Provides a basis for further research in mathematical statistics
Clarifies the structures of higher order asymptotics and non-regular estimation
Can serve as a research source and a fundamental tool for practitioners
This book is a reconstruction of Masafumi Akahira’s works on statistical estimation and its related fields, especially higher order asymptotics and non-regular cases with consideration based on information amounts. There have been few books on higher order asymptotics and non-regular cases, but the book helps the reader to understand the meaning and implications of the hierarchical structure of higher order asymptotics and gives some insights into the structure of non-regular estimation.
After the results on the second and third order asymptotic efficiency in the volume entitled “Joint Statistical Papers of Akahira and Takeuchi” are summarized, the positive resolution of the conjecture “third order efficiency implies fourth order efficiency” of J. K. Ghosh is described, from which the fourth order asymptotic efficiency of the bias-adjusted maximum likelihood estimator and the bias-adjusted generalized Bayes estimator is shown. In non-regular situations, the (maximum) order of consistency and the (second order) asymptotic sufficiency are discussed including the view of loss of information, and in regular cases, the asymptotic deficiency of asymptotically efficient estimators is stated. For a truncated family of distributions, the influence of a nuisance parameter on the estimation of the interest parameter is investigated through maximum likelihood estimators. Also discussed is the higher order sequential estimation with the Bhattacharyya type bound, including the presence of a nuisance parameter.
In interval estimation, a systematic method of the construction of a confidence interval for the difference between means is discussed including the Behrens–Fisher type problem, and also ordinary, Bayesian, likelihood ratio and combined Bayesian–frequentist type confidence intervals for a positive parameter are provided and compared. Further, the higher order approximations to percentage points of the non-central t-distribution and the distribution of a non-central t-statistic without the normality assumptions are given. Finally, the large deviation efficiency and large deviation approximations are discussed up to the higher order.
Masafumi Akahira is Professor Emeritus at the University of Tsukuba. He has served as Vice President of the University of Tsukuba and President of the Japan Statistical Society.
Conference proceedings
Apr 2026
Includes the papers presented at the 7th World Congress on the Square of Opposition in Leuven, in September 2022
Ranges over an array of topics, like logic, mathematics, philosophy, theology, and art
Explores different applications of the square of opposition
Part of the book series: Studies in Universal Logic (SUL)
This proceedings volume stems from the 7th World Congress on the Square of Opposition, which took place in Leuven, Belgium, in September 2022, after the previous editions in Montreux, Switzerland in 2007, Corté, Corsica, in 2010, Beirut, Lebanon in 2012, Vatican City in 2014, Easter Island in 2016, and Crete in 2018. This interdisciplinary event gathered logicians, philosophers, mathematicians, semioticians, theologians, cognitivists, artists, and computer scientists.
The theory of the square of opposition was developed based on a logical structure coming from Aristotelian logic, a square, which was expanded into more complex geometrical objects: hexagons, octagons, polyhedra, and even four-dimensional objects. It has been continuously studied for two thousand years and it can also be found in works by Gottlob Frege. Such a logical construction can be applied to many fields, ranging from metalogic to highway code, through economics, music, physics, color theory, and theology. This volume contains new advances on the different aspects of this theory: its history, philosophy, application, and mathematical shapes.
The articles in this volume will be of interest to researchers and students in logic, mathematics, and philosophy alike.
Book
Mar 2026
offers a comprehensive compilation of an interesting field in mathematics
covers the theory of bicontinuous semigroups of operators
provides an overview to anyone starting their research in bicontinuous semigroups
Part of the book series: Frontiers in Mathematics (FM)
This monograph serves two primary purposes. First, it provides a comprehensive treatment of bi-continuous semigroup theory and its applications, ranging from network theory to control theory. The exposition proceeds systematically from fundamental theory through approximation methods, extrapolation techniques as well as perturbation theory, concluding with applications to networks, mean ergodicity and control systems. Each chapter includes illustrative examples to reinforce the theoretical concepts.
Second, this work aims at supporting emerging researchers and doctoral students by offering a structured foundation for further investigation in operator semigroups. By synthesizing current knowledge, it provides a reference point for new research projects in this domain.
