The Conway–Maxwell–Poisson Distribution

Cover
Half-title page
Series page
Title page
Copyright page
Dedication
Contents
List of Figures
List of Tables
Preface
Acknowledgments
1 Introduction: Count Data Containing Dispersion
	1.1 Poisson Distribution
		1.1.1 R Computing
	1.2 Data Over-dispersion
		1.2.1 R Computing
	1.3 Data Under-dispersion
		1.3.1 R Computing
	1.4 Weighted Poisson Distributions
	1.5 Motivation, and Summary of the Book
2 The Conway–Maxwell–Poisson (COM–Poisson) Distribution
	2.1 The Derivation/Motivation: A Flexible Queueing Model
	2.2 The Probability Distribution
		2.2.1 R Computing
	2.3 Distributional and Statistical Properties
		2.3.1 R Computing
	2.4 Parameter Estimation and Statistical Inference
		2.4.1 Combining COM–Poissonness Plot with Weighted Least Squares
		2.4.2 Maximum Likelihood Estimation
		2.4.3 Bayesian Properties and Estimation
		2.4.4 R Computing
		2.4.5 Hypothesis Tests for Dispersion
	2.5 Generating Data
		2.5.1 Inversion Method
		2.5.2 Rejection Sampling
		2.5.3 R Computing
	2.6 Reparametrized Forms
	2.7 COM–Poisson Is a Weighted Poisson Distribution
	2.8 Approximating the Normalizing Term, Z(λ, ν)
	2.9 Summary
3 Distributional Extensions and Generalities
	3.1 The Conway–Maxwell–Skellam (COM–Skellam or CMS) Distribution
	3.2 The Sum-of-COM–Poissons (sCMP) Distribution
	3.3 Conway–Maxwell Inspired Generalizations of the Binomial Distribution
		3.3.1 The Conway–Maxwell–binomial (CMB) Distribution
		3.3.2 The Generalized Conway–Maxwell–Binomial Distribution
		3.3.3 The Conway–Maxwell–multinomial (CMM) Distribution
		3.3.4 CMB and CMM as Sums of Dependent Bernoulli Random Variables
		3.3.5 R Computing
	3.4 CMP-Motivated Generalizations of the Negative Binomial Distribution
		3.4.1 The Generalized COM–Poisson (GCMP) Distribution
		3.4.2 The COM–Negative Binomial (COMNB) Distribution
		3.4.3 The COM-type Negative Binomial (COMtNB) Distribution
		3.4.4 Extended CMP (ECMP) Distribution
	3.5 Conway–Maxwell Katz (COM–Katz) Class of Distributions
	3.6 Flexible Series System Life-Length Distributions
		3.6.1 The Exponential-CMP (ExpCMP) Distribution
		3.6.2 The Weibull–CMP (WCMP) Distribution
	3.7 CMP-Motivated Generalizations of the Negative Hypergeometric Distribution
		3.7.1 The COM-negative Hypergeometric (COMNH) Distribution, Type I
		3.7.2 The COM–Poisson-type Negative Hypergeometric (CMPtNH) Distribution
		3.7.3 The COM-Negative Hypergeometric (CMNH) Distribution, Type II
	3.8 Summary
4 Multivariate Forms of the COM–Poisson Distribution
	4.1 Trivariate Reduction
		4.1.1 Parameter Estimation
		4.1.2 Hypothesis Testing
		4.1.3 Multivariate Generalization
	4.2 Compounding Method
		4.2.1 Parameter Estimation
		4.2.2 Hypothesis Testing
		4.2.3 R Computing
		4.2.4 Multivariate Generalization
	4.3 The Sarmanov Construction
		4.3.1 Parameter Estimation and Hypothesis Testing
		4.3.2 Multivariate Generalization
	4.4 Construction with Copulas
	4.5 Real Data Examples
		4.5.1 Over-dispersed Example: Number of Shunter Accidents
		4.5.2 Under-dispersed Example: Number of All-Star Basketball Players
	4.6 Summary
5 COM–Poisson Regression
	5.1 Introduction: Generalized Linear Models
		5.1.1 Logistic Regression
		5.1.2 Poisson Regression
		5.1.3 Addressing Data Over-dispersion: Negative Binomial Regression
		5.1.4 Addressing Data Over- or Under-dispersion: Restricted Generalized Poisson Regression
