The arctan family of distributions: New results with applications

Abstract

In this paper, we explore the family of arctan transformation of a distribution function. We get some general properties such as those related to the right tail and scale transformation, among others. The results obtained are used to generalize the Pareto Type II (also known as Lomax) distribution, giving us a distribution with a long right-tail that admits the zero value in its support. We show some properties and provide closed-form expressions for the raw moments, the quantile function, the tail value at risk, and other analytical forms that can be helpful in financial and actuarial settings, such as the limited expected value, the mean excess function, and the integrated tail distribution. We also show three numerical illustrations including health expenditure for outpatients, automobile insurance claim size and to see how the new model works as compared to other distributions used in the applied statistical literature.