On discriminating between lognormal and Pareto tail: an unsupervised mixture-based approach

Many stochastic models in economics and finance are described by distributions with a lognormal body. Testing for a possible Pareto tail and estimating the parameters of the Pareto distribution in these models is an important topic. Although the problem has been extensively studied in the literature, most applications are characterized by some weaknesses. We propose a method that exploits all the available information by taking into account the data generating process of the whole population. After estimating a lognormal-Pareto mixture with a known threshold via the EM algorithm, we exploit this result to develop an unsupervised tail estimation approach based on the maximization of the profile likelihood function. Monte Carlo experiments and two empirical applications to the size of US metropolitan areas and of firms in an Italian district confirm that the proposed method works well and outperforms two commonly used techniques. Simulation results are available in an online supplementary appendix.