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Nonparametric Estimation Under Shape Constraints
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Book Synopsis Nonparametric Estimation under Shape Constraints by : Piet Groeneboom
Download or read book Nonparametric Estimation under Shape Constraints written by Piet Groeneboom and published by Cambridge University Press. This book was released on 2014-12-11 with total page 429 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces basic concepts of shape constrained inference and guides the reader to current developments in the subject.
Book Synopsis Nonparametric Estimation under Shape Constraints by : Piet Groeneboom
Download or read book Nonparametric Estimation under Shape Constraints written by Piet Groeneboom and published by Cambridge University Press. This book was released on 2014-12-11 with total page 429 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book treats the latest developments in the theory of order-restricted inference, with special attention to nonparametric methods and algorithmic aspects. Among the topics treated are current status and interval censoring models, competing risk models, and deconvolution. Methods of order restricted inference are used in computing maximum likelihood estimators and developing distribution theory for inverse problems of this type. The authors have been active in developing these tools and present the state of the art and the open problems in the field. The earlier chapters provide an introduction to the subject, while the later chapters are written with graduate students and researchers in mathematical statistics in mind. Each chapter ends with a set of exercises of varying difficulty. The theory is illustrated with the analysis of real-life data, which are mostly medical in nature.
Book Synopsis Nonparametric Survival Analysis Under Shape Restrictions by : Shabnam Fani
Download or read book Nonparametric Survival Analysis Under Shape Restrictions written by Shabnam Fani and published by . This book was released on 2014 with total page 126 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main problem studied in this thesis is to analyse and model time-to- event data, particularly when the survival times of subjects under study are not exactly observed. One of the primary tasks in the analysis of survival data is to study the distribution of the event times of interest. In order to avoid strict assumptions associated with a parametric model, we resort to nonparametric methods for estimating a function. Although other nonparametric approaches, such as Kaplan-Meier, kernel-based, and roughness penalty methods, are popular tools for solving function estimation problems, they suffer from some non-trivial issues like the loss of some important information about the true underlying function, difficulties with bandwidth or tuning parameter selection. In contrast, one can avoid these issues at the cost of enforcing some qualitative shape constraints on the function to be estimated. We confine our survival analysis studies to estimating a hazard function since it may make a lot of practical sense to impose certain shape constraints on it. Specifically, we study the problem of nonparametric estimation of a hazard function subject to convex shape restrictions, which naturally entails monotonicity constraints. In this thesis, three main objectives are addressed. Firstly, the problem of nonparametric maximum-likelihood estimation of a hazard function under convex shape restrictions is investigated. We introduce a new nonparametric approach to estimating a convex hazard function in the case of exact observations, the case of interval-censored observations, and the mixed case of exact and interval-censored observations. A new idea to handle the problem of choosing the minimum of a convex hazard function estimate is proposed. Based on this, a new fast algorithm for nonparametric hazard function estimation under convexity shape constraints is developed. Theoretical justification for the convergence of the new algorithm is provided. Secondly, nonparametric estimation of a hazard function under smoothness and convex shape assumptions is studied. Particularly, our nonparametric maximum-likelihood approach is generalized for smooth estimation of a function by applying a higher-order smoothness assumption of an estimator. We also evaluate the performance of the estimators using simulation studies and real-world data. Numerical studies suggest that the shape-constrained estimators generally outperform their unconstrained competitors. Moreover, the empirical results indicate that the smooth shape-restricted estimator has more capability to model human mortality data compared to the piecewise linear continuous estimator, specifically in the infant mortality phase. Lastly, our nonparametric estimation of a hazard function approach under convex shape restrictions is extended to the Cox proportional hazards model. A new algorithm is also developed to estimate both convex baseline hazard function and the effects of covariates on survival times. Numerical studies reveal that our new approaches generally dominate the traditional partial likelihood method in the case of right-censored data and the fully semiparametric maximum likelihood estimation method in the case of interval-censored data. Overall, our series of studies show that the shape-restricted approach tends to provide more accurate estimation than its unconstrained competitors, and further investigations in this direction can be highly fruitful.
