Beyond Beta

The following sections are included:

• Preface

• Contents

The following sections are included:

• An Historical Overview

• Deriving the CDF utilizing a Geometric Argument

• Moments of Triangular Distributions

• Maximum Likelihood Method for the Threshold Parameter θ
  • An illustrative example

• Three Parameter Maximum Likelihood Estimation
  • Some details about the functions G(a, b) and
  • ML estimation procedure in pseudo Pascal
    • The search routine Bsearch
    • The search routine ABSearch

• Solving for a and b using a Lower and Upper Quantile Estimate

• Concluding Remarks

The following sections are included:

• The Topp and Leone Distribution
  • Location estimates of Topp and Leone distributions
  • Variance of Topp and Leone distributions
  • Maximum likelihood estimation

• The Trapezoidal Distribution
  • Moments of the trapezoidal distribution
  • Inverse cumulative distribution function

• A Linear Combination of Uniform Variables
  • Elementary derivation of the cdf of X given by (2.34)
  • Algorithm for evaluating the cdf given by (2.36)

• Concluding Remarks

The following sections are included:

• Introduction: The Leading Example

• The Standard Two-Sided Power Distributions
  • Moments
  • Properties of the cdf
  • Quantile properties
  • Some limiting distributions
  • Relative entropy

• Maximum Likelihood Method of Estimating Parameters
  • An illustrative example

• Method of Moments

• Moment Ratio Diagram Comparison with the Beta Family

• Musings on STSP and Beta Families and Concluding Remarks

The following sections are included:

• Introduction: The Four-Parameter TSP Distribution

• Four-Parameter Maximum Likelihood Estimation
  • An illustrative example

• Elicitation Methods for TSP Distributions
  • Indirect elicitation of the shape parameter n
  • The PERT “controversy” via an illustrative example
    • Sources for the PERT “controversy” in Sec. 4.3.2
    • Attempts to improve the mean and variance estimation using expert judgment
  • Indirect elicitation of the parameters a, b and n
    • Description of the numerical algorithm
    • Assessment of the effect of the elicitation procedure in a PERT example
    • Some mathematical details regarding the algorithm in Sec. 4.3.3.1

• Concluding Remarks

The following sections are included:

• Illustrative Example

• The Functional Form of the Generalized Trapezoidal Density
  • Construction of the probability density function
  • Mixing behavior of the component density functions

• Basic Properties of the Generalized Trapezoidal Distribution
  • Cumulative distribution function
  • Moments

• Concluding Remarks

The following sections are included:

• Motivation

• Derivation of UTSP Family by a Single Limiting Operation

• Some Properties of the Uneven STSP Distribution

• ML Estimation Procedure for USTSP Distributions
  • Mathematical details of the ML estimation procedure
    • Step 1: maximizing over the LHS power parameter n₁
    • Step 2: maximizing over the RHS power parameter n₃
    • Step 3: maximizing over the jump parameter α
    • Step 4: maximizing over the threshold parameter θ

• Illustrative Example

The following sections are included:

• Introduction

• Cumulative Distribution Function and Density Function

• Properties of Standard RGTL Distributions
  • Limiting distributions
  • Some stochastic dominance properties
  • Mode analysis of standard RGTL distributions
  • Failure rate function
  • Cumulative moments
  • Inverse cumulative distribution function

• Maximum Likelihood Estimation of SRGTL Parameters

• Fitting 2001 US Household Income Distribution Data

The following sections are included:

• Standard Two-Sided Families of Distributions

• The Two-Sided (Linear) Slope Distribution
  • Moment estimation for ST-SS distributions
  • Maximum likelihood estimation of ST-SS parameters

The following sections are included:

• Epilogue

• Appendix A: Graphical Overview of Continuous Univariate Families of Distributions possessing a Bounded Domain

• Appendix B: The Johnson S_B Distribution
  • Motivation and Representation
  • Some Properties of the Johnson S_B Family
    • Median value
    • Mode analysis
    • Moments and parameter estimation
    • Some thoughts on limiting distributions
  • Concluding Remarks

• Bibliography

• Author Index

• Subject Index