Serves as a solid modern classical text for a course in measure theory
Accessible to a wide audience of students from various disciplines
Text is self-contained and requires only modest prerequisites
Part of the book series: Universitext (UTX)
Classical in its approach, this textbook is thoughtfully designed and composed in two parts. Part I is meant for a one-semester beginning graduate course in measure theory, proposing an “abstract” approach to measure and integration, where the classical concrete cases of Lebesgue measure and Lebesgue integral are presented as an important particular case of general theory. Part I may be also accessible to advanced undergraduates who fulfill the prerequisites which include an introductory course in analysis, linear algebra (Chapter 5 only), and elementary set theory. Part II of the text is more advanced and is addressed to a more experienced reader. The material is designed to cover another one-semester graduate course subsequent to a first course, dealing with measure and integration in topological spaces. With modest prerequisites, this text is intended to meet the needs of a contemporary course in measure theory for mathematics students and is also accessible to a wider student audience, namely those in statistics, economics, engineering, and physics.
The final section of each chapter in Part I presents problems that are integral to each chapter, the majority of which consist of auxiliary results, extensions of the theory, examples, and counterexamples. Problems which are highly theoretical have accompanying hints. The last section of each chapter of Part II consists of Additional Propositions containing auxiliary and complementary results. The entire book contains collections of suggested readings at the end of each chapter in order to highlight alternate approaches, proofs, and routes toward additional results. This second edition adds a new discussion on probability measures, some of which are scattered among proposed problems in Part I and all of them summarized in the Appendix to Part I. Chapters on decomposition of measures and representation theorems include substantially more material. A comprehensive discussion on the Cantor–Lebesque measure can be found in problems 7.15 and 7.16. Rajchman measures have been considered in Problems 7.17 and 7.18. There is a new subsection on Borel regular measures on topological spaces in Section 12.4.
Corresponds to discrete data for a small sample size
Treats improved transformation based on the theory of asymptotic expansion
Requires only a small amount of computation of statistics, unlike exact and Monte Carlo methods
Part of the book series: SpringerBriefs in Statistics (BRIEFSSTATIST)
Part of the book sub series: JSS Research Series in Statistics (JSSRES)
This book provides a guide for improving test statistics that are based on phi-divergence for discrete models, which include various kinds of independence models of contingency tables as well as generalized linear models of binary data. The improvements are based on the theory of asymptotic expansion and lead to correct conclusions of a test even when sample sizes are not large. Without such an improvement, there is a risk that the results of a test will lead to the opposite conclusion, as a limiting distribution is used for an approximated distribution of test statistics.
Mainly, for the phi-divergence family of statistics that include Pearson’s chi-square statistic, the log-likelihood ratio statistic, and the power divergence family of statistics as a special case, the book derives the improvement of statistics as transformed statistics. This accomplishment is achieved by using the expression of approximation of the distribution of original phi-divergence statistics based on Edgeworth expansion. For an independence model of a contingency table, a complete independence model, an independence model among a group of factors, and a conditional independence model are considered. The test statistics of a contingency table for a log-linear model are also presented for consideration. Additionally, the selection of statistics for which the distribution is close to the limiting distribution is discussed using the evaluation of second-order correction of moments.
Dedicated to undergrads and math lovers without cutting corners in the argumentation
Provides gentle answers to the typical question why normal subgroups and ideals matter
Provides lots of figures and tables, which visualize various abstract concepts
Part of the book series: Springer Asia Pacific Mathematics Series (SAPACM, volume 10)
This book is designed as an undergraduate textbook for students in science and engineering, rather than for mathematics majors, yet it maintains full mathematical rigor. It covers groups, rings, modules over rings, finite fields, polynomial rings over finite fields, and error-correcting codes. Even in mathematics departments, undergraduates often wonder why concepts like normal subgroups and ideals matter, and standard textbooks may not provide satisfying answers. This book addresses such questions with both intuition and precision. For example: (1) A normal subgroup is the kernel of a group homomorphism and gives rise to a factor group; a non-normal subgroup does neither. (2) An ideal is a special additive subgroup that serves as the kernel of a ring homomorphism and yields a factor ring; a non-ideal additive subgroup does not. The reader will appreciate the elegant parallelism between these ideas. Key features include:
A prerequisite chapter that subtly introduces module theory through an elementary presentation of the Euclidean algorithm, accessible even to high school students.
Recurring use of orbits and clusters, with intuitive illustrations, to clarify the operational meaning of normal subgroups and ideals via homomorphisms.