Book Synopsis Nonparametric Estimation of Additive Models with Shape Constraints by : Lu Wang
Download or read book Nonparametric Estimation of Additive Models with Shape Constraints written by Lu Wang and published by . This book was released on 2016 with total page 97 pages. Available in PDF, EPUB and Kindle. Book excerpt: Monotone additive models are useful in estimating productivity curves or analyzing disease risk where the predictors are known to have monotonic effects on the response. Existing literature mainly focuses on univariate monotone smoothing. Available methods for the estimation of monotone additive models are either difficult to interpret or have no asymptotic guarantees. In the first part of this dissertation, we propose a one-step backfitted constrained polynomial spline method for the estimation of monotone additive models. In our proposed method, we obtain monotone estimators by imposing a set of linear constraints on the spline coefficients for each additive component. In the second part of the dissertation, we extend the constrained polynomial spline method to estimate the production frontier that is used to quantify the maximum production output in econometrics. The estimation of frontier functions is more challenging since it is the boundary of the support rather than the mean output function to be estimated. Here, we develop a two-step shape constrained polynomial spline method for the frontier estimation. The first step is to capture the shape of frontier while the second step is to estimate the location of frontier. Both proposed methods in this dissertation give smooth estimators with the desired shape constraints (monotonicity or/and concavity). They are easily implementable and computationally efficient by taking advantage of linear programming. Most importantly, our methods are applicable for multi-dimensions where some existing methods fail to work. For the assessment of properties of the proposed estimators, asymptotic theory is also developed. In addition, the simulation studies and application of our methods to analyze Norwegian Farm data in both parts suggest that our proposed methods have better numerical performance than the existing methods, especially when the data has outliers.
Book Synopsis Shape Constrained Nonparametric Estimation in the Cox Model by : Gabriela Florentina Nane
Download or read book Shape Constrained Nonparametric Estimation in the Cox Model written by Gabriela Florentina Nane and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Nonparametric Estimation and Inference Under Shape Restrictions by : Joel Horowitz
Download or read book Nonparametric Estimation and Inference Under Shape Restrictions written by Joel Horowitz and published by . This book was released on 2016 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Economic theory often provides shape restrictions on functions of interest in applications, such as monotonicity, convexity, non-increasing (non-decreasing) returns to scale, or the Slutsky inequality of consumer theory; but economic theory does not provide finite-dimensional parametric models. This motivates nonparametric estimation under shape restrictions. Nonparametric estimates are often very noisy. Shape restrictions stabilize nonparametric estimates without imposing arbitrary restrictions, such as additivity or a single-index structure, that may be inconsistent with economic theory and the data. This paper explains how to estimate and obtain an asymptotic uniform confidence band for a conditional mean function under possibly nonlinear shape restrictions, such as the Slutsky inequality. The results of Monte Carlo experiments illustrate the finite-sample performance of the method, and an empirical example illustrates its use in an application.
Book Synopsis Missing and Modified Data in Nonparametric Estimation by : Sam Efromovich
Download or read book Missing and Modified Data in Nonparametric Estimation written by Sam Efromovich and published by CRC Press. This book was released on 2018-03-12 with total page 867 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a systematic and unified approach for modern nonparametric treatment of missing and modified data via examples of density and hazard rate estimation, nonparametric regression, filtering signals, and time series analysis. All basic types of missing at random and not at random, biasing, truncation, censoring, and measurement errors are discussed, and their treatment is explained. Ten chapters of the book cover basic cases of direct data, biased data, nondestructive and destructive missing, survival data modified by truncation and censoring, missing survival data, stationary and nonstationary time series and processes, and ill-posed modifications. The coverage is suitable for self-study or a one-semester course for graduate students with a prerequisite of a standard course in introductory probability. Exercises of various levels of difficulty will be helpful for the instructor and self-study. The book is primarily about practically important small samples. It explains when consistent estimation is possible, and why in some cases missing data should be ignored and why others must be considered. If missing or data modification makes consistent estimation impossible, then the author explains what type of action is needed to restore the lost information. The book contains more than a hundred figures with simulated data that explain virtually every setting, claim, and development. The companion R software package allows the reader to verify, reproduce and modify every simulation and used estimators. This makes the material fully transparent and allows one to study it interactively. Sam Efromovich is the Endowed Professor of Mathematical Sciences and the Head of the Actuarial Program at the University of Texas at Dallas. He is well known for his work on the theory and application of nonparametric curve estimation and is the author of Nonparametric Curve Estimation: Methods, Theory, and Applications. Professor Sam Efromovich is a Fellow of the Institute of Mathematical Statistics and the American Statistical Association.
