A new heavy-tailed Gumbel-G family of distributions with risk measures and applications

In this study, we introduce a new heavy-tailed distribution by adding an additional parameter to the Gumbel-G family of distributions. The new distribution is named, the type I heavy-tailed Gumbel-G family of distributions. Several statistical properties including hazard rate function, quantile function, moments, moments of residual life, distribution of the order statistics, and Rényi entropy are discussed. Risk measures such as value at risk, tail value at risk, tail variance, and tail variance premium are also derived. To estimate the unknown parameters of the new distribution, we adopt the maximum likelihood estimation method and assess the consistency property via a Monte Carlo simulation. Finally, we illustrate the goodness-of-fit of the new family of distributions by fitting three real life data sets including biomedical and insurance data.