Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk

9.4.3 PARAMETER ESTIMATION FOR THE G-AND-H SEVERITY IN OPRISK

There have been many proposed methods for performing parameter estimation in the g-and-h family of distributions. In this section, we survey a few of the possibilities. Dutta and Babbel (2002) suggest a method of parameter estimation for constant g and h parameters based on percentile matching. This involves recognizing the relationships between data percentiles and the g-and-h parameters given in Proposition 9.6.

Proposition 9.6 (g-and-h Distribution Percentile Matching Estimation)

Consider a g-and-h distributed random variable X ~ GH (a, b, g, h) and a sample of n loss data points with order statistics that will be used to fit the g-and-h distribution. Then the location parameter is given by the following percentile:

(9.53)

Then the value of the g parameter is given by the following estimator (based on the p-th percentile given by p = for one or more i ∈ {1,2,…, n}) given by

(9.54)

with W_p = inf{w : Φ(w) > p} the p-th percentile of a standard Normal and X_p = inf{x : F(x) > p} denotes the p-th percentile of the sample ...