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INTRODUCTION

Leon Ehrenpreis

in The Universality of the Radon Transform

This introductory chapter presents a detailed summary of the highlights of the book. In particular, it explains in various contexts why the Radon transform leads to a “basis” for all functions, and ... More

This introductory chapter presents a detailed summary of the highlights of the book. In particular, it explains in various contexts why the Radon transform leads to a “basis” for all functions, and the origins of John-like equations.Less

Statistical Alternatives and Supplements to Random Sampling

Patrick Dattalo

in Strategies to Approximate Random Sampling and Assignment

This chapter describes the following alternatives and complements to RS in terms of their assumptions, implementations, strengths, and weaknesses: (1) randomization tests; (2) multiple imputation; ... More

This chapter describes the following alternatives and complements to RS in terms of their assumptions, implementations, strengths, and weaknesses: (1) randomization tests; (2) multiple imputation; and (3) mean-score logistic regression. Randomization tests are statistical alternatives to RS. Multiple imputation is a statistical supplement RS. Mean-score logistic regression is a statistical alternative or supplement to RS.Less

Statistical Inference in Nonparametric Frontier Models: Recent Developments and Perspectives

Léopold Simar Paul W. Wilson

in The Measurement of Productive Efficiency and Productivity Change

This chapter recasts the parametric and statistical approach of Chapter 2, and the nonparametric and deterministic approach of Chapter 3 into a nonparametric and statistical approach. It presents in ... More

This chapter recasts the parametric and statistical approach of Chapter 2, and the nonparametric and deterministic approach of Chapter 3 into a nonparametric and statistical approach. It presents in a unified notation the basic assumptions needed to define the data-generating process (DGP) and shows how the nonparametric estimators [free disposal hull (FDH) and data envelopment analysis (DEA)] can be described easily in this framework. It then discusses bootstrap methods for inference based on DEA and FDH estimates. After that it discusses two ways FDH estimators can be improved, using bias corrections and interpolation. The chapter proposes a way for defining robust nonparametric estimators of the frontier, based on a concept of “partial frontiers” (order-m frontiers or order-α quantile frontiers). The next section surveys the most recent techniques allowing investigation of the effects of these external factors on efficiency. The two approaches are reconciled with each other, and a nonparametric method is shown to be particularly useful even if in the end a parametric model is desired. This mixed “semiparametric” approach seems to outperform the usual parametric approaches based on regression ideas. The last section concludes with a discussion of still-important, open issues and questions for future research.Less

Nonparametric Bayesian Networks *

Katja Ickstadt, Bjöorn Bornkamp, Marco Grzegorczyk, Jakob Wieczorek, Malik R. Sheriff, Hernáan E. Grecco, and Eli Zamir

in Bayesian Statistics 9

A convenient way of modelling complex interactions is by employing graphs or networks which correspond to conditional independence structures in an underlying statistical model. One main class of ... More

A convenient way of modelling complex interactions is by employing graphs or networks which correspond to conditional independence structures in an underlying statistical model. One main class of models in this regard are Bayesian networks, which have the drawback of making parametric assumptions. Bayesian nonparametric mixture models offer a possibility to overcome this limitation, but have hardly been used in combination with networks. This manuscript bridges this gap by introducing nonparametric Bayesian network models. We review (parametric) Bayesian networks, in particular Gaussian Bayesian networks, from a Bayesian perspective as well as nonparametric Bayesian mixture models. Afterwards these two modelling approaches are combined into nonparametric Bayesian networks. The new models are compared both to Gaussian Bayesian networks and to mixture models in a simulation study, where it turns out that the nonparametric network models perform favourably in non‐Gaussian situations. The new models are also applied to an example from systems biology, namely finding modules within the MAPK cascade.Less

Particle Learning for Sequential Bayesian Computation *

Hedibert F. Lopes, Michael S. Johannes, Carlos M. Carvalho, and Nicholas G. Polson

in Bayesian Statistics 9

Particle learning provides a simulation‐based approach to sequential Bayesian computation. To sample from a posterior distribution of interest we use an essential state vector together with a ... More

Particle learning provides a simulation‐based approach to sequential Bayesian computation. To sample from a posterior distribution of interest we use an essential state vector together with a predictive distribution and propagation rule to build a resampling‐sampling framework. Predictive inference and sequential Bayes factors are a direct by‐product. Our approach provides a simple yet powerful framework for the construction of sequential posterior sampling strategies for a variety of commonly used models.Less