Book Synopsis A Mixture-based Framework for Nonparametric Density Estimation by : Chew-Seng Chee
Download or read book A Mixture-based Framework for Nonparametric Density Estimation written by Chew-Seng Chee and published by . This book was released on 2011 with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt: The primary goal of this thesis is to provide a mixture-based framework for nonparametric density estimation. This framework advocates the use of a mixture model with a nonparametric mixing distribution to approximate the distribution of the data. The implementation of a mixture-based nonparametric density estimator generally requires the specification of parameters in a mixture model and the choice of the bandwidth parameter. Consequently, a nonparametric methodology consisting of both the estimation and selection steps is described. For the estimation of parameters in mixture models, we employ the minimum disparity estimation framework within which there exist several estimation approaches differing in the way smoothing is incorporated in the disparity objective function. For the selection of the bandwidth parameter, we study some popular methods such as cross-validation and information criteria-based model selection methods. Also, new algorithms are developed for the computation of the mixture-based nonparametric density estimates. A series of studies on mixture-based nonparametric density estimators is presented, ranging from the problems of nonparametric density estimation in general to estimation under constraints. The problem of estimating symmetric densities is firstly investigated, followed by an extension in which the interest lies in estimating finite mixtures of symmetric densities. The third study utilizes the idea of double smoothing in defining the least squares criterion for mixture-based nonparametric density estimation. For these problems, numerical studies whether using both simulated and real data examples suggest that the performance of the mixture-based nonparametric density estimators is generally better than or at least competitive with that of the kernel-based nonparametric density estimators. The last but not least concern is nonparametric estimation of continuous and discrete distributions under shape constraints. Particularly, a new model called the discrete k-monotone is proposed for estimating the number of unknown species. In fact, the discrete k- monotone distribution is a mixture of specific discrete beta distributions. Empirica results indicate that the new model outperforms the commonly used nonparametric Poisson mixture model in the context of species richness estimation. Although there remain issues to be resolved, the promising results from our series of studies make the mixture-based framework a valuable tool for nonparametric density estimation.
Author :Alexandre B. Tsybakov Publisher :Springer Science & Business Media ISBN 13 :0387790527 Total Pages :222 pages Book Rating :4.3/5 (877 download)
Book Synopsis Introduction to Nonparametric Estimation by : Alexandre B. Tsybakov
Download or read book Introduction to Nonparametric Estimation written by Alexandre B. Tsybakov and published by Springer Science & Business Media. This book was released on 2008-10-22 with total page 222 pages. Available in PDF, EPUB and Kindle. Book excerpt: Developed from lecture notes and ready to be used for a course on the graduate level, this concise text aims to introduce the fundamental concepts of nonparametric estimation theory while maintaining the exposition suitable for a first approach in the field.
Book Synopsis Shape-restricted Density Estimation for Financial Data by : Yu Liu
Download or read book Shape-restricted Density Estimation for Financial Data written by Yu Liu and published by . This book was released on 2016 with total page 179 pages. Available in PDF, EPUB and Kindle. Book excerpt: The motivation of the study in this thesis is about how to estimate an asset return distribution in finance that is often skewed, high-peaked and heavy-tailed. To avoid misspecification which is possible for a parametric model, we turn to nonparametric methods to estimating a density function. There are many nonparametric approaches, such as kernel-based and penalty methods, to solving estimation problems, but they may easily fail to satisfy some practically known properties or have difficulty in choosing the value of the bandwidth or tuning parameter. By contrast, one can avoid these issues by imposing certain shape constraints on the density function, that appear very reasonable from a practical point of view. Nonparametric density estimation under shape restrictions offer many advantages, such as having the required shapes, easily described fitted models and possibly a higher estimation efficiency. Specifically, we are interested in estimating a density function that is log-concave, or unimodal with heavy tails. Three main objectives are addressed in this thesis. Firstly, nonparametric maximum likelihood estimation of a log-concave density function is investigated. In particular, a new fast algorithm is proposed and studied for computing the nonparametric maximum likelihood estimate of a log-concave density. Theoretically, the characterization of the nonparametric maximum likelihood estimate is studied and the algorithm is guaranteed to converge to the unique maximum likelihood estimate under log-concavity constraints. Numerical studies show that it outperforms other algorithms that are available in the literature. Tests for log-concavity based on the new algorithm are also developed. Secondly, nonparametric estimation under smoothness and log-concavity shape assumptions is studied. We propose several new smooth estimators based on the maximum likelihood approach by employing piecewise quadratic functions for the log-density function. This leads us to define a log-concave distribution family that allows the second derivative of the log-density to change the direction of monotonicity at most once. Algorithms for these likelihood maximization problems are developed. Numerical studies of simulated and real-world data show that the new smooth estimator has the best performance of all nonparametric estimators studied. We also apply our smooth estimator to the receiver operating characteristic curve estimation, with good results obtained. Finally, we study the problem of estimating a unimodal, highly heavy-tailed distribution, as normally seen in financial data. A novel idea is proposed that it imposes log-concavity on the main body, and log-convexity on the tails. With the corresponding algorithm developed, the new shape-restricted estimator very much dominates the other ones for both simulated and real-world financial data, by providing excellent, nonparametric fits to the data in both the center and tails of the distribution. Bootstrap testing for identifying the function form implied by the new estimator has been developed. Tail performance is further studied in great detail and an application to Value-at-risk estimation is investigated. As a matter of fact, the study provides a very general approach to nonparametric density estimation under shape restrictions. Different pieces of shape restrictions can be combined easily in a seamless way, with fast computing algorithms available. Shape-restricted estimation is able to provide more accurate estimates compared with unconstrained estimates, and the work reported in this thesis lies a promising foundation for many more shape-restricted estimation methods to be developed and applied in the future.
