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E-raamat: Introduction to Statistical Limit Theory

  • Formaat: 645 pages
  • Ilmumisaeg: 07-Jan-2011
  • Kirjastus: Chapman & Hall/CRC
  • Keel: eng
  • ISBN-13: 9781439884577
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Introduction to Statistical Limit TheorySuurem pilt
  • Formaat: 645 pages
  • Ilmumisaeg: 07-Jan-2011
  • Kirjastus: Chapman & Hall/CRC
  • Keel: eng
  • ISBN-13: 9781439884577
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"Helping students develop a good understanding of asymptotic theory, Introduction to Statistical Limit Theory provides a thorough yet accessible treatment of common modes of convergence and their related tools used in statistics. It also discusses how the results can be applied to several common areas in the field.The author explains as much of the background material as possible and offers a comprehensive account of the modes of convergence of random variables, distributions, and moments, establishing afirm foundation for the applications that appear later in the book. The text includes detailed proofs that follow a logical progression of the central inferences of each result. It also presents in-depth explanations of the results and identifies important tools and techniques. Through numerous illustrative examples, the book shows how asymptotic theory offers deep insight into statistical problems, such as confidence intervals, hypothesis tests, and estimation. With an array of exercises and experimentsin each chapter, this classroom-tested book gives students the mathematical foundation needed to understand asymptotic theory. It covers the necessary introductory material as well as modern statistical applications, exploring how the underlying mathematical and statistical theories work together"--

"Those moving on to advanced statistics typically lack the mathematical foundation that allows them to make full use of statistical limit theory. This accessible resource reviews approximation theory and limit theory for sequences of functions and basic notions of functional analysis. It provides detailed arguments that show how underlying mathematical and statistical theory work together. Among its unique qualities, the text covers expansion theory, which is becoming increasingly important in modern applications. It also discusses bootstrap, kernel smoothing, and Markov chain Monte Carlo and includes a wide array of examples and problems from the fundamental to very advanced"--

Provided by publisher.

Arvustused

This book is clearly aimed at the student learning the material. The prose is clear and graphs are used well to illustrate concepts. Students will really like the large number of worked concrete examples. ... Each chapter has exercises and R projects that would be very helpful to solidify the understanding of the material in the chapter. These and the worked examples would make this an excellent choice for the student attempting to learn the material through self-study ... -Peter Rabinovitch, MAA Reviews, June 2011