Emphasis on proof design patterns, inspired by fields like architecture and software engineering.
Extensive use of diagrams to support conceptual understanding. Readers are encouraged to draw, compute, design reasoning flows, and then write proofs.
Complete answers to quizzes and exercises are provided, allowing readers to check their understanding after thoughtful attempts.
Presents all material required to understand the solution of the famous Kato conjecture
Includes lecture videos presenting the proofs to theorems and providing further insight
Contains original exercises and comprehensive solutions enriching each lecture
Part of the book series: Operator Theory: Advances and Applications (OT, volume 313)
Part of the book sub series: Advances in Partial Differential Equations (APDE)
The study of the Laplacian through the Fourier transform lies at the center of classical harmonic analysis. It is Plancherel’s theorem that intimately links square-integrable functions with the theory of weak derivatives and a symbolic calculus for the Laplacian. Examples include Littlewood–Paley inequalities, Riesz transform estimates and Calderón–Zygmund extrapolation. Over the last decades, the quest to generalize these properties to elliptic operators L in divergence form with bounded measurable coefficients has triggered the development of new techniques that led to a surge of spectacular results in elliptic and parabolic PDE theory.
Assuming only undergraduate knowledge in analysis and some background on Hilbert spaces and the Fourier transform, the authors develop the cornerstones of this ‘L-adapted Fourier analysis’ over 14 consecutive lectures. As they delve deeper into the topic, readers make first encounters with maximal functions, Carleson measures and a T(b) theorem. The lectures culminate in a self-contained presentation of the solution to the Kato conjecture, a challenging problem that resisted solution for 40 years until it was finally solved in 2001.
This book can serve as a fully developed curriculum for a first graduate course in harmonic analysis and PDEs. Based on the 27th Internet Seminar on Evolution Equations, organized by the authors in the 2023/24 academic year, each lecture is enriched with original exercises, detailed solutions, and video presentations guiding through each theorem’s proof and offering additional insights.
Introduces new tools for treating elliptic boundary value problems for systems of PDEs in rough domains
Develops a Calderón-Zygmund theory for singular integral operators acting on Herz-type spaces
Demonstrates the use of boundary layer potential methods to establish well-posedness results for boundary value problems
Part of the book series: Progress in Mathematics (PM, volume 362)
This monograph presents state-of-the-art results at the intersection of Harmonic Analysis, Functional Analysis, Geometric Measure Theory, and Partial Differential Equations, providing tools for treating elliptic boundary value problems for systems of PDE’s in rough domains. Largely self-contained, it develops a comprehensive Calderón-Zygmund theory for singular integral operators on many Herz-type spaces, and their associated Hardy and Sobolev spaces, in the optimal geometric-measure theoretic setting of uniformly rectifiable sets. The present work highlights the effectiveness of boundary layer potential methods as a means of establishing well-posedness results for a wide family of boundary value problems, including Dirichlet, Neumann, Regularity, and Transmission Problems. Graduate students, researchers, and professional mathematicians interested in harmonic analysis and boundary problems will find this monograph a valuable resource in the field.
Conference proceedings
Mar 2026
Collect in a unique book papers from outstanding number theorists working in closely related areas
Present in a unique book the most recent developments in a quickly evolving area of modern number theory
Provide an introduction to some of the research themes connected with the work of M. Bertolini
Part of the book series: Springer Proceedings in Mathematics & Statistics (PROMS, volume 527)
This book arises from the conference “Elliptic Curves and Modular Forms in Arithmetic Geometry, Celebrating Massimo Bertolini’s 60th birthday” held in Milano in September 2022. Massimo Bertolini is one of the most influential number theorists of the last 30 years, whose results and ideas have been a source of inspiration for many mathematicians working in the fascinating area of the Birch and Swinnerton-Dyer conjecture, a Millennium Problem of the Clay Institute. The beauty of the subject, combined with the deep mathematics involved, attracts some of the most brilliant mathematicians in all the world. The book of Massimo Bertolini opened the way to study these problems, using several different techniques, especially of p-adic nature, and is recognized as a leading mathematician in this area. Because of the special position of Massimo in this area of number theory, many influential mathematicians attended the conference in Milano and the Summer School in Essen in his honor on the occasion of his 60th birthday.