Showing papers on "Parametric statistics published in 2005"

Nonlinear Time Series: Nonparametric and Parametric Methods

Abstract: Introduction.- Stationary Time Series.- Smoothing in Time Series.- ARMA Modeling and Forecasting.- Parametric Nonlinear Time Series Models.- Nonparametric Models.- Hypothesis Testing.- Continuous Time Models in Finance.- Nonlinear Prediction.

A Parametric approach to the Estimation of Cointegration Vectors in Panel Data

Abstract: In this article, a parametric framework for estimation and inference in cointegrated panel data models is considered that is based on a cointegrated VAR(p) model. A convenient two-step estimator is suggested where, in the first step, all individual specific parameters are estimated, and in the second step, the long-run parameters are estimated from a pooled least-squares regression. The two-step estimator and related test procedures can easily be modified to account for contemporaneously correlated errors, a feature that is often encountered in multi-country studies. Monte Carlo simulations suggest that the two-step estimator and related test procedures outperform semiparametric alternatives such as the fully modified OLS approach, especially if the number of time periods is small.

Learning Gaussian processes from multiple tasks

Abstract: We consider the problem of multi-task learning, that is, learning multiple related functions. Our approach is based on a hierarchical Bayesian framework, that exploits the equivalence between parametric linear models and nonparametric Gaussian processes (GPs). The resulting models can be learned easily via an EM-algorithm. Empirical studies on multi-label text categorization suggest that the presented models allow accurate solutions of these multi-task problems.

Statistical Analysis of Geographic Information with ArcView GIS And ArcGIS

Abstract: PREFACE. ACKNOWLEDGMENTS. 1 INTRODUCTION. 1.1 Why Statistics and Sampling? 1.2 What Are Special about Spatial Data? 1.3 Spatial Data and the Need for Spatial Analysis/ Statistics. 1.4 Fundamentals of Spatial Analysis and Statistics. 1.5 ArcView Notes-Data Model and Examples. PART I: CLASSICAL STATISTICS. 2 DISTRIBUTION DESCRIPTORS: ONE VARIABLE (UNIVARIATE). 2.1 Measures of Central Tendency. 2.2 Measures of Dispersion. 2.3 ArcView Examples. 2.4 Higher Moment Statistics. 2.5 ArcView Examples. 2.6 Application Example. 2.7 Summary. 3 RELATIONSHIP DESCRIPTORS: TWO VARIABLES (BIVARIATE). 3.1 Correlation Analysis. 3.2 Correlation: Nominal Scale. 3.3 Correlation: Ordinal Scale. 3.4 Correlation: Interval /Ratio Scale. 3.5 Trend Analysis. 3.6 ArcView Notes. 3.7 Application Examples. 4 HYPOTHESIS TESTERS. 4.1 Probability Concepts. 4.2 Probability Functions. 4.3 Central Limit Theorem and Confidence Intervals. 4.4 Hypothesis Testing. 4.5 Parametric Test Statistics. 4.6 Difference in Means. 4.7 Difference Between a Mean and a Fixed Value. 4.8 Significance of Pearson's Correlation Coefficient. 4.9 Significance of Regression Parameters. 4.10 Testing Nonparametric Statistics. 4.11 Summary. PART II: SPATIAL STATISTICS. 5 POINT PATTERN DESCRIPTORS. 5.1 The Nature of Point Features. 5.2 Central Tendency of Point Distributions. 5.3 Dispersion and Orientation of Point Distributions. 5.4 ArcView Notes. 5.5 Application Examples. 6 POINT PATTERN ANALYZERS. 6.1 Scale and Extent. 6.2 Quadrat Analysis. 6.3 Ordered Neighbor Analysis. 6.4 K-Function. 6.5 Spatial Autocorrelation of Points. 6.6 Application Examples. 7 LINE PATTERN ANALYZERS. 7.1 The Nature of Linear Features: Vectors and Networks. 7.2 Characteristics and Attributes of Linear Features. 7.3 Directional Statistics. 7.4 Network Analysis. 7.5 Application Examples. 8 POLYGON PATTERN ANALYZERS. 8.1 Introduction. 8.2 Spatial Relationships. 8.3 Spatial Dependency. 8.4 Spatial Weights Matrices. 8.5 Spatial Autocorrelation Statistics and Notations. 8.6 Joint Count Statistics. 8.7 Spatial Autocorrelation Global Statistics. 8.8 Local Spatial Autocorrelation Statistics. 8.9 Moran Scatterplot. 8.10 Bivariate Spatial Autocorrelation. 8.11 Application Examples. 8.12 Summary. APPENDIX: ArcGIS Spatial Statistics Tools. ABOUT THE CD-ROM. INDEX.

Motion Competition: A Variational Approach to Piecewise Parametric Motion Segmentation

Abstract: We present a novel variational approach for segmenting the image plane into a set of regions of parametric motion on the basis of two consecutive frames from an image sequence. Our model is based on a conditional probability for the spatio-temporal image gradient, given a particular velocity model, and on a geometric prior on the estimated motion field favoring motion boundaries of minimal length. Exploiting the Bayesian framework, we derive a cost functional which depends on parametric motion models for each of a set of regions and on the boundary separating these regions. The resulting functional can be interpreted as an extension of the Mumford-Shah functional from intensity segmentation to motion segmentation. In contrast to most alternative approaches, the problems of segmentation and motion estimation are jointly solved by continuous minimization of a single functional. Minimizing this functional with respect to its dynamic variables results in an eigenvalue problem for the motion parameters and in a gradient descent evolution for the motion discontinuity set. We propose two different representations of this motion boundary: an explicit spline-based implementation which can be applied to the motion-based tracking of a single moving object, and an implicit multiphase level set implementation which allows for the segmentation of an arbitrary number of multiply connected moving objects. Numerical results both for simulated ground truth experiments and for real-world sequences demonstrate the capacity of our approach to segment objects based exclusively on their relative motion.

Sampling and reconstruction of signals with finite rate of innovation in the presence of noise

Abstract: Recently, it was shown that it is possible to develop exact sampling schemes for a large class of parametric nonbandlimited signals, namely certain signals of finite rate of innovation. A common feature of such signals is that they have a finite number of degrees of freedom per unit of time and can be reconstructed from a finite number of uniform samples. In order to prove sampling theorems, Vetterli et al. considered the case of deterministic, noiseless signals and developed algebraic methods that lead to perfect reconstruction. However, when noise is present, many of those schemes can become ill-conditioned. In this paper, we revisit the problem of sampling and reconstruction of signals with finite rate of innovation and propose improved, more robust methods that have better numerical conditioning in the presence of noise and yield more accurate reconstruction. We analyze, in detail, a signal made up of a stream of Diracs and develop algorithmic tools that will be used as a basis in all constructions. While some of the techniques have been already encountered in the spectral estimation framework, we further explore preconditioning methods that lead to improved resolution performance in the case when the signal contains closely spaced components. For classes of periodic signals, such as piecewise polynomials and nonuniform splines, we propose novel algebraic approaches that solve the sampling problem in the Laplace domain, after appropriate windowing. Building on the results for periodic signals, we extend our analysis to finite-length signals and develop schemes based on a Gaussian kernel, which avoid the problem of ill-conditioning by proper weighting of the data matrix. Our methods use structured linear systems and robust algorithmic solutions, which we show through simulation results.

Direct reconstruction of kinetic parameter images from dynamic PET data

Abstract: Our goal in this paper is the estimation of kinetic model parameters for each voxel corresponding to a dense three-dimensional (3-D) positron emission tomography (PET) image. Typically, the activity images are first reconstructed from PET sinogram frames at each measurement time, and then the kinetic parameters are estimated by fitting a model to the reconstructed time-activity response of each voxel. However, this "indirect" approach to kinetic parameter estimation tends to reduce signal-to-noise ratio (SNR) because of the requirement that the sinogram data be divided into individual time frames. In 1985, Carson and Lange proposed, but did not implement, a method based on the expectation-maximization (EM) algorithm for direct parametric reconstruction. The approach is "direct" because it estimates the optimal kinetic parameters directly from the sinogram data, without an intermediate reconstruction step. However, direct voxel-wise parametric reconstruction remained a challenge due to the unsolved complexities of inversion and spatial regularization. In this paper, we demonstrate and evaluate a new and efficient method for direct voxel-wise reconstruction of kinetic parameter images using all frames of the PET data. The direct parametric image reconstruction is formulated in a Bayesian framework, and uses the parametric iterative coordinate descent (PICD) algorithm to solve the resulting optimization problem. The PICD algorithm is computationally efficient and is implemented with spatial regularization in the domain of the physiologically relevant parameters. Our experimental simulations of a rat head imaged in a working small animal scanner indicate that direct parametric reconstruction can substantially reduce root-mean-squared error (RMSE) in the estimation of kinetic parameters, as compared to indirect methods, without appreciably increasing computation.

