While the Poisson distribution is a classical statistical model for count data, the distributional model hinges on the constraining property that its mean equal its variance. This text instead introduces the Conway-Maxwell-Poisson distribution and motivates its use in developing flexible statistical methods based on its distributional form. This two-parameter model not only contains the Poisson distribution as a special case but, in its ability to account for data over- or under-dispersion, encompasses both the geometric and Bernoulli distributions. The resulting statistical methods serve in a multitude of ways, from an exploratory data analysis tool, to a flexible modeling impetus for varied statistical methods involving count data. The first comprehensive reference on the subject, this text contains numerous illustrative examples demonstrating R code and output. It is essential reading for academics in statistics and data science, as well as quantitative researchers and data analysts in economics, biostatistics and other applied disciplines.
Author(s): Kimberly F. Sellers
Series: Institute of Mathematical Statistics Monographs, 8
Edition: 1
Publisher: Cambridge University Press
Year: 2023
Language: English
Pages: 250
City: Cambridge
Tags: Conway; Maxwell; Poisson; Conway Maxwell Poisson; Conway–Maxwell–Poisson; Distribution; Probability Theory; Statistics; Statistics Theory; Statistics Methods; Economics; Econometrics; Mathematical Methods; R Programming Language
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
