On a Combination of Gamma and Generalized Error Distributions with Application to Aircraft Flight Path Deviations

Abstract

The probability distribution of certain rare events, such as large flight path deviations of aircraft in air traffic management (ATM) scenarios, can be modeled by a generalized error distribution with parameter k = 0.5, intermediate between the Laplace k = 1.0 and Gaussian k = 2.0 distribution. To model not only the “tail” shape of the distribution for rare events, but also the “body” shape for frequent events, a further extension is introduced—namely, the combined Gamma and generalized error distribution. It includes as particular cases the Gamma, Laplace, and Gauss distributions. Its general properties are obtained, including the normalization constant, the variance and all other central and noncentral moments of any order, as well as the cumulative probability distribution, and its series expansion for small variable and its asymptotic expansion for large variable. As an example, a mixture of generalized error distribution and combined Gamma and generalized error distribution reasonably models the whole range of small to large flight path deviations for aircraft flying in an ATM scenario.

Key Words:

• Probability distributions
• Rare events
• Air traffic management

Mathematics Subject Classification:

• GOE05

Acknowledgments

The first author wants to acknowledge the benefit of discussion on the statistics of rare events with Prof. M.L.C. Loura. Also, we are grateful for the comments of the referee that led to better clarification of several points. The second author acknowledges the Ph.D. grant SFRH/BD/3104/2000 from “Fundação para a Cieˆncia e Tecnologia”.