Adaptive neural control for a class of nonlinearly parametric time-delay systems

Abstract: In this paper, an adaptive neural controller for a class of time-delay nonlinear systems with unknown nonlinearities is proposed. Based on a wavelet neural network (WNN) online approximation model, a state feedback adaptive controller is obtained by constructing a novel integral-type Lyapunov-Krasovskii functional, which also efficiently overcomes the controller singularity problem. It is shown that the proposed method guarantees the semiglobal boundedness of all signals in the adaptive closed-loop systems. An example is provided to illustrate the application of the approach.

Geostatistical motion interpolation

Abstract: A common motion interpolation technique for realistic human animation is to blend similar motion samples with weighting functions whose parameters are embedded in an abstract space. Existing methods, however, are insensitive to statistical properties, such as correlations between motions. In addition, they lack the capability to quantitatively evaluate the reliability of synthesized motions. This paper proposes a method that treats motion interpolations as statistical predictions of missing data in an arbitrarily definable parametric space. A practical technique of geostatistics, called universal kriging, is then introduced for statistically estimating the correlations between the dissimilarity of motions and the distance in the parametric space. Our method statistically optimizes interpolation kernels for given parameters at each frame, using a pose distance metric to efficiently analyze the correlation. Motions are accurately predicted for the spatial constraints represented in the parametric space, and they therefore have few undesirable artifacts, if any. This property alleviates the problem of spatial inconsistencies, such as foot-sliding, that are associated with many existing methods. Moreover, numerical estimates for the reliability of predictions enable motions to be adaptively sampled. Since the interpolation kernels are computed with a linear system in real-time, motions can be interactively edited using various spatial controls.

Comparison of the Performance of Nonparametric and Parametric MANOVA Test Statistics when Assumptions Are Violated

Abstract: . Multivariate analysis of variance (MANOVA) is a useful tool for social scientists because it allows for the comparison of response-variable means across multiple groups. MANOVA requires that the observations are independent, the response variables are multivariate normally distributed, and the covariance matrix of the response variables is homogeneous across groups. When the assumptions of normality and homogeneous covariance matrices are not met, past research has shown that the type I error rate of the standard MANOVA test statistics can be inflated while their power can be attenuated. The current study compares the performance of a nonparametric alternative to one of the standard parametric test statistics when these two assumptions are not met. Results show that when the assumption of homogeneous covariance matrices is not met, the nonparametric approach has a lower type I error rate and higher power than the most robust parametric statistic. When the assumption of normality is untenable, th...

Passivity analysis of neural networks with time delay

Abstract: The passivity conditions for delayed neural networks (DNNs) are considered in this paper. We firstly derive the passivity condition for DNNs without uncertainties, and then extend the result to the case with time-varying parametric uncertainties. The proposed approach is based on a Lyapunov-Krasovskii functional construction. The passivity conditions are presented in terms of linear matrix inequalities, which can be easily solved by using the effective interior-point algorithm. Numerical examples are also given to demonstrate the effectiveness of the theoretical results.

Parametric study for an ant algorithm applied to water distribution system optimization