Preface xvii
1 Sequences of Real Numbers and Functions
1(52)
1.1 Introduction
1(1)
1.2 Sequences of Real Numbers
1(11)
1.3 Sequences of Real Functions
12(6)
1.4 The Taylor Expansion
18(10)
1.5 Asymptotic Expansions
28(11)
1.6 Inversion of Asymptotic Expansions
39(3)
1.7 Exercises and Experiments
42(11)
1.7.1 Exercises
42(8)
1.7.2 Experiments
50(3)
2 Random Variables and Characteristic Functions
53(48)
2.1 Introduction
53(1)
2.2 Probability Measures and Random Variables
53(6)
2.3 Some Important Inequalities
59(6)
2.4 Some Limit Theory for Events
65(9)
2.5 Generating and Characteristic Functions
74(17)
2.6 Exercises and Experiments
91(10)
2.6.1 Exercises
91(6)
2.6.2 Experiments
97(4)
3 Convergence of Random Variables
101(58)
3.1 Introduction
101(1)
3.2 Convergence in Probability
102(5)
3.3 Stronger Modes of Convergence
107(10)
3.4 Convergence of Random Vectors
117(4)
3.5 Continuous Mapping Theorems
121(3)
3.6 Laws of Large Numbers
124(11)
3.7 The Glivenko-Cantelli Theorem
135(5)
3.8 Sample Moments
140(7)
3.9 Sample Quantiles
147(5)
3.10 Exercises and Experiments
152(7)
3.10.1 Exercises
152(5)
3.10.2 Experiments
157(2)
4 Convergence of Distributions
159(70)
4.1 Introduction
159(1)
4.2 Weak Convergence of Random Variables
159(23)
4.3 Weak Convergence of Random Vectors
182(13)
4.4 The Central Limit Theorem
195(6)
4.5 The Accuracy of the Normal Approximation
201(13)
4.6 The Sample Moments
214(1)
4.7 The Sample Quantiles
215(6)
4.8 Exercises and Experiments
221(8)
4.8.1 Exercises
221(6)
4.8.2 Experiments
227(2)
5 Convergence of Moments
229(26)
5.1 Convergence in rth Mean
229(8)
5.2 Uniform Integrability
237(6)
5.3 Convergence of Moments
243(5)
5.4 Exercises and Experiments
248(7)
5.4.1 Exercises
248(4)
5.4.2 Experiments
252(3)
6 Central Limit Theorems
255(28)
6.1 Introduction
255(1)
6.2 Non-Identically Distributed Random Variables
255(8)
6.3 Triangular Arrays
263(2)
6.4 Transformed Random Variables
265(13)
6.5 Exercises and Experiments
278(5)
6.5.1 Exercises
278(3)
6.5.2 Experiments
281(2)
7 Asymptotic Expansions for Distributions
283(56)
7.1 Approximating a Distribution
283(1)
7.2 Edgeworth Expansions
284(20)
7.3 The Cornish-Fisher Expansion
304(7)
7.4 The Smooth Function Model
311(3)
7.5 General Edgeworth and Cornish-Fisher Expansions
314(5)
7.6 Studentized Statistics
319(5)
7.7 Saddlepoint Expansions
324(6)
7.8 Exercises and Experiments
330(9)
7.8.1 Exercises
330(5)
7.8.2 Experiments
335(4)
8 Asymptotic Expansions for Random Variables
339(20)
8.1 Approximating Random Variables
339(1)
8.2 Stochastic Order Notation
340(8)
8.3 The Delta Method
348(2)
8.4 The Sample Moments
350(4)
8.5 Exercises and Experiments
354(5)
8.5.1 Exercises
354(1)
8.5.2 Experiments
355(4)
9 Differentiable Statistical Functionals
359(24)
9.1 Introduction
359(1)
9.2 Functional Parameters and Statistics
359(4)
9.3 Differentiation of Statistical Functionals
363(6)
9.4 Expansion Theory for Statistical Functionals
369(6)
9.5 Asymptotic Distribution
375(3)
9.6 Exercises and Experiments
378(5)
9.6.1 Exercises
378(3)
9.6.2 Experiments
381(2)
10 Parametric Inference
383(92)
10.1 Introduction
383(1)
10.2 Point Estimation
383(31)
10.3 Confidence Intervals
414(10)
10.4 Statistical Hypothesis Tests
424(14)
10.5 Observed Confidence Levels
438(9)
10.6 Bayesian Estimation
447(12)
10.7 Exercises and Experiments
459(16)
10.7.1 Exercises
459(11)
10.7.2 Experiments
470(5)
11 Nonparametric Inference
475(90)
11.1 Introduction
475(3)
11.2 Unbiased Estimation and U-Statistics
478(17)
11.3 Linear Rank Statistics
495(19)
11.4 Pitman Asymptotic Relative Efficiency
514(10)
11.5 Density Estimation
524(17)
11.6 The Bootstrap
541(12)
11.7 Exercises and Experiments
553(12)
11.7.1 Exercises
553(7)
11.7.2 Experiments
560(5)
A Useful Theorems and Notation
565(12)
A.1 Sets and Set Operators
565(1)
A.2 Point-Set Topology
566(1)
A.3 Results from Calculus
567(1)
A.4 Results from Complex Analysis
568(1)
A.5 Probability and Expectation
569(1)
A.6 Inequalities
569(1)
A.7 Miscellaneous Mathematical Results
570(1)
A.8 Discrete Distributions
570(2)
A.8.1 The Bernoulli Distribution
570(1)
A.8.2 The Binomial Distribution
570(1)
A.8.3 The Geometric Distribution
571(1)
A.8.4 The Multinomial Distribution
571(1)
A.8.5 The Poisson Distribution
571(1)
A.8.6 The (Discrete) Uniform Distribution
572(1)
A.9 Continuous Distributions
572(5)
A.9.1 The Beta Distribution
572(1)
A.9.2 The Cauchy Distribution
572(1)
A.9.3 The Chi-Squared Distribution
573(1)
A.9.4 The Exponential Distribution
573(1)
A.9.5 The Gamma Distribution
573(1)
A.9.6 The LaPlace Distribution
574(1)
A.9.7 The Logistic Distribution
574(1)
A.9.8 The Lognormal Distribution
574(1)
A.9.9 The Multivariate Normal Distribution
574(1)
A.9.10 The Non-Central Chi-Squared Distribution
575(1)
A.9.11 The Normal Distribution
575(1)
A.9.12 Student's t Distribution
575(1)
A.9.13 The Triangular Distribution
575(1)
A.9.14 The (Continuous) Uniform Distribution
576(1)
A.9.15 The Wald Distribution
576(1)
B Using R for Experimentation
577(24)
B.1 An Introduction to R
577(1)
B.2 Basic Plotting Techniques
577(5)
B.3 Complex Numbers
582(1)
B.4 Standard Distributions and Random Number Generation
582(10)
B.4.1 The Bernoulli and Binomial Distributions
583(1)
B.4.2 The Beta Distribution
583(1)
B.4.3 The Cauchy Distribution
584(1)
B.4.4 The Chi-Squared Distribution
584(1)
B.4.5 The Exponential Distribution
585(1)
B.4.6 The Gamma Distribution
585(1)
B.4.7 The Geometric Distribution
586(1)
B.4.8 The LaPlace Distribution
587(1)
B.4.9 The Lognormal Distribution
587(1)
B.4.10 The Multinomial Distribution
588(1)
B.4.11 The Normal Distribution
588(1)
B.4.12 The Multivariate Normal Distribution
589(1)
B.4.13 The Poisson Distribution
589(1)
B.4.14 Student's t Distribution
590(1)
B.4.15 The Continuous Uniform Distribution
590(1)
B.4.16 The Discrete Uniform Distribution
591(1)
B.4.17 The Wald Distribution
592(1)
B.5 Writing Simulation Code
592(2)
B.6 Kernel Density Estimation
594(1)
B.7 Simulating Samples from Normal Mixtures
594(1)
B.8 Some Examples
595(6)
B.8.1 Simulating Flips of a Fair Coin
595(1)
B.8.2 Investigating the Central Limit Theorem
595(2)
B.8.3 Plotting the Normal Characteristic Function
597(1)
B.8.4 Plotting Edgeworth Corrections
598(1)
B.8.5 Simulating the Law of the Iterated Logarithm
599(2)
References 601(12)
Author Index 613(4)
Subject Index 617
Alan M. Polansky is an associate professor in the Division of Statistics at Northern Illinois University. Dr. Polansky is the author of Observed Confidence Levels: Theory and Application (CRC Press, October 2007). His research interests encompass nonparametric statistics and industrial applications of statistics.
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