Ramachandran / Tsokos Mathematical Statistics with Applications in R

2. Auflage 2014
ISBN: 978-0-12-417132-9
Verlag: Academic Press
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 826 Seiten, Format (B × H): 191 mm x 235 mm

Mathematical Statistics with Applications in R, Second Edition, offers a modern calculus-based theoretical introduction to mathematical statistics and applications. The book covers many modern statistical computational and simulation concepts that are not covered in other texts, such as the Jackknife, bootstrap methods, the EM algorithms, and Markov chain Monte Carlo (MCMC) methods such as the Metropolis algorithm, Metropolis-Hastings algorithm and the Gibbs sampler. By combining the discussion on the theory of statistics with a wealth of real-world applications, the book helps students to approach statistical problem solving in a logical manner.This book provides a step-by-step procedure to solve real problems, making the topic more accessible. It includes goodness of fit methods to identify the probability distribution that characterizes the probabilistic behavior or a given set of data. Exercises as well as practical, real-world chapter projects are included, and each chapter has an optional section on using Minitab, SPSS and SAS commands. The text also boasts a wide array of coverage of ANOVA, nonparametric, MCMC, Bayesian and empirical methods; solutions to selected problems; data sets; and an image bank for students.Advanced undergraduate and graduate students taking a one or two semester mathematical statistics course will find this book extremely useful in their studies. - Step-by-step procedure to solve real problems, making the topic more accessible - Exercises blend theory and modern applications - Practical, real-world chapter projects - Provides an optional section in each chapter on using Minitab, SPSS and SAS commands - Wide array of coverage of ANOVA, Nonparametric, MCMC, Bayesian and empirical methods

Kandethody M. Ramachandran is Professor of Mathematics and Statistics at the University of South Florida. His research interests are concentrated in the areas of applied probability, statistics, machine learning, and generative AI. His research publications span a variety of areas such as control of heavy traffic queues, stochastic delay systems, machine learning methods applied to game theory, finance, cyber security, health sciences, and other emerging areas. He is also co-author of three books. He is the founding director of the Interdisciplinary Data Sciences Consortium (IDSC). He is extensively involved in activities to improve statistics and mathematics education. He is a recipient of the Teaching Incentive Program award at the University of South Florida. He is also the PI of a two million dollar grant from NSF, and a co_PI of a 1.4 million grant from HHMI to improve STEM education at USF.

Zielgruppe

Advanced undergraduate and graduate students taking a one or two semester mathematical statistics course

Autoren/Hrsg.

Ramachandran, K.M.

Ramachandran, Kandethody M.

Tsokos, Chris P.

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik Mathematische Statistik

Inhaltsverzeichnis

- Descriptive Statistics
- Basic Concepts from Probability Theory
- Additional Topics in Probability
- Sampling Distributions
- Estimation
- Properties of Point Estimation, Hypothesis Testing
- Linear Regression Models
- Design of Experiments
- Analysis of variance
- Bayesian Estimation and Inference
- Nonparametric tests
- Empirical Methods
- Time-series Analysis
- Overview of Statistical Applications
- Appendices
- Selected Solutions to Exercises

Leseproben

Chapter 1

Descriptive Statistics

Abstract

In today’s society, decisions are made on the basis of data. Most scientific or industrial studies and experiments produce data, and the analysis of these data and drawing useful conclusions from them become one of the central issues. Statistics is an integral part of the quantitative approach to knowledge. The field of statistics is concerned with the scientific study of collecting, organizing, analyzing, and drawing conclusions from data. Statistics benefits all of us because of its ability to predict the future based on data we have previously gathered. Statistical methods help us to transform data to information and knowledge. Statistical concepts enable us to solve problems in a diversity of contexts, add substance to decisions, and reduce guesswork. The discipline of statistics stemmed from the need to place knowledge management on a systematic evidence base. Earlier works on statistics dealt only with the collection, organization, and presentation of data in the form of tables and charts. In order to place statistical knowledge on a systematic evidence base, we require a study of the laws of probability. In mathematical statistics we create a probabilistic model and view the data as a set of random outcomes from that model. Advances in probability theory enable us to draw valid conclusions and to make reasonable decisions on the basis of data. In the present chapter we briefly review some of the basic concepts of descriptive statistics. Such concepts will give us a visual and descriptive presentation of the problem under investigation.

Keywords

Data collection

Elementary statistics

Probabilistic model

Data representation

Numerical summary

Chapter Contents

1.1 Introduction   2

1.2 Basic Concepts   4

1.3 Sampling Schemes   8

1.4 Graphical Representation of Data   13

1.5 Numerical Description of Data   26

1.6 Computers and Statistics   39

1.7 Chapter Summary   39

1.8 Computer Examples   41

Projects for Chapter 1   51

Objective

Review the basic concepts of elementary statistics.

1.1 Introduction

In today’s society, decisions are made on the basis of data. Most scientific or industrial studies and experiments produce data, and the analysis of these data and drawing useful conclusions from them become one of the central issues. Statistics is an integral part of the quantitative approach to knowledge. The field of statistics is concerned with the scientific study of collecting, organizing, analyzing, and drawing conclusions from data. Statistics benefits all of us because of its ability to predict the future based on data we have previously gathered. Statistical methods help us to transform data to information and knowledge. Statistical concepts enable us to solve problems in a diversity of contexts, add substance to decisions, and reduce guesswork. The discipline of statistics stemmed from the need to place knowledge management on a systematic evidence base. Earlier works on statistics dealt only with the collection, organization, and presentation of data in the form of tables and charts. In order to place statistical knowledge on a systematic evidence base, we require a study of the laws of probability. In mathematical statistics we create a probabilistic model and view the data as a set of random outcomes from that model. Advances in probability theory enable us to draw valid conclusions and to make reasonable decisions on the basis of data.

