Mathematical Statistics.

Format:

Book

Author/Creator:

Tian, Guoliang.

Series:

Textbooks for Tomorrow's Scientists Series:Textbooks for Tomorrow's Scientists Introduces Advanced Textbooks on a Wide Range of Topics in STM Subject Areas. These Books Offer Students, Researchers, Faculty and Professionals the Resources They Need to Lear

Language:

English

Physical Description:

1 online resource (336 pages)

Edition:

1st ed.

Place of Publication:

Les Ulis : EDP Sciences, 2026.

Summary:

This book covers core topics of mathematical statistics with logical rigor--from foundational probability theory to point estimation, interval estimation, hypothesis testing, and key concepts like sufficient statistics and Fisher information.

Contents:

Cover-1

Cover-2

Mathematical Statistics

Preface

Contents

Chapter 1 Probability and Distributions

1.1 Probability

1.1.1 Permutation, combination and binomial coefficients

1.1.2 Sample space

1.1.3 Events

1.1.4 Properties of probability

1.2 Conditional Probability

1.3 Bayes Theorem

1.4 Probability Distributions

1.5 Bivariate Distributions

1.5.1 Joint distribution

1.5.2 Marginal and conditional distributions

1.5.3 Independency of two random variables

1.6 Expectation, Variance and Moments

1.6.1 Moments

1.6.2 Some probability inequalities

1.6.3 Conditional expectation

1.6.4 Compound random variables

1.6.5 Calculation of (conditional) probability via (conditional) expectation

1.7 Moment Generating Function

1.8 Beta and Gamma Distributions

1.8.1 Beta distribution

1.8.2 Gamma distribution

1.9 Bivariate Normal Distribution

1.9.1 Univariate normal distribution

1.9.2 Correlation coefficient

1.9.3 Joint density

1.9.4 Stochastic representation of random variables or random vectors

1.10 Inverse Bayes Formulae

1.10.1 Three inverse Bayes Formulae

1.10.2 Understanding IBF

1.10.3 Two examples

1.11 Categorical Distribution

1.12 Zero-inflated Poisson Distribution

Exercise 1

Chapter 2 Sampling Distributions

2.1 Distribution of the Function of Random Variables

2.1.1 Cumulative distribution function technique

2.1.2 Transformation technique

2.1.3 Moment generating function technique

2.2 Statistics, Sample Mean and Sample Variance

2.2.1 Distribution of the sample mean

2.2.2 Distribution of the sample variance

2.3 The t and F Distributions

2.3.1 The t distribution

2.3.2 The F distribution

2.4 Order Statistics

2.4.1 Distribution of single order statistic

2.4.2 Joint distribution of more order statistics.

2.5 Limit Theorems

2.5.1 Convergency of sequences of distribution functions

2.5.2 Convergence in probability

2.5.3 Relationships of four classes of convergency

2.5.4 Law of large number

2.5.5 Central limit theorem

2.6 Some Challenging Questions

Exercise 2

Chapter 3 Point Estimation

3.1 Maximum Likelihood Estimator

3.1.1 Point estimator and point estimate

3.1.2 Joint density and likelihood function

3.1.3 Maximum likelihood estimate and maximum likelihood estimator

3.1.4 The invariance property of MLE

3.2 Moment Estimator

3.3 Bayesian Estimator

3.4 Properties of Estimators

3.4.1 Unbiasedness

3.4.2 Efficiency

3.4.3 Sufficiency

3.4.4 Completeness

3.5 Limiting Properties of MLE

3.6 Some Challenging Questions

Exercise 3

Chapter 4 Confidence Interval Estimation

4.1 Introduction

4.2 The Confidence Interval of Normal Mean

4.2.1 The variance is known

4.2.2 The variance is unknown

4.3 The Confidence Interval of the Difference of Two Normal Means

4.4 The Confidence Interval of Normal Variance

4.4.1 The mean is known

4.4.2 The mean is unknown

4.5 The Confidence Interval of the Ratio of Two Normal Variances

4.6 Large-Sample Confidence Intervals

4.7 The Shortest Confidence Interval

Exercise 4

Chapter 5 Hypothesis Testing

5.1 Introduction

5.1.1 Several basic notions

5.1.2 Type I error and Type II error

5.1.3 Power function

5.2 The Neyman-Pearson Lemma

5.2.1 Simple null hypothesis versus simple alternative

5.2.2 Composite hypotheses

5.3 Likelihood Ratio Test

5.3.1 Likelihood ratio statistic

5.3.2 Likelihood ratio test

5.4 Tests on Normal Means

5.4.1 One-sample normal test when variance is known

5.4.2 One-sample t test

5.4.3 Two-sample t test

5.5 Goodness of Fit Test

5.5.1 Introduction.

5.5.2 The chi-square test for totally known distribution

5.5.3 The chi-square test for known distribution family with unknown parameters

Exercise 5

Chapter 6 Critical Regions and p-values for Skew Null Distributions

6.1 One-sample Chi-square Test on Normal Variances

6.2 Two-sample F Test on Normal Variances

Appendix A Basic Statistical Distributions

A.1 Discrete Distributions

A.2 Continuous Distributions

Appendix B A Unified Expectation Technique

B.1 Continuous Random Variables

B.2 Discrete Random Variables

Appendix C The Newton-Raphson and Fisher Scoring Algorithms

C.1 Newton's Method for Root Finding

C.2 Newton's Method for Calculating MLE

C.3 The Newton-Raphson Algorithm for High-dimensional Cases

C.4 The Fisher Scoring Algorithm

List of Figures

List of Tables

List of Acronyms

List of Symbols

References

Subject Index.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

2-7598-3949-4

9782759839490

OCLC:

1572092618