The Generalized Moment-Transmuted Cauchy Distribution: A Heavy-Tailed Family with Controllable Finite Moments

<title>Abstract</title> <p>The Cauchy distribution is a fundamental heavy-tailed model whose practical use is often limited by the non-existence of finite moments. In this paper, we propose the Generalized Moment-Transmuted Cauchy (GMTC) distribution by applying a generalized survival transformation to the classical Cauchy distribution. The proposed family includes the standard Cauchy distribution as a special case and introduces an additional shape parameter that governs the tail behaviour and the existence of moments. Several distributional properties, including the probability density function, cumulative distribution function, survival function, hazard rate, quantile function, stochastic ordering, and moment conditions, are derived. Parameter estimation is carried out using the maximum likelihood method, and the asymptotic properties of the estimators are established. Simulation studies and real data applications illustrate the flexibility and usefulness of the proposed model for analysing heavy-tailed data. MSC (2020) Classifications: 62E15; 62F10; 60E05; 60G70.</p>