	5.2 Conway–Maxwell–Poisson (COM–Poisson) Regression
		5.2.1 Model Formulations
		5.2.2 Parameter Estimation
			Maximum Likelihood Estimation
			Moment-based Estimation
			Bayesian Estimation
		5.2.3 Hypothesis Testing
		5.2.4 R Computing
			Maximum Likelihood Estimation for MCMP1 Regression
			Bayesian Estimation for ACMP Regression
		5.2.5 Illustrative Examples
			Example: Number of Children in a Subset of German Households
			Example: Airfreight Breakage Study
			Example: Number of Faults in Textile Fabrics
	5.3 Accounting for Excess Zeroes: Zero-inflated COM–Poisson Regression
		5.3.1 Model Formulations
			A Further Extension: The ZISCMP Regression
		5.3.2 Parameter Estimation
			Frequentist Approach
			Bayesian Formulation
		5.3.3 Hypothesis Testing
		5.3.4 A Word of Caution
		5.3.5 Alternative Approach: Hurdle Model
	5.4 Clustered Data Analysis
	5.5 R Computing for Excess Zeroes and/or Clustered Data
		5.5.1 Examples
			Example: Unwanted Pursuit Behavior Perpetrations
			Example: Epilepsy and Progabide
	5.6 Generalized Additive Model
	5.7 Computing via Alternative Softwares
		5.7.1 MATLAB Computing
		5.7.2 SAS Computing
	5.8 Summary
6 COM–Poisson Control Charts
	6.1 CMP-Shewhart Charts
		6.1.1 CMP Control Chart Probability Limits
		6.1.2 R Computing
		6.1.3 Example: Nonconformities in Circuit Boards
		6.1.4 Multivariate CMP-Shewhart Chart
	6.2 CMP-inspired EWMA Control Charts
		6.2.1 COM–Poisson EWMA (CMP-EWMA) Chart
		6.2.2 CMP-EWMA Chart with Multiple Dependent State Sampling
		6.2.3 CMP-EWMA Chart with Repetitive Sampling
		6.2.4 Modified CMP-EWMA Chart
		6.2.5 Double EWMA Chart for CMP Attributes
		6.2.6 Hybrid EWMA Chart
	6.3 COM–Poisson Cumulative Sum (CUSUM) Charts
		6.3.1 CMP-CUSUM charts
			The λ-CUSUM chart
			The ν-CUSUM Chart
			The s-CUSUM Chart
			CUSUM Chart Design, and Comparisons
		6.3.2 Mixed EWMA-CUSUM for CMP Attribute Data
	6.4 GenerallyWeighted Moving Average
	6.5 COM–Poisson Chart Via Progressive Mean Statistic
	6.6 Summary
7 COM–Poisson Models for Serially Dependent Count Data
	7.1 CMP-motivated Stochastic Processes
		7.1.1 The Homogeneous CMP Process
			Parameter Estimation
			R Computing
		7.1.2 Copula-based CMP Markov Models
			Statistical Inference
		7.1.3 CMP-Hidden Markov Models
			R Computing
	7.2 Intensity Parameter Time Series Modeling
		7.2.1 ACMP-INGARCH
		7.2.2 MCMP1-ARMA
	7.3 Thinning-Based Models
		7.3.1 Autoregressive Models
			The CMPAR(1) Model
			The SCMPAR(1) Model
			Bivariate COM–Poisson Autoregressive Model
		7.3.2 Moving Average Models
			INMA(1) Models with COM–Poisson Innovations
			The SCMPMA(1) Model
			Bivariate MCMP2MA(1) Model
	7.4 CMP Spatio-temporal Models
	7.5 Summary
8 COM–Poisson Cure Rate Models
	8.1 Model Background and Notation
	8.2 Right Censoring
		8.2.1 Parameter Estimation Methods
			Maximum Likelihood Estimation
			Bayesian Approach
			EM Algorithm
		8.2.2 Quantifying Variation
		8.2.3 Simulation Studies
		8.2.4 Hypothesis Testing and Model Discernment
	8.3 Interval Censoring
		8.3.1 Parameter Estimation
			EM Algorithm Approach
		8.3.2 Variation Quantification
		8.3.3 Simulation Studies
		8.3.4 Hypothesis Testing and Model Discernment
	8.4 Destructive CMP Cure Rate Model
		8.4.1 Parameter Estimation
		8.4.2 Hypothesis Testing and Model Discernment
	8.5 Lifetime Distributions
	8.6 Summary
References
Index