Abstract: Much research has been carried out on the optimization of water distribution systems (WDSs). Within the last decade, the focus has shifted from the use of traditional optimization methods, such as linear and nonlinear programming, to the use of heuristics derived from nature (HDNs), namely, genetic algorithms, simulated annealing and more recently, ant colony optimization (ACO), an optimization algorithm based on the foraging behavior of ants. HDNs have been seen to perform better than more traditional optimization methods and amongst the HDNs applied to WDS optimization, a recent study found ACO to outperform other HDNs for two well-known case studies. One of the major problems that exists with the use of HDNs, particularly ACO, is that their searching behavior and, hence, performance, is governed by a set of user-selected parameters. Consequently, a large calibration phase is required for successful application to new problems. The aim of this paper is to provide a deeper understanding of ACO parameters and to develop parametric guidelines for the application of ACO to WDS optimization. For the adopted ACO algorithm, called AS/sub i-best/ (as it uses an iteration-best pheromone updating scheme), seven parameters are used: two decision policy control parameters /spl alpha/ and /spl beta/, initial pheromone value /spl tau//sub 0/, pheromone persistence factor /spl rho/, number of ants m, pheromone addition factor Q, and the penalty factor (PEN). Deterministic and semi-deterministic expressions for Q and PEN are developed. For the remaining parameters, a parametric study is performed, from which guidelines for appropriate parameter settings are developed. Based on the use of these heuristics, the performance of AS/sub i-best/ was assessed for two case studies from the literature (the New York Tunnels Problem, and the Hanoi Problem) and an additional larger case study (the Doubled New York Tunnels Problem). The results show that AS/sub i-best/ achieves the best performance presented in the literature, in terms of efficiency and solution quality, for the New York Tunnels Problem. Although AS/sub i-best/ does not perform as well as other algorithms from the literature for the Hanoi Problem (a notably difficult problem), it successfully finds the known least cost solution for the larger Doubled New York Tunnels Problem.

Coupled parametric active contours

Abstract: We propose an extension of parametric active contours designed to track nonoccluding objects transiently touching each other, a task where both parametric and single level set-based methods usually fail. Our technique minimizes a cost functional that depends on all contours simultaneously and includes a penalty for contour overlaps. This scheme allows us to take advantage of known constraints on object topology, namely, that objects cannot merge. The coupled contours preserve the identity of previously isolated objects during and after a contact event, thus allowing segmentation and tracking to proceed as desired.

Inference of non-overlapping camera network topology by measuring statistical dependence

Abstract: We present an approach for inferring the topology of a camera network by measuring statistical dependence between observations in different cameras. Two cameras are considered connected if objects seen departing in one camera is seen arriving in the other. This is captured by the degree of statistical dependence between the cameras. The nature of dependence is characterized by the distribution of observation transformations between cameras, such as departure to arrival transition times, and color appearance. We show how to measure statistical dependence when the correspondence between observations in different cameras is unknown. This is accomplished by non-parametric estimates of statistical dependence and Bayesian integration of the unknown correspondence. Our approach generalizes previous work which assumed restricted parametric transition distributions and only implicitly dealt with unknown correspondence. Results are shown on simulated and real data. We also describe a technique for learning the absolute locations of the cameras with Global Positioning System (GPS) side information

Identification of Parametric Variations of Structures Based on Least Squares Estimation and Adaptive Tracking Technique

Abstract: An important objective of health monitoring systems for civil infrastructures is to identify the state of the structure and to detect the damage when it occurs. System identification and damage detection, based on measured vibration data, have received considerable attention recently. Frequently, the damage of a structure may be reflected by a change of some parameters in structural elements, such as a degradation of the stiffness. Hence it is important to develop data analysis techniques that are capable of detecting the parametric changes of structural elements during a severe event, such as the earthquake. In this paper, we propose a new adaptive tracking technique, based on the least-squares estimation approach, to identify the time-varying structural parameters. In particular, the new technique proposed is capable of tracking the abrupt changes of system parameters from which the event and the severity of the structural damage may be detected. The proposed technique is applied to linear structures, including the Phase I ASCE structural health monitoring benchmark building, and a nonlinear elastic structure to demonstrate its performance and advantages. Simulation results demonstrate that the proposed technique is capable of tracking the parametric change of structures due to damages.

Combining information from independent sources through confidence distributions

Abstract: This paper develops new methodology, together with related theories, for combining information from independent studies through confidence distributions. A formal definition of a confidence distribution and its asymptotic counterpart (i.e., asymptotic confidence distribution) are given and illustrated in the context of combining information. Two general combination methods are developed: the first along the lines of combining p-values, with some notable differences in regard to optimality of Bahadur type efficiency; the second by multiplying and normalizing confidence densities. The latter approach is inspired by the common approach of multiplying likelihood functions for combining parametric information. The paper also develops adaptive combining methods, with supporting asymptotic theory which should be of practical interest. The key point of the adaptive development is that the methods attempt to combine only the correct information, downweighting or excluding studies containing little or wrong information about the true parameter of interest. The combination methodologies are illustrated in simulated and real data examples with a variety of applications.