Book Synopsis Nonparametric Estimation and Model Selection Using Constrained Splines in Linear Inversion Problems by : Davide Verotta
Download or read book Nonparametric Estimation and Model Selection Using Constrained Splines in Linear Inversion Problems written by Davide Verotta and published by . This book was released on 1992 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Nonparametric Estimation Subject to Shape Restrictions by : Yazhen Wang
Download or read book Nonparametric Estimation Subject to Shape Restrictions written by Yazhen Wang and published by . This book was released on 1992 with total page 302 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Nonparametric Functional Estimation by : B. L. S. Prakasa Rao
Download or read book Nonparametric Functional Estimation written by B. L. S. Prakasa Rao and published by Academic Press. This book was released on 2014-07-10 with total page 539 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonparametric Functional Estimation is a compendium of papers, written by experts, in the area of nonparametric functional estimation. This book attempts to be exhaustive in nature and is written both for specialists in the area as well as for students of statistics taking courses at the postgraduate level. The main emphasis throughout the book is on the discussion of several methods of estimation and on the study of their large sample properties. Chapters are devoted to topics on estimation of density and related functions, the application of density estimation to classification problems, and the different facets of estimation of distribution functions. Statisticians and students of statistics and engineering will find the text very useful.
Book Synopsis Shape Restricted Inference with Applications to Nonparametric Regression, Smooth Nonparametric Function Estimation, and Density Estimation by : Mary Catherine Meyer
Download or read book Shape Restricted Inference with Applications to Nonparametric Regression, Smooth Nonparametric Function Estimation, and Density Estimation written by Mary Catherine Meyer and published by . This book was released on 1996 with total page 230 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Contributions to Nonparametric Estimation in Reliability by : Lyn Rehder Whitaker
Download or read book Contributions to Nonparametric Estimation in Reliability written by Lyn Rehder Whitaker and published by . This book was released on 1985 with total page 186 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Stochastic Nonparametric Envelopment of Data by : Timo Kuosmanen
Download or read book Stochastic Nonparametric Envelopment of Data written by Timo Kuosmanen and published by . This book was released on 2007 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The field of production frontier estimation is divided between the parametric Stochastic Frontier Analysis (SFA) and the deterministic, nonparametric Data Envelopment Analysis (DEA). This paper explores an amalgam of DEA and SFA that melds a nonparametric frontier with a stochastic composite error. Our model imposes the standard SFA assumptions for the inefficiency and noise terms. The frontier is estimated nonparametrically, imposing monotonicity and convexity as in DEA. For estimation, we propose two alternative methods based on shape constrained nonparametric least squares. The performance of the proposed estimation techniques is examined using Monte Carlo simulations and an illustrative application.
Book Synopsis Methods for Shape-constrained Kernel Density Estimation by : Mark Anthony Wolters
Download or read book Methods for Shape-constrained Kernel Density Estimation written by Mark Anthony Wolters and published by . This book was released on 2012 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonparametric density estimators are used to estimate an unknown probability density while making minimal assumptions about its functional form. Although the low reliance of nonparametric estimators on modelling assumptions is a benefit, their performance will be improved if auxiliary information about the density's shape is incorporated into the estimate. Auxiliary information can take the form of shape constraints, such as unimodality or symmetry, that the estimate must satisfy. Finding the constrained estimate is usually a difficult optimization problem, however, and a consistent framework for finding estimates across a variety of problems is lacking. It is proposed to find shape-constrained density estimates by starting with a pilot estimate obtained by standard methods, and subsequently adjusting its shape until the constraints are satisfied. This strategy is part of a general approach, in which a constrained estimation problem is defined by an estimator, a method of shape adjustment, a constraint, and an objective function. Optimization methods are developed to suit this approach, with a focus on kernel density estimation under a variety of constraints. Two methods of shape adjustment are examined in detail. The first is data sharpening, for which two optimization algorithms are proposed: a greedy algorithm that runs quickly but can handle a limited set of constraints, and a particle swarm algorithm that is suitable for a wider range of problems. The second is the method of adjustment curves, for which it is often possible to use quadratic programming to find optimal estimates. The methods presented here can be used for univariate or higher-dimensional kernel density estimation with shape constraints. They can also be extended to other estimators, in both the density estimation and regression settings. As such they constitute a step toward a truly general optimizer, that can be used on arbitrary combinations of estimator and constraint.
