| Description |
1 online resource (xxiii, 666 pages) : illustrations |
| Series |
Frontiers in Probability and the Statistical Sciences Series |
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Frontiers in probability and the statistical sciences.
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| Contents |
Intro -- Preface -- Introduction -- Contents -- 1 Poisson Processes -- 1.1 Inhomogeneous Poisson Processes -- 1.1.1 Poisson Processes -- 1.1.2 Stochastic Integral -- 1.1.3 Likelihood Ratio -- 1.2 Limit Theorems -- 1.2.1 Law of Large Numbers -- 1.2.2 Central Limit Theorem -- 1.2.3 Weak Convergence in calC and calD -- 1.2.4 Asymptotic Expansions -- 1.3 Exercises -- 1.4 Notes -- 2 Parameter Estimation -- 2.1 On Efficiency of Estimators -- 2.1.1 Cramér-Rao and Van Trees Lower Bounds -- 2.1.2 Hajek-Le Cam's Bound -- 2.1.3 General Lower Bounds -- 2.2 Regular Case |
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2.2.1 Method of Moments Estimators -- 2.2.2 Minimum Distance Estimators -- 2.2.3 Maximum Likelihood and Bayesian Estimators -- 2.2.4 Multi-step MLE -- 2.2.5 On Non-asymptotic Expansions -- 2.3 Non Regular Cases -- 2.3.1 Non-smooth Intensities -- 2.3.2 Null Fisher Information -- 2.3.3 Discontinuous Fisher Information -- 2.3.4 Border of a Parametric Set -- 2.3.5 Multiple True Models -- 2.4 Exercises -- 2.5 Notes -- 3 Nonparametric Estimation -- 3.1 Mean Function Estimation -- 3.1.1 Empirical Mean Function -- 3.1.2 First Order Optimality -- 3.1.3 Second Order Optimality |
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3.2 Intensity Function Estimation -- 3.2.1 Kernel-Type Estimator -- 3.2.2 Rate Optimality -- 3.2.3 Pinsker's Optimality -- 3.2.4 Estimation of the Derivatives -- 3.3 Estimation of Functionals -- 3.3.1 Linear Functionals -- 3.3.2 Nonlinear Functionals -- 3.4 Exercises -- 3.5 Notes -- 4 Hypothesis Testing -- 4.1 Notations and Definitions -- 4.2 Two Simple Hypotheses -- 4.2.1 The Most Powerful Test -- 4.2.2 Asymptotics of the Most Powerful Test -- 4.3 Parametric Alternatives -- 4.3.1 Regular Case -- 4.3.2 Non-regular Cases -- 4.4 Goodness-of-Fit Tests |
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4.4.1 Cramér-Von Mises and Kolmogorov-Smirnov Type Tests -- 4.4.2 Close Alternatives -- 4.4.3 Parametric Null Hypothesis -- 4.5 Two Related Models -- 4.5.1 Poisson Type Processes -- 4.5.2 Poisson Versus Hawkes -- 4.5.3 Poisson Versus Stress-Release -- 4.6 Exercises -- 4.7 Notes -- 5 Applications -- 5.1 Misspecification -- 5.1.1 ̀̀Consistency'' -- 5.1.2 Smooth Intensity Functions -- 5.1.3 Cusp-Type Singularity -- 5.1.4 Change-Point Problem -- 5.1.5 Misspecifications in Regularity -- 5.2 Phase and Frequency Modulations -- 5.2.1 Phase Modulation -- 5.2.2 Frequency Modulation |
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5.2.3 Choosing the Model and the Estimator -- 5.3 Poisson Source Identification -- 5.3.1 Models of Observations -- 5.3.2 MLE and BE -- 5.3.3 LSE -- 5.3.4 Two Sources -- 5.4 Exercises -- 5.5 Notes -- 6 Likelihood Ratio and Properties of MLE and BE -- 6.1 Parameter Estimation. General Results -- 6.1.1 MLE and BE. Method of Study -- 6.1.2 Two Theorems on Asymptotic Behaviour of Estimators -- 6.2 Smooth and Cusp Cases. MLE -- 6.2.1 Smooth Case -- 6.2.2 Cusp Case -- 6.3 Change-Point Case. MLE -- 6.4 Smooth, Cusp and Change-Point Cases. BE -- 6.4.1 Smooth Case -- 6.4.2 Cusp Case -- 6.4.3 Change-Point Case |
| Summary |
This book covers an extensive class of models involving inhomogeneous Poisson processes and deals with their identification, i.e. the solution of certain estimation or hypothesis testing problems based on the given dataset. These processes are mathematically easy-to-handle and appear in numerous disciplines, including astronomy, biology, ecology, geology, seismology, medicine, physics, statistical mechanics, economics, image processing, forestry, telecommunications, insurance and finance, reliability, queuing theory, wireless networks, and localisation of sources. Beginning with the definitions and properties of some fundamental notions (stochastic integral, likelihood ratio, limit theorems, etc.), the book goes on to analyse a wide class of estimators for regular and singular statistical models. Special attention is paid to problems of change-point type, and in particular cusp-type change-point models, then the focus turns to the asymptotically efficient nonparametric estimation of the mean function, the intensity function, and of some functionals. Traditional hypothesis testing, including some goodness-of-fit tests, is also discussed. The theory is then applied to three classes of problems: misspecification in regularity (MiR),corresponding to situations where the chosen change-point model and that of the real data have different regularity; optical communication with phase and frequency modulation of periodic intensity functions; and localization of a radioactive (Poisson) source on the plane using K detectors. Each chapter concludes with a series of problems, and state-of-the-art references are provided, making the book invaluable to researchers and students working in areas which actively use inhomogeneous Poisson processes |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Description based on online resource; title from digital title page (viewed on October 18, 2023) |
| Subject |
Poisson processes.
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Poisson processes.
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| Form |
Electronic book
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| ISBN |
9783031370540 |
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3031370546 |
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