An efficient algorithm for statistical minimization of total power under timing yield constraints

Abstract: Power minimization under variability is formulated as a rigorous statistical robust optimization program with a guarantee of power and timing yields. Both power and timing metrics are treated probabilistically. Power reduction is performed by simultaneous sizing and dual threshold voltage assignment. An extremely fast run-time is achieved by casting the problem as a second-order conic problem and solving it using efficient interior-point optimization methods. When compared to the deterministic optimization, the new algorithm, on average, reduces static power by 31% and total power by 17% without the loss of parametric yield. The run time on a variety of public and industrial benchmarks is 30/spl times/ faster than other known statistical power minimization algorithms.

Choosing a robustness tuning parameter

Abstract: A novel method is proposed for choosing the tuning parameter associated with a family of robust estimators. It consists of minimising estimated mean squared error, an approach that requires pilot estimation of model parameters. The method is explored for the family of minimum distance estimators proposed by [Basu, A., Harris, I.R., Hjort, N.L. and Jones, M.C., 1998, Robust and efficient estimation by minimising a density power divergence. Biometrika, 85, 549–559.] Our preference in that context is for a version of the method using the L 2 distance estimator [Scott, D.W., 2001, Parametric statistical modeling by minimum integrated squared error. Technometrics, 43, 274–285.] as pilot estimator.

Parametric analysis of oscillatory activity as measured with EEG/MEG

Abstract: We assess the suitability of conventional parametric statistics for analyzing oscillatory activity, as measured with electroencephalography/magnetoencephalography (EEG/MEG). The approach we consider is based on narrow-band power time-frequency decompositions of single-trial data. The ensuing power measures have a chi(2)-distribution. The use of the general linear model (GLM) under normal error assumptions is, therefore, difficult to motivate for these data. This is unfortunate because the GLM plays a central role in classical inference and is the standard estimation and inference framework for neuroimaging data. The key contribution of this work is to show that, in many circumstances, one can appeal to the central limit theorem and assume normality for generative models of power. If this is not appropriate, one can transform the data to render the error terms approximately normal. These considerations allow one to analyze induced and evoked oscillations using standard frameworks like statistical parametric mapping. We establish the validity of parametric tests using synthetic and real data and compare its performance to established nonparametric procedures.

On the generation of a regular multi-input multi-output technology using parametric output dis-tance functions

Abstract: Monte-Carlo experimentation is a well-known approach to test the performance of alternative methodologies under different hypothesis. In the frontier analysis framework, whatever parametric or non-parametric methods tested, most experiments have been developed up to now assuming single output multi-input production functions and data generated using a Cobb-Douglas technology. The aim of this paper is to show how reliable multi-output multi-input production data can be generated using a parametric output distance function approach. A flexible translog technology is used for this purpose that satisfies regularity conditions. Two meaningful outcomes of this analysis are the identification of a valid range of parameters values satisfying monotonicity and curvature restrictions and of a rule of thumb to be applied in empirical studies. JEL Classification: C14, C15, C24 Keywords: Output distance function; technical efficiency, Monte-Carlo experiments. Acknowledgments : The authors are grateful to Chris O’Donnell, David Roibas, Luis Orea and participants in the

Estimating Semiparametric ARCH(∞) Models by Kernel Smoothing Methods

Abstract: We investigate a class of semiparametric ARCH(∞) models that includes as a special case the partially nonparametric (PNP) model introduced by Engle and Ng (1993) and which allows for both flexible dynamics and flexible function form with regard to the “news impact” function. We show that the functional part of the model satisfies a type II linear integral equation and give simple conditions under which there is a unique solution. We propose an estimation method that is based on kernel smoothing and profiled likelihood. We establish the distribution theory of the parametric components and the pointwise distribution of the nonparametric component of the model. We also discuss efficiency of both the parametric part and the nonparametric part. We investigate the performance of our procedures on simulated data and on a sample of S&P500 index returns. We find evidence of asymmetric news impact functions, consistent with the parametric analysis.

