistics written by Christophe Giraud and published by CRC Press. This book was released on 2021-08-25 with total page 410 pages. Available in PDF, EPUB and Kindle. Book excerpt: Praise for the first edition: "[This book] succeeds singularly at providing a structured introduction to this active field of research. ... it is arguably the most accessible overview yet published of the mathematical ideas and principles that one needs to master to enter the field of high-dimensional statistics. ... recommended to anyone interested in the main results of current research in high-dimensional statistics as well as anyone interested in acquiring the core mathematical skills to enter this area of research." —Journal of the American Statistical Association Introduction to High-Dimensional Statistics, Second Edition preserves the philosophy of the first edition: to be a concise guide for students and researchers discovering the area and interested in the mathematics involved. The main concepts and ideas are presented in simple settings, avoiding thereby unessential technicalities. High-dimensional statistics is a fast-evolving field, and much progress has been made on a large variety of topics, providing new insights and methods. Offering a succinct presentation of the mathematical foundations of high-dimensional statistics, this new edition: Offers revised chapters from the previous edition, with the inclusion of many additional materials on some important topics, including compress sensing, estimation with convex constraints, the slope estimator, simultaneously low-rank and row-sparse linear regression, or aggregation of a continuous set of estimators. Introduces three new chapters on iterative algorithms, clustering, and minimax lower bounds. Provides enhanced appendices, minimax lower-bounds mainly with the addition of the Davis-Kahan perturbation bound and of two simple versions of the Hanson-Wright concentration inequality. Covers cutting-edge statistical methods including model selection, sparsity and the Lasso, iterative hard thresholding, aggregation, support vector machines, and learning theory. Provides detailed exercises at the end of every chapter with collaborative solutions on a wiki site. Illustrates concepts with simple but clear practical examples.
Book Synopsis Multivariate Nonparametric Methods with R by : Hannu Oja
Download or read book Multivariate Nonparametric Methods with R written by Hannu Oja and published by Springer Science & Business Media. This book was released on 2010-03-25 with total page 239 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers a new, fairly efficient, and robust alternative to analyzing multivariate data. The analysis of data based on multivariate spatial signs and ranks proceeds very much as does a traditional multivariate analysis relying on the assumption of multivariate normality; the regular L2 norm is just replaced by different L1 norms, observation vectors are replaced by spatial signs and ranks, and so on. A unified methodology starting with the simple one-sample multivariate location problem and proceeding to the general multivariate multiple linear regression case is presented. Companion estimates and tests for scatter matrices are considered as well. The R package MNM is available for computation of the procedures. This monograph provides an up-to-date overview of the theory of multivariate nonparametric methods based on spatial signs and ranks. The classical book by Puri and Sen (1971) uses marginal signs and ranks and different type of L1 norm. The book may serve as a textbook and a general reference for the latest developments in the area. Readers are assumed to have a good knowledge of basic statistical theory as well as matrix theory. Hannu Oja is an academy professor and a professor in biometry in the University of Tampere. He has authored and coauthored numerous research articles in multivariate nonparametrical and robust methods as well as in biostatistics.
Book Synopsis Nonparametric Inference by : Z. Govindarajulu
Download or read book Nonparametric Inference written by Z. Govindarajulu and published by World Scientific Publishing Company Incorporated. This book was released on 2007-01-01 with total page 669 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a solid foundation on nonparametric inference for students taking a graduate course in nonparametric statistics and serves as an easily accessible source for researchers in the area. With the exception of some sections requiring familiarity with measure theory, readers with an advanced calculus background will be comfortable with the material.
