In this article we consider the problem of estimating the parameters of the Fréchet distribution from both frequentist and Bayesian points of view. First we briefly describe different frequentist approaches, namely, maximum likelihood, method of moments, percentