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A Concise Introduction to Mathematical Statistics

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This book gives a thorough introduction to mathematical statistics. The text is unique as an introductory text, mainly by the use of the Riemann-Stieltjes integral. This enables a unified treatment of basic concepts in probability and inference theory, in a mathematically rigorous manner, without the use of measure theory. The approach differentiates this book from other introductory texts, where one does not give a unified approach to basic concepts, as well as from advanced texts, where on...

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This book gives a thorough introduction to mathematical statistics. The text is unique as an introductory text, mainly by the use of the Riemann-Stieltjes integral. This enables a unified treatment of basic concepts in probability and inference theory, in a mathematically rigorous manner, without the use of measure theory. The approach differentiates this book from other introductory texts, where one does not give a unified approach to basic concepts, as well as from advanced texts, where one does give a unified approach relying on advanced mathematics. The treatment of probability theory differs from comparable books in that one discusses basic concepts rigorously but without the use of Lebesgue integration. Thereby it allows one to concentrate on the basic concepts of mathematical statistics, without sacrifice of mathematical stringency. The approach also enables a concise definition of the plug-in estimator in inference theory. Arguably, the plug-in estimator is the most natural and intuitive estimator possible. The introduction of it is however mathematically advanced, and typically covered in PhD level texts. Using the Riemann-Stieltjes integral the introduction of it becomes elementary. The book is intended for students at the Faculty of Science and Faculty of Engineering that have taken a full year of basic mathematics courses, including real analysis and linear algebra.