Bayesian Modelling and Inference on Mixtures of Distributions

Abstract: Publisher Summary Mixture distributions comprise a finite or infinite number of components, possibly of different distributional types, that can describe different features of data. The Bayesian paradigm allows for probability statements to be made directly about the unknown parameters, prior or expert opinion to be included in the analysis, and hierarchical descriptions of both local-scale and global features of the model. This chapter aims to introduce the prior modeling, estimation, and evaluation of mixture distributions in a Bayesian paradigm. The chapter shows that mixture distributions provide a flexible, parametric framework for statistical modeling and analysis. Focus is on the methods rather than advanced examples, in the hope that an understanding of the practical aspects of such modeling can be carried into many disciplines. It also points out the fundamental difficulty in doing inference with such objects, along with a discussion about prior modeling, which is more restrictive than usual, and the constructions of estimators, which also is more involved than the standard posterior mean solution. Finally, this chapter gives some pointers to the related models and problems like mixtures of regressions and hidden Markov models as well as Dirichlet priors.

A parametric scheme for the retrieval of two-dimensional ocean wave spectra from synthetic aperture radar look cross spectra

Abstract: [1] A parametric inversion scheme for the retrieval of two-dimensional (2-D) ocean wave spectra from look cross spectra acquired by spaceborne synthetic aperture radar (SAR) is presented. The scheme uses SAR observations to adjust numerical wave model spectra. The Partition Rescaling and Shift Algorithm (PARSA) is based on a maximum a posteriori approach in which an optimal estimate of a 2-D wave spectrum is calculated given a measured SAR look cross spectrum (SLCS) and additional prior knowledge. The method is based on explicit models for measurement errors as well as on uncertainties in the SAR imaging model and the model wave spectra used as prior information. Parameters of the SAR imaging model are estimated as part of the retrieval. Uncertainties in the prior wave spectrum are expressed in terms of transformation variables, which are defined for each wave system in the spectrum, describing rotations and rescaling of wave numbers and energy as well as changes of directional spreading. Technically, the PARSA wave spectra retrieval is based on the minimization of a cost function. A Levenberg-Marquardt method is used to find a numerical solution. The scheme is tested using both simulated SLCS and ERS-2 SAR data. It is demonstrated that the algorithm makes use of the phase information contained in SLCS, which is of particular importance for multimodal sea states. Statistics are presented for a global data set of 11,000 ERS-2 SAR wave mode SLCS acquired in southern winter 1996.

Statistical modeling of speech signals based on generalized gamma distribution

Abstract: In this letter, we propose a new statistical model, two-sided generalized gamma distribution (G/spl Gamma/D) for an efficient parametric characterization of speech spectra. G/spl Gamma/D forms a generalized class of parametric distributions, including the Gaussian, Laplacian, and Gamma probability density functions (pdfs) as special cases. We also propose a computationally inexpensive online maximum likelihood (ML) parameter estimation algorithm for G/spl Gamma/D. Likelihoods, coefficients of variation (CVs), and Kolmogorov-Smirnov (KS) tests show that G/spl Gamma/D can model the distribution of the real speech signal more accurately than the conventional Gaussian, Laplacian, Gamma, or generalized Gaussian distribution (GGD).

Gate sizing using incremental parameterized statistical timing analysis

Abstract: As technology scales into the sub-90 nm domain, manufacturing variations become an increasingly significant portion of circuit delay. As a result, delays must be modeled as statistical distributions during both analysis and optimization. This paper uses incremental, parametric statistical static timing analysis (SSTA) to perform gate sizing with a required yield target. Both correlated and uncorrelated process parameters are considered by using a first-order linear delay model with fitted process sensitivities. The fitted sensitivities are verified to be accurate with circuit simulations. Statistical information in the form of criticality probabilities are used to actively guide the optimization process which reduces run-time and improves area and performance. The gate sizing results show a significant improvement in worst slack at 99.86% yield over deterministic optimization.

