1st Edition

Probability and Statistical Models with Applications




ISBN 9781584881247
Published September 21, 2000 by Chapman and Hall/CRC
664 Pages

USD $185.00

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Book Description

This monograph of carefully collected articles reviews recent developments in theoretical and applied statistical science, highlights current noteworthy results and illustrates their applications; and points out possible new directions to pursue. With its enlightening account of statistical discoveries and its numerous figures and tables, Probability and Statistical Models with Applications is a must read for probabilists and theoretical and applied statisticians.

Table of Contents

PREFACE
LIST OF CONTRIBUTORS
LIST OF TABLES
LIST OF FIGURES
THEOPHILOS N. CACOULLOS - A VIEW OF HIS CAREER
PUBLICATIONS OF THEOPHILOS N. CACOULLOS
THE TEN COMMANDMENTS FOR A STATISTICIAN
PART I APPROXIMATION, BOUNDS AND INEQUALITIES.
NON-UNIFORM BOUNDS IN PROBABILITY APPROXIMATIONS USING STEIN'S METHOD
Introduction
Poisson Approximation
Binomial Approximation: Binary Expansion of a Random Integer
Normal Approximation
Conclusion
PROBABILITY INEQUALITIES FOR MULTIVARIATE DISTRIBUTIONS WITH APPLICATIONS TO STATISTICS
Introduction and Summary
Positive Dependence and Product-Type Inequalities
Negative Dependence and Product-Type Inequalities
Bonferroni-Type Inequalities
Applications
APPLICATIONS OF COMPOUND POISSON APPROXIMATION
Introduction
First Applications
Word Counts
Discussion and Numerical Examples
COMPOUND POISSON APPROXIMATION FOR SUMS OF DEPENDENT RANDOM VARIABLES
Introduction
Preliminaries and Notations
Main Results
Examples of Applications
UNIFIED VARIANCE BOUNDS AND A STEIN-TYPE IDENTITY
Introduction
Properties of the Transformation
Application to Variance Bounds
PROBABILITY INEQUALITIES FOR U-STATISTICS
Introduction
Preliminaries
Probability Inequalities
PART II PROBABILITY AND STOCHASTIC PROCESSES
THEORY AND APPLICATIONS OF DECOUPLING
Complete Decoupling of Marginal Laws and One-Sided Bounds
Tangent Sequences and Conditionally Independent Variables
Basic Decoupling Inequalities for Tangent Sequences
Applications to Martingale Inequalities and Exponential Tail Probability Bounds
Decoupling of Multilinear Forms, U-Statistics and U-Processes
Total Decoupling of Stopping Times
Principle of Conditioning in Weak Convergence
Conclusion
A NOTE ON THE PROBABILITY OF RAPID EXTINCTION OF ALLELES IN A WRIGHT-FISHER PROCESS
Introduction
The Lower Bound for Boundary Sets
STOCHASTIC INTEGRAL FUNCTIONALS IN AN ASYMPTOTIC SPLIT STATE SPACE
Introduction
Preliminaries
Phase Merging Scheme for Markov Jump Processes
Average of Stochastic Integral Functional
Diffusion Approximation of Stochastic Integral Function
Integral Functional with Perturbed Kernel
BUSY PERIODS FOR SOME QUEUES WITH DETERMINISTIC INTERARRIVAL OR SERVICE TIMES
Introduction
Preliminaries: A Basic Class of Polynomials
The Dg/M(Q)/1 Queue
The M(Q)/Dg /1 Queue
THE EVOLUTION OF POPULATION STRUCTURE OF THE PERTURBED NON-HOMOGENOUS SEMI-MARKOV SYSTEM
Introduction
The Perturbed Non-Homogeneous Semi-Markov System
The Expected Population Structure with Respect to the First Passage Time Probabilities
The Expected Population Structure with Respect to the Duration of a Membership in a State
The Expected Population Structure with Respect to the State Occupancy of a Membership
The Expected Population Structure with Respect to the Counting Transition Probabilities
PART III DISTRIBUTIONS, CHARACTERIZATIONS, AND APPLICATIONS
CHARACTERIZATIONS OF SOME EXPONENTIAL FAMILIES BASED ON SURVIVAL DISTRIBUTIONS AND MOMENTS
Introduction
An Auxiliary Lemma
Characterizations Based on Survival Distributions
Characterizations Based on Moments
BIVARIATE DISTRIBUTIONS COMPATIBLE OR NEARLY COMPATIBLE WITH GIVEN CONDITIONAL INFORMATION
Introduction
Imprecise Specification
Precise Specification
An Example
A CHARACTERIZATION OF A DISTRIBUTION ARISING FROM ABSORPTION SAMPLING
Introduction
The Characterization Theorem
An Application
REFINEMENTS OF INEQUALITIES FOR SYMMETRIC FUNCTIONS
GENERAL OCCUPANCY DISTRIBUTIONS
Introduction
A General Random Occupancy Model
Special Occupancy Distributions
A SKEW t Distribution
Introduction
Derivation of Skew t Density
Properties of Skew t Distribution
A First Bivariate Skew t Distribution
A Second Bivariate Skew t Distribution
ON THE POSTERIOR MOMENTS FOR TRUNCATION PARAMETER DISTRIBUTIONS AND IDENTIFIABILITY BY POSTERIOR MEAN FOR EXPONENTIAL DISTRIBUTION WITH LOCATION PARAMETERS
Introduction
Posterior Moments
Examples
Identifiability by Posterior Mean
An Illustrative Example
DISTRIBUTIONS OF RADON VOLUMES WITHOUT USING INTEGRAL GEOMETRY TECHNIQUES
Introduction