11 Nonlinear and nonstationary models

Timo Teräsvirta, Dag Tjøstheim, and W. J. Granger

in Modelling Nonlinear Economic Time Series

Long memory, unit root models and cointegration are important in linear modelling of nonstationary processes, not the least in econometrics. Recently, nonlinear generalizations of these concepts have ... More

Long memory, unit root models and cointegration are important in linear modelling of nonstationary processes, not the least in econometrics. Recently, nonlinear generalizations of these concepts have been attempted. The framework is mathematically demanding, requiring tools that can handle both nonstationarity and nonlinearity. Two such tools are local times and null recurrent Markov chains. These are reviewed in parametric and non‐parametric cases.Less

13 Basic nonparametric estimates

Timo Teräsvirta, Dag Tjøstheim, and W. J. Granger

in Modelling Nonlinear Economic Time Series

There are several books on the topics treated in this chapter. For completeness and ease of reference, in the present chapter a brief summary of some results in this area is presented. Among other ... More

There are several books on the topics treated in this chapter. For completeness and ease of reference, in the present chapter a brief summary of some results in this area is presented. Among other things, kernel estimation, choice of bandwidth and local polynomial estimation are briefly discussed.Less

16 Building nonlinear models

Timo Teräsvirta, Dag Tjøstheim, and W. J. Granger

in Modelling Nonlinear Economic Time Series

The topic of this chapter is nonlinear model building. Building non‐parametric models is considered first, followed by building various types of parametric nonlinear models. The latter include smooth ... More

The topic of this chapter is nonlinear model building. Building non‐parametric models is considered first, followed by building various types of parametric nonlinear models. The latter include smooth transition, switching regression, and artificial neural network models. The three stages of model building: specification, estimation and evaluation, are illustrated by a number of empirical examples involving both economic and non‐economic time series and data sets.Less

Interlude

M. D. Edge

in Statistical Thinking from Scratch: A Primer for Scientists

This chapter marks a turning point. The preceding two chapters considered probability theory, which describes the kinds of data that result from specified processes. The remainder of the book ... More

This chapter marks a turning point. The preceding two chapters considered probability theory, which describes the kinds of data that result from specified processes. The remainder of the book consider statistical estimation and inference, which starts with data and attempts to make conclusions about the process that produced them. First, general concepts in statistical estimation and inference are discussed, and then simple linear regression from nonparametric/semiparametric, parametric frequentist, and Bayesian perspectives.Less

A Primer on Nonparametric Methods and Their Application to Research in Human Capital and Economic Growth

in Human Capital and Economic Growth

This chapter outlines basic nonparametric techniques, looking at their specific applications in testing the human capital–growth relationship. The second section provides an overview of nonparametric ... More

This chapter outlines basic nonparametric techniques, looking at their specific applications in testing the human capital–growth relationship. The second section provides an overview of nonparametric econometric methods. The third section discusses a commonly used semiparametric model, the semiparametric partially linear (PLR) model. The fourth section describes the modeling of economic growth using the PLR model. It notes that a semiparametric PLR specification of the cross-country growth regression function is a particularly useful way of studying the contribution of human capital to economic growth from a nonlinear perspective.Less

Graph Kernels by Spectral Transforms

Zhu Xiaojin, Kandola Jaz, Lafferty John, and Ghahramani Zoubin

in Semi-Supervised Learning

This chapter develops an approach to searching over a nonparametric family of spectral transforms by using convex optimization to maximize kernel alignment to the labeled data. Order constraints are ... More

This chapter develops an approach to searching over a nonparametric family of spectral transforms by using convex optimization to maximize kernel alignment to the labeled data. Order constraints are imposed to encode a preference for smoothness with respect to the graph structure. This results in a flexible family of kernels that is more data-driven than the standard parametric spectral transforms. This approach relies on a quadratically constrained quadratic program (QCQP) and is computationally practical for large data sets. Many graph-based semi-supervised learning methods can be viewed as imposing smoothness conditions on the target function with respect to a graph representing the data points to be labeled. The smoothness properties of the functions are encoded in terms of Mercer kernels over the graph. The central quantity in such regularization is the spectral decomposition of the graph Laplacian, a matrix derived from the graph’s edge weights.Less

The Lifetime Risk of Nursing Home Use

Michael D. Hurd, Pierre-Carl Michaud, and Susann Rohwedder

in Discoveries in the Economics of Aging

This paper estimates the lifetime risk and distribution of stays in nursing homes using 10 waves of data from the Health and Retirement Study covering the population over the age of 50. Using both ... More