Book Synopsis Latent Source Models for Nonparametric Inference by : George H. Chen
Download or read book Latent Source Models for Nonparametric Inference written by George H. Chen and published by . This book was released on 2015 with total page 101 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nearest-neighbor inference methods have been widely and successfully used in numerous applications such as forecasting which news topics will go viral, recommending products to people in online stores, and delineating objects in images by looking at image patches. However, there is little theoretical understanding of when, why, and how well these nonparametric inference methods work in terms of key problem-specific quantities relevant to practitioners. This thesis bridges the gap between theory and practice for these methods in the three specific case studies of time series classification, online collaborative filtering, and patch-based image segmentation. To do so, for each of these problems, we prescribe a probabilistic model in which the data appear generated from unknown "latent sources" that capture salient structure in the problem. These latent source models naturally lead to nearest-neighbor or nearest-neighbor-like inference methods similar to ones already used in practice. We derive theoretical performance guarantees for these methods, relating inference quality to the amount of training data available and problems-specific structure modeled by the latent sources.
Book Synopsis Statistical Inference of Properties of Distributions by : Jiantao Jiao
Download or read book Statistical Inference of Properties of Distributions written by Jiantao Jiao and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Modern data science applications--ranging from graphical model learning to image registration to inference of gene regulatory networks--frequently involve pipelines of exploratory analysis requiring accurate inference of a property of the distribution governing the data rather than the distribution itself. Notable examples of properties include mutual information, Kullback--Leibler divergence, total variation distance, the entropy rate, among others. This thesis makes contributions to the performance, structure, and deployment of minimax rate-optimal estimators for a large variety of properties in high-dimensional and nonparametric settings. We present general methods for constructing information theoretically near-optimal estimators, and identify the corresponding limits in terms of the parameter dimension, the mixing rate (for processes with memory), and smoothness of the underlying density (in the nonparametric setting). We employ our schemes on the Google 1 Billion Word Dataset to estimate the fundamental limit of perplexity in language modeling, and to improve graphical model learning. The estimators are efficiently computable and exhibit a ``sample size enlargement'' phenomenon, i.e., they attain with $n$ samples what prior methods would have needed $n\log n$ samples to achieve. We provide a brief survey on our utilization and development of relevant tools from approximation theory, probability theory, and functional analysis.
Book Synopsis All of Nonparametric Statistics by : Larry Wasserman
Download or read book All of Nonparametric Statistics written by Larry Wasserman and published by Springer Science & Business Media. This book was released on 2006-09-10 with total page 272 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text provides the reader with a single book where they can find accounts of a number of up-to-date issues in nonparametric inference. The book is aimed at Masters or PhD level students in statistics, computer science, and engineering. It is also suitable for researchers who want to get up to speed quickly on modern nonparametric methods. It covers a wide range of topics including the bootstrap, the nonparametric delta method, nonparametric regression, density estimation, orthogonal function methods, minimax estimation, nonparametric confidence sets, and wavelets. The book’s dual approach includes a mixture of methodology and theory.
Book Synopsis Nonparametric Econometrics by : Qi Li
Download or read book Nonparametric Econometrics written by Qi Li and published by Princeton University Press. This book was released on 2011-10-09 with total page 769 pages. Available in PDF, EPUB and Kindle. Book excerpt: A comprehensive, up-to-date textbook on nonparametric methods for students and researchers Until now, students and researchers in nonparametric and semiparametric statistics and econometrics have had to turn to the latest journal articles to keep pace with these emerging methods of economic analysis. Nonparametric Econometrics fills a major gap by gathering together the most up-to-date theory and techniques and presenting them in a remarkably straightforward and accessible format. The empirical tests, data, and exercises included in this textbook help make it the ideal introduction for graduate students and an indispensable resource for researchers. Nonparametric and semiparametric methods have attracted a great deal of attention from statisticians in recent decades. While the majority of existing books on the subject operate from the presumption that the underlying data is strictly continuous in nature, more often than not social scientists deal with categorical data—nominal and ordinal—in applied settings. The conventional nonparametric approach to dealing with the presence of discrete variables is acknowledged to be unsatisfactory. This book is tailored to the needs of applied econometricians and social scientists. Qi Li and Jeffrey Racine emphasize nonparametric techniques suited to the rich array of data types—continuous, nominal, and ordinal—within one coherent framework. They also emphasize the properties of nonparametric estimators in the presence of potentially irrelevant variables. Nonparametric Econometrics covers all the material necessary to understand and apply nonparametric methods for real-world problems.
