ON BAYESIAN PREDICTIVE MOMENTS OF NEXT RECORD VALUE USING THREE-PARAMETER GAMMA PRIORS

Abstract

A forecasting model of next record value proposed by Hill [1] assumes the underlying distribution F(x) is of an algebraic functional form with a shape parameter α for large x. That is, 1 − F(x) ≃Cx ^−α, for large x. In this article, we extend his model by incorporating a three-parameter Gamma prior of α to derive analytical solutions of the predictive distribution and moments of X given that X is a new record value. These closed-form formulas can be represented as ratios of moments of Gamma distributions. We apply the proposed model to a real-life data set that consists of the insured property losses of 33 catastrophes caused by tropical storms in the United States in 1995. The example illustrates the importance of incorporating prior experience and accounting for uncertainty in parameter estimation when forecasting record values. Both considerations are the main ingredients in the development of the proposed model.

Keywords:

• Bayesian forecasting
• Record values
• Long-tailed distribution
• Binomial theorem

Acknowledgments