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The Continuous Rank Probability Score of a Generalized Beta-Prime Distribution and Some Special Cases

Matthew LeDuc

Abstract

This working paper describes new results in derivations of the Continuous Ranked Probability Score of a generalized beta-prime distribution and several special cases, such as the Dagum distribution and Singh-Maddala distribution. Comparison with Monte Carlo estimates is also presented.

The Continuous Rank Probability Score of a Generalized Beta-Prime Distribution and Some Special Cases

Abstract

This working paper describes new results in derivations of the Continuous Ranked Probability Score of a generalized beta-prime distribution and several special cases, such as the Dagum distribution and Singh-Maddala distribution. Comparison with Monte Carlo estimates is also presented.
Paper Structure (8 sections, 3 theorems, 31 equations, 1 table)

This paper contains 8 sections, 3 theorems, 31 equations, 1 table.

Table of Contents

  1. Introduction
  2. The CRPS of a Generalized Beta-Prime Distribution
  3. The CRPS of Several Special Cases of the Generalized Beta-Prime Distribution
  4. The CRPS of the Singh-Maddala Distribution
  5. The CRPS of the Dagum Distribution
  6. The CRPS of the Log-Logistic Distribution
  7. Numerical Validation
  8. Conclusion

Key Result

Proposition 2.1

Suppose that the predictive distribution is a generalized Beta-prime distribution with finite mean $\mu$. Then given observation $y$ the CRPS of the predictive distribution is given by where $B(x;\alpha,\beta)$ is the incomplete Beta function and $w=\frac{y^p}{q^p+y^p}$.

Theorems & Definitions (6)

  • Proposition 2.1
  • proof
  • Corollary 3.1
  • proof
  • Corollary 3.2
  • proof