Permutation, Parametric, and Bootstrap Tests of Hypotheses

Abstract: Kenneth Lange’s latest book deftly blends theoretical and practical concepts about optimization theory in the field of statistics. In the foreward, Lange states that the text is intended for graduate students in statistics, although I think that it may also be accessible for upper-level undergraduate students with a rigorous background in pure mathematics. The book can be divided into two sections. The first five chapters of the text contain very little information on practical optimization methods, but rather act as a primer on the mathematical analysis necessary to understand the analytic underpinnings of modern optimization. Although it may not have been the author’s intent, the first five chapters may have significant educational value beyond computing as a primer that could be titled “A Review of Real Analysis for Statisticians.” Lange gives very brief, but clear descriptions of analytical ideas such as convergence, connectedness, and differentiation that would be useful for any first-year statistics doctoral student lacking the proper undergraduate real analysis courses needed for advanced statistics courses; the only necessary topic missing is that of measure theory. This book helps develop the student’s intuition for abstract concepts in analysis much better than many undergraduate analysis texts. Chapter 2 contains an overview of six of the “seven C’s of analysis” (convergence, complete, closed, compact, continuous, and connected). Although the chapter is only 21 pages long, any student who devotes proper attention to them will learn very quickly whether such material constitutes a review of the basic analysis necessary for statistical inference or whether he or she needs to seek outside reference materials for more details on the fundamental theorems and ideas presented (e.g., Bolzano–Weierstrass, intermediate mean value theorem, uniform continuity). Next comes a chapter is devoted to the concepts of differentiation that are essential for proofs in optimization. Because the bulk of the optimization algorithms used in statistics require these tools, the material covered is absolutely essential (although hopefully covered in a previous advanced calculus course). Chapter 4 covers a topic even more specific to optimization, Karush–Kuhn– Tucker (KKT) theory. Although the material in this chapter is extremely useful for describing of the algorithms to come, the steep gradient of mathematical sophistication was a bit jarring; students may quickly find the nature of the course changed after becoming accustomed to the more introductory material in the first three chapters. Chapter 5, on convexity (the seventh “C”), is a jewel of the text. Convexity can be a unifying concept in statistics, yet is rarely given sufficient attention as such. The author returns to the style of the first three chapters and presents clear definitions and examples in the area of convexity. Students who have a firm grasp of the material in this chapter should gain maximum benefit from the presentation of the algorithms in the second half of the book. The second half of the text gives more practical direction in the art of statistical optimization. The usual suspects (e.g., the EM algorithm, Newton’s and quasi-Newton methods) receive their own chapters, although even in the discussion of these standard optimization methods, Lange embeds the methodology in a more general framework. His discussion of the majorization and minimization algorithm allows for a common framework in which to discuss other methods (which have seen significantly greater use in statistics). Chapter 6, on the majorization–minimization (MM) algorithms, brings the first real applications of the material in the first half to statistical optimization problems. The philosophy of the MM algorithm is presented in a straightforward manner, and relevant examples of potential implementations of the algorithm are presented for linear regression and the Bradley–Terry model of ranking. As someone who has taught optimization, I found the examples in the text interesting but wished more applications like those had been provided as exercises for students. Chapters 7 and 8 cover the two most important optimization algorithms in statistics: the EM algorithm and Newton’s method-based maximization. The one unique feature of these chapters (compared to other texts) is the consideration of both algorithms in light of the principles behind the MM algorithm. One can find interesting, but accessible examples throughout both chapters, including, but not limited to, factor analysis, image analysis, and generalized linear models. From a researcher’s perspective, I found the material in Chapters 9–11 the most relevant. Conjugate gradient, convex programming, interior point methods, and duality have not been extensively covered in statistical computing textbooks. Such modern optimization methods have been explored in statistical optimization practice only very recently. I was glad to see them given a fair bit of explanation in the last three chapters. The concept of analyzing convergence of optimization algorithms has been covered only in the basic presentation of new statistical optimization algorithms; this should be useful to someone interested in exploring the limits of current and future statistical optimization algorithms. The material in Chapter 10 facilitates theoretical (rather than simulated) comparison of algorithms in different contexts. Although I found Optimization to be an extremely engaging textbook, I find it difficult to advocate its use as a single graduate text for a statistical computing course. The book does not cover stochastic versions of the EM algorithm nor simulated annealing, two of the more popular optimization algorithms used by statisticians. The exercises are entirely theoretical in nature, which may not be appropriate for most courses in graduate-level statistical computing. The number of graduate programs in statistics that could devote an entire semester-long computing course to the finer (and theoretical) points of maximization algorithms is most likely small. However, the text is ideal for graduate students or researchers beginning research on optimization problems in statistics. There is little doubt that someone who worked through the text as part of a reading course or a specialized graduate seminar would benefit greatly from the author’s perspective, giving him or her a more intricate understanding of why optimization algorithms work, rather than how to implement them.