Goodness of fit test statistics for the zeta family

Abstract

Goodness of fit test procedures for the zeta parametric family based on quadratic distances and the Box--Cox transform are developed. Test statistics based on quadratic distances are shown to follow a chi-square distribution asymptotically. Test procedures based on the Box-Cox transform make use of the estimator of the parameter introduced by the Box-Cox transform, and numerical computations are based on the nonlinear weighted least squares algorithms.

Estimators of the regression parameters of the zeta distribution

“…Under the composite hypothesis H 0 : β ∈ the full parameter space, it has an asymptotic χ 2 distribution with number of degrees of freedom which is equal to dim(Y ) − dim(β). The proof is analogous to the one provided in Luong and Doray (1996) for the case without covariates. In our example, the above statistic has a value χ 2 = 12.4381, smaller than the critical value χ 2 20, 0.95 = 31.4, indicating again a good fit of the zeta model with age as a covariate.…”

Discrete power law with exponential cutoff and Lotka's law

“…This method has its flaws too. Luong and Doray () point out it is efficient, but it is not robust, sensitive to outliers. Its score function depends on the sample mean, so its influence function is unbounded (Huber, ).…”