Statistical methods are used in almost every discipline, including agriculture, astronomy, biology, business, communications, economics, education, electronics, geology, health sciences, and many other fields of science and engineering, and can aid us in several ways. Modern applications of statistical techniques include statistical communication theory and signal processing, information theory, network security and denial of service problems, clinical trials, artificial and biological intelligence, quality control of manufactured items, software reliability, and survival analysis. The first of these is to assist us in designing experiments and surveys. We desire our experiment to yield adequate answers to the questions that prompted the experiment or survey. We would like the answers to have good precision without involving a lot of expenditure. Statistically designed experiments facilitate development of robust products that are insensitive to changes in the environment and internal component variation. Another way that statistics assists us is in organizing, describing, summarizing, and displaying experimental data. This is termed descriptive statistics. A third use of statistics is in drawing inferences and making decisions based on data. For example, scientists may collect experimental data to prove or disprove an intuitive conjecture or hypothesis. Through the proper use of statistics we can conclude whether the hypothesis is valid or not. In the process of solving a real-life problem using statistics, the following three basic steps may be identified. First, consistent with the objective of the problem, we identify the model—the appropriate statistical method. Then, we justify the applicability of the selected model to fulfill the aim of our problem. Last, we properly apply the related model to analyze the data and make the necessary decisions, which results in answering the question of our problem with minimum risk. Starting with Chapter 2, we will study the necessary background material to proceed with the development of statistical methods for solving real-world problems.

In the present chapter we briefly review some of the basic concepts of descriptive statistics. Such concepts will give us a visual and descriptive presentation of the problem under investigation. Now, we proceed with some basic definitions.

1.1.1 Data Collection

One of the first problems that a statistician faces is obtaining data. The inferences that we make depend critically on the data that we collect and use. Data collection involves the following important steps.

General Procedure for Data Collection

1. Define the objectives of the problem and proceed to develop the experiment or survey.

2. Define the variables or parameters of interest.

3. Define the procedures of data collection and measuring techniques. This includes sampling procedures, sample size, and data-measuring devices (questionnaires, telephone interviews, etc.).

Example 1.1.1

We may be interested in estimating the average household income in a certain community. In this case, the parameter of interest is the average income of a typical household in the community. To acquire the data, we may send out a questionnaire or conduct a telephone interview. Once we have the data, we may first want to represent the data in graphical or tabular form to better understand its distributional behavior. Then we will use appropriate analytical techniques to estimate the parameter(s) of interest, in this case the average household income.

Very often a statistician is confined to data that have already been collected, possibly even collected for other purposes. This makes it very difficult to determine the quality of data. Planned collection of data, using proper techniques, is much preferred.

1.2 Basic Concepts

Statistics is the science of data. This involves collecting, classifying, summarizing, organizing, analyzing, and interpreting data. It also involves model building. Suppose we wish to study household incomes in a certain neighborhood. We may decide to randomly select, say, 50 families and examine their household incomes. As another example, suppose we wish to determine the diameter of a rod, and we take 10 measurements of the diameter. When we consider these two examples, we note that in the first case the population (the household incomes of all families in the neighborhood) really exists, whereas in the second, the population (set of all possible measurements of the diameter) is only conceptual. In either case we can visualize the totality of the population values, of which our sample data are only a small part. Thus, we define a population to be the set of all measurements or objects that are of interest and a sample to be a subset of that population. The population acts as the sampling frame from which a sample is selected. Now we introduce some basic notions commonly used in statistics.

Definition 1.2.1

A population is the collection or set of all objects or measurements that are of interest to the collector.

Example 1.2.1

Suppose we wish to study the heights of all female students at a certain university. The population will be the...

Über Autor(innen)

Tsokos, Chris P.
Chris P. Tsokos is Distinguished University Professor of Mathematics and Statistics at the University of South Florida. Dr. Tsokos' research has extended into a variety of areas, including stochastic systems, statistical models, reliability analysis, ecological systems, operations research, time series, Bayesian analysis, and mathematical and statistical modelling of global warming, both parametric and nonparametric survival analysis, among others. He is the author of more than 300 research publications in these areas, including Random Integral Equations with Applications to Life Sciences and Engineering, Probability Distribution: An Introduction to Probability Theory with Applications, Mainstreams of Finite Mathematics with Applications, Probability with the Essential Analysis, Applied Probability Bayesian Statistical Methods with Applications to Reliability, and Mathematical Statistics with Applications, among others.

Dr. Tsokos is the recipient of many distinguished awards and honors, including Fellow of the American Statistical Association, USF Distinguished Scholar Award, Sigma Xi Outstanding Research Award, USF Outstanding Undergraduate Teaching Award, USF Professional Excellence Award, URI Alumni Excellence Award in Science and Technology, Pi Mu Epsilon, election to the International Statistical Institute, Sigma Pi Sigma, USF Teaching Incentive Program, and several humanitarian and philanthropic recognitions and awards. He is also a member of several academic and professional societies, and serves as Honorary Editor, Chief-Editor, Editor or Associate Editor for more than twelve academic research journals.