This paper estimates the lifetime risk and distribution of stays in nursing homes using 10 waves of data from the Health and Retirement Study covering the population over the age of 50. Using both nonparametric and parametric approaches which account for censoring, we estimate that a 50 year old has a 53% to 59% chance of ever entering a nursing home before he dies and that, conditional on any stay, the average duration is just over a year. We show that stays at the end of life which are typically not captured in core interviews are very important for assessing lifetime exposure. The HRS performs exit interviews with proxies for those who died. Excluding exit interviews yields lifetime risk under 40%. Being female, white and a nonsmoker are associated with higher lifetime risk due to lower (competing) mortality risk and higher nursing home risk at older ages.Less

Geometric weight priors and their applications

Ramsés H. Mena

in Bayesian Theory and Applications

This chapter discusses random probability measures (r.p.m.s) that result in robust choice of nonparametric priors due to their simpler weight structure. The key idea is that having simpler weights ... More

This chapter discusses random probability measures (r.p.m.s) that result in robust choice of nonparametric priors due to their simpler weight structure. The key idea is that having simpler weights results in a more efficient use of the infinite collection of locations to assign the required mass to a particular set B ∈ Χ. Having simpler weights also results in easier ways to estimate models and extend them to non-exchangeable contexts.Less

Revisiting Bayesian curve fitting using multivariate normal mixtures ∗

Stephen G Walker and George Karabatsos

in Bayesian Theory and Applications

This chapter develops a Bayesian nonparametric regression model which relies on a standard Bayesian nonparametric form for the joint distribution of both the dependent and independent variables. The ... More

This chapter develops a Bayesian nonparametric regression model which relies on a standard Bayesian nonparametric form for the joint distribution of both the dependent and independent variables. The regression model then is available as a conditional density which can only be written as a ratio of two infinite-dimensional mixture models. The chapter is organized as follows. Section 15.2 describes the regression model and the methods for sampling the posterior distribution of the model. To obtain full posterior inference of the model, a reversible-jump sampling algorithm is used to deal with the uncomputable normalizing constant. Section 15.3 illustrates the model using data analysis.Less

Bayesian regression structure discovery

Hugh A. Chipman, Edward I George, and Robert E. McCulloch

in Bayesian Theory and Applications

This chapter describes two different Bayesian approaches that illustrate the vast potential of Bayesian methods to extract information hidden in high-dimensional data. The first is based on the ... More

This chapter describes two different Bayesian approaches that illustrate the vast potential of Bayesian methods to extract information hidden in high-dimensional data. The first is based on the classical parametric form of the normal linear model, while the second is based on an approach called BART (Bayesian Additive Regression Trees). It shows that although the overall BART sum-of-trees model is complex, the simple structure of the individual tree components enables us to uncover structure with inferential posterior summaries. In particular, it is shown how BART provides a novel approach to model-free variable selection, the search for interesting variables, and model-free interaction detection and the search for interesting pairs of variables.Less

Surviving fully Bayesian nonparametric regression models

Timothy E. Hanson and Alejandro Jara

in Bayesian Theory and Applications

This chapter compares two Bayesian nonparametric models that generalize the accelerated failure time model, based on recent work on probability models for predictor-dependent probability ... More

This chapter compares two Bayesian nonparametric models that generalize the accelerated failure time model, based on recent work on probability models for predictor-dependent probability distributions. It begins by reviewing commonly used semiparametric survival models. It then discusses the Bayesian nonparametric priors used in the generalizations of the accelerated failure time (AFT) model. Next, the two generalizations of the accelerated failure time model are introduced and compared by means of real-life data analyses. The models correspond to generalizations of AFT models based on dependent extensions of the Dirichlet process (DP) and Polya tree (PT) priors. Advantages of the induced survival regression models include ease of interpretability and computational tractability.Less

Flexible Bayesian modelling for clustered categorical responses in developmental toxicology

Kottas Athanasios and Fronczyk Kassandra

in Bayesian Theory and Applications

Developmental toxicity studies investigate birth defects caused by toxic chemicals. This chapter develops a Bayesian nonparametric modelling approach for risk assessment in developmental toxicity ... More

Developmental toxicity studies investigate birth defects caused by toxic chemicals. This chapter develops a Bayesian nonparametric modelling approach for risk assessment in developmental toxicity studies. The model is built from a mixture with a product Binomial kernel, to capture the nested structure of the responses, and a dependent Dirichlet process (DDP) prior for the dose-dependent mixing distributions. The resulting nonparametric DDP mixture model provides rich inference for the response distributions as well as for the dose-response curves. Data from a toxicity experiment involving a plasticizing agent were used to illustrate the scientifically relevant features of the DDP mixture model with regard to estimation of different dose-response relationships for different endpoints, including non-monotonic dose-response curves.Less