Evaluation of Arbitrary Moments of the Random Volumes
PART IV TIME SERIES, LINEAR, AND NON-LINEAR SERIES
COINTEGRATION OF ECONOMIC TIME SERIES
Introduction
Regression Models
Simultaneous Equation Models
Canonical Analysis and the Reduced Rank Regression Estimator
Autoregressive Processes
Nonstationary Models
Cointegrated Models
Asymptotic Distribution of Estimators and Test Criterion
ON SOME POWER PROPERTIES OF GOODNESS-OF-FIT TESTS IN TIME SERIES ANALYSIS
Testing Spectral Density Fits
Local Power Considerations
Comparison
LINEAR CONSTRAINTS ON A LINEAR MODEL
Introduction
Geometric Interpretation of the Role of the Linear Constraints
M-METHODS IN GENERALIZED NONLINEAR MODELS
Introduction
Definitions and Assumptions
Asymptotic Results
Test of Significance and Computational Algorithm
PART V INFERENCE AND APPLICATIONS
EXTENTIONS OF A VARIATION OF THE ISOPERIMETRIC PROBLEM
Introduction
Information Retrieval Problem
Information Retrieval without Measurement Error
Useful Information in a Variable
Allocation of Storage Space
The Isoperimetric Problem
Extensions
ON FINDING A SINGLE POSITIVE UNIT IN GROUP-TESTING
Introduction
Description of Properties, Numerical Results
Some Formulas for Procedure RDH
The Greedy Procedure RG
Conclusions
Changing the Prior with Procedure RDH
Robustness of Procedure RDH for q Known
TESTING HYPOTHESES ON VARIANCES IN THE PRESENCE OF CORRELATIONS
Bivariate Normal Population
Modifying the Hypothesis
Non-Null Moments
Null Case
The Conditional Hypothesis
ESTIMATING THE SMALLEST SCALE PARAMETER: UNIVERSAL DOMINATION RESULTS
Introduction
Main Results
ON SENSITIVITY OF EXPONENTIAL RATE OF CONVERGENCE FOR THE MAXIMUM LIKELIHOOD ESTIMATOR
Introduction
Main Results
Some Applications
Discussion
A CLOSER LOOK AT WEIGHTED LIKELIHOOD IN THE CONTEXT OF MIXTURES
Introduction
Background
Simulation Experiments and Results
Model Selection
Conclusions
ON NONPARAMETRIC FUNCTION ESTIMATION WITH INFINITE-ORDER FLAT-TOP KERNELS
Introduction: A General Family of Flat-Top Kernels of Infinite Order
Multivariate Density Estimation: A Review
Further Issues on Density Estimation
MULTIPOLISHING LARGE TWO-WAY TABLES
Introduction
Bilinear Multipolishers
Matrix Approximations
Displays
Examplel
Concluding Remarks
ON DISTANCES AND MEASURES OF INFORMATION: A CASE OF DIVERSITY
Introduction
Measuring Information-Measures of Information
Properties of Measures of Information
Measures of Information and Inference
Applications
Conclusions
REPRESENTATION FORMULAE FOR PROBABILITIES OF CORRECT CLASSIFICATION
Introduction
Vector Algebraic Preliminaries
Distributional Results
ESTIMATION OF CYCLING EFFECT ON RELIABILITY
Models
Semiparametric Estimation
PART VI APPLICATIONS TO BIOLOGY AND MEDICINE
A NEW TEST FOR TREATMENT VS. CONTROL IN AN ORDERED 2 X 3 CONTINGENCY TABLE
Introduction
New Test, Implementation and Example
Simulation Study
Theoretical Properties
Appendix
AN EXPERIMENTAL STUDY OF THE OCCURRENCE TIMES OF RARE SPECIES
Statement of the Problem
The Design of the Experiment
Stage 2 of the Experiment
Findings
A DISTRIBUTION FUNCTIONAL ARISING IN EPIDEMIC CONTROL
Introduction
Properties of the Functional
Proof of the Theorem
Application to Epidemic Control
BIRTH AND DEATH URN FOR TERNARY OUTCOMES: STOCHASTIC PROCESSES APPLIED TO URN MODELS
Introduction
A Birth and Death Urn with Immigration for Ternary Outcomes
Embedding the Urn Scheme in a Continuous-Time Birth and Death Process
The Probabiity Generating Function for the Number of Success on Treatment i in the Continuous-Time Birth and Death Process
The Probability Generating Function for the Number of Trials on Treatment i in the Continuous-Time Birth and Death Process
The Number of Trials on Treatment i in the Continuous -Time Birth and Death Process
The Joint Probability Generating Function for the Number of Successes and the Number of Trials in the Continuous-Time Birth and Death Process
Adopting a Stopping Rule to Convert Continuous-Time Statistics to the Urn Design
Limiting Results for the Proportion of Trials on Treatment i
AUTHOR INDEX
SUBJECT INDEX
NOTE: References at the end of each chapter.

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Editor(s)

Biography

CH. A. Charalambides, M. V. Koutras, N. Balakrishnan

Reviews

"… the book contains some chapters of interest for probabilists and theoretical and applied statisticians. Most of the articles end with a complete list of references of recent developments, which will make it easier for graduate students and researchers in finding articles of interest."
-Short Book Reviews, Vol. 21, No. 2, August 2001


"…deserves a place in the central reference section of any university mathematics library."
- The Mathematical Gazette, March 1997