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      • 11
        Preface
      • 15
        Mathematical Statistics
        • 15
          What is unique about the subject
        • 16
          Very short historical overview and the development of the subject
        • 16
          Two important results
        • 17
          Some important areas in Mathematical Statistics
        • 19
          Important applications and connections to other areas
          • 19
            Philosophy
          • 20
            Physics
          • 20
            Chemistry
          • 20
            Engineering Science
          • 21
            Medicine
          • 21
            Economy
          • 22
            Biology
          • 22
            Law
      • 23
        The Nobel Prize and the Fields medal
  •  PART I Probability Theory
      • 27
        CHAPTER 1 An introductory example and overview
        • 27
          The rolling of a dice
      • 33
        CHAPTER 2 Set theory and introduction to probability
        • 33
          Set theory and events
        • 37
          Probabilities
        • 43
          Probabilities, σ -algebras
        • 50
          A σ -algebra as a container of information about an experi- ment
        • 53
          Exercises
      • 55
        CHAPTER 3 Conditional probability and independence
        • 55
          Conditional probability
        • 59
          Independent events
        • 61
          Exercises
      • 63
        CHAPTER 4 Random variables
        • 66
          The distribution of a random variable
        • 70
          Counting measure, length measure, and integrals with such
        • 73
          The density of a rv
        • 78
          The quantiles
        • 80
          Exercises
      • 83
        CHAPTER 5 Random vectors
        • 87
          Counting measures and length measures on Rn and integrals with such
        • 90
          The density of a random vector
        • 93
          Discrete random vectors 91 Continuous random vectors
        • 96
          Mixed discrete and continuous rv
        • 98
          Independent rv’s
        • 100
          Conditional distributions
        • 105
          Exercises
      • 107
        CHAPTER 6 Functions of random variables and of random vec- tors
        • 108
          Functions of one random variable
        • 116
          Functions of a random vector
        • 118
          Maxima and minima
        • 122
          Sums and convolutions
        • 126
          General real valued functions of a random vector 125 Vector valued functions of random vectors
        • 128
          Exercises
      • 129
        CHAPTER 7 Expectation
        • 130
          The Riemann-Stieltjes integral on R
        • 138
          Application to probability theory
        • 147
          The Riemann-Stieltjes integral on Rn
        • 151
          Applications to probability theory
        • 162
          Expectations and covariances of random vectors
        • 164
          The expectation of positive rv’s
        • 165
          Exercises
      • 169
        CHAPTER 8 Conditional expectation
        • 169
          The definition and properties of a conditional expectation
        • 176
          Exercises
      • 179
        CHAPTER 9 Examples of distributions
        • 179
          The Bernoulli distribution
        • 180
          The binomial distribution
        • 183
          The geometric distribution
        • 184
          The exponential distribution
        • 185
          The discrete uniform distribution
        • 186
          The continuous uniform distribution
        • 186
          The Poisson distribution
        • 188
          The Gaussian (normal) distribution
        • 192
          The multinomial distribution
        • 195
          The multivariate Gaussian (normal) distribution
        • 199
          Exercises
      • 201
        CHAPTER 10 Stochastic convergence
        • 202
          Convergence of random variables
        • 202
          Sure convergence
        • 203
          Almost sure convergence
        • 203
          Convergence in p’th mean
        • 204
          Convergence in probabiltiy
        • 206
          The interpretation of a probability: consequences of of the LLN
        • 207
          Convergence in distribution
        • 210
          The prevalence of the Gaussian distribution: consequences of the central limit theorem
        • 211
          The prevalence of the Poisson distribution
        • 213
          Exercises
      • 215
        CHAPTER 11 Stochastic processes
        • 215
          Introduction
        • 215
          Stochastic processes
        • 220
          The distribution of a stochastic process
        • 221
          Three important classes of processes 221 Stationary processes
        • 222
          Markov processes
        • 223
          Martingales
        • 224
          Two important processes 224 The partial sum process
        • 224
          The empirical process
        • 226
          Exercises
  •  PART II Inference Theory
      • 229
        CHAPTER 12 An introductory example and overview
        • 230
          A coin toss experiment
      • 235
        CHAPTER 13 Statistics
        • 236
          A statistic seen as a function of the data sample
        • 238
          A statistic seen as a function(al) of the distribution function
        • 245
          Properties of estimators
        • 245
          Finite-sample properties
        • 249
          Asymptotic properties
        • 252
          The standard error
        • 253
          Exercises
      • 257
        CHAPTER 14 Methods to obtain estimators
        • 257
          The plug-in estimator
        • 257
          The Maximum Likelihood method
        • 266
          The Least Squares estimator
        • 269
          Extensions and modifications
        • 269
          The ML estimator
        • 272
          The LS estimator
        • 274
          Exercises
      • 277
        CHAPTER 15 Confidence intervals
        • 279
          Pivot functions
        • 284
          Joint confidence intervals for several parameters
        • 286
          Exercises
      • 289
        CHAPTER 16 Tests
        • 293
          The power of a test and the power function
        • 295
          Composite null hypothesis
        • 298
          p-values
        • 301
          Multiple testing
        • 303
          Exercises
      • 307
        CHAPTER 17 Normal approximation of estimators
        • 307
          Normal approximation of linear functionals
        • 310
          Normal approximation of the binomial distribution
        • 312
          Normal approximation of the Poisson distribution 311 A note on the Gaussian approximation
        • 313
          Exercises
      • 315
        CHAPTER 18 Applications to some common situations
        • 315
          Data from a Gaussian distribution 315 One sample
        • 318
          Several samples 316 Observation in pairs
        • 319
          Binomial data 319 One sample
        • 321
          Two samples, inference for difference in success probability
        • 322
          Poisson data
        • 324
          Exercises
      • 327
        CHAPTER 19 Test-based intervals and the confidence interval method
        • 327
          The confidence interval method
        • 329
          Test based confidence intervals
        • 331
          Exercises
      • 333
        CHAPTER 20 Parametric, semi-parametric and non-parametric estimation problems
      • 337
        CHAPTER 21 The empirical distribution function
        • 21.1
          339
          Some more advanced properties of the empirical distribution function
      • 343
        CHAPTER 22 Some nonparametric inference problems
        • 343
          Introduction
        • 345
          Density function estimation
        • 350
          Regression function estimation
        • 352
          One and k-sample tests
        • 353
          Why a nonparametric test
        • 355
          Estimating the survival function in survival analysis
        • 359
          Exercises
      • 363
        CHAPTER 23 Linear regression
        • 365
          The least squares estimator
        • 367
          Normal linear model
        • 370
          The ML estimator in the Normal linear model
        • 371
          Test for and confidence interval at a point y xO on the plane in a Normal linear model
        • 372
          Residual analysis and model fit
        • 375
          Testing of a model and model choice
        • 376
          All subsets regression
        • 377
          Stepwise forward or backward regression
        • 379
          Prediction intervals
        • 380
          Some further interesting regression problems
        • 381
          Dichotomous response and logistic regression
        • 382
          Time to an event as response variable and regression models in survival analysis
        • 384
          Exercises
      • 387
        CHAPTER 24 Introduction to Inference for Stochastic processes
        • 387
          Introduction
        • 389
          The inference problem
        • 390
          Inference for Poisson processes 389 The definition and further properties 389 Inference for
        • 391
          Inference for Markov chains
        • 391
          The definition and further properties
        • 394
          Inference for the transition probabilities
        • 397
          Exercises
      • 399
        APPENDIX A Some useful results from analysis
        • 399
          Maps of sets
        • 400
          Continuity
        • 401
          Measurability of a rv
      • 405
        APPENDIX B Distributions arising from the Gaussian distribution
      • 413
        APPENDIX C The Riemann integral
        • 413
          Integration on R
        • 416
          Integration on Rn
      • 419
        Bibliography
Information

Författare:

Dragi Anevski

Språk:

Engelska

ISBN:

9789144115757

Utgivningsår:

2017

Artikelnummer:

39387-01

Upplaga:

Första

Sidantal:

420

Köp- och leveransvillkor

Förlagskontakt

Jens Fredholm

Förläggare Teknik Teknik, matematik och statistik

046-31 21 58 E-post
Fritjof Janson

Förlagskoordinator Kurslitteratur och Kompetensutveckling

046-31 22 57 E-post
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