Halphen Distribution System. II: Parameter and Quantile Estimation

In this paper, the second in a series of two, procedures for the estimation of parameters and quantiles of the Halphen type A, B, and IB distributions are presented. Because the Halphen distributions are members of the exponential class of distributions, parameters can be estimated from sufficient statistics, and maximum likelihood estimators should possess certain optimality characteristics. In some cases, the maximum likelihood system of equations does not allow a solution, and the limiting forms of the Halphen distributions, the gamma and inverse gamma, should alternatively by considered. The asymptotic variance of parameter estimators may be obtained by inverting the Fisher information matrix. The asymptotic variance of quantile estimators is obtained by classical first-order approximations. Practical experience with fitting the Halphen distributions is reported.