An extension to empirical likelihood for evaluating probability weighted moments

- Nonparametric inference methods for probability weighted moments (PWM) are proposed.
- This approach extends the classical Empirical Likelihood (EL) technique.
- An asymptotic extension of a nonparametric version of Wilks theorem is derived.
- The novel EL nonparametric confidence interval estimation of the PWM is obtained.
- The proposed methodology is applied towards inference of the Gini index.

The scientific literature has addressed widely the theoretical and applied framework based on probability weighted moments (PWMs). PWMs generalize the concept of conventional moments of a probability function. These methods are commonly applied for modeling extremes of natural phenomena. We propose and examine empirical likelihood (EL) inference methods for PWMs. This approach extends the classical EL technique for evaluating usual moments, including the population mean. We provide an asymptotic proposition, extending a well-known nonparametric version of Wilks theorem used to evaluate the Type I error rates of EL ratio tests. This result is applied in order to develop a powerful nonparametric EL ratio test and the corresponding distribution-free confidence interval (CI) estimation of the PWMs. We show that the proposed method can be easily employed towards inference of the Gini index, a widely used measure for assessing distributional inequality. An extensive Monte Carlo (MC) study shows that the proposed technique provides a well-controlled Type I error rate, as well as very accurate CI estimation, that outperforms the CI estimation based on the classical schemes to analyze the PWMs. These results are clearly observed in the cases when underlying data are skewed and/or consist of a relatively small number of data points. A real data example of myocardial infarction disease is used to illustrate the applicability of the proposed method.