# Probability and Statistical Models with Applications

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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.

## 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