I work on estimation of income distributions. When I have very high inequality situation I have these two options: Estimate a conventional distribution (e.g. generalize beta) where the parameters go to values that make the second order moment infinite [one issue is then how to do inference] but I can also estimate other distributions (e.g. mixture of lognormals) that can handle fat tails while keeping second order moment finite. My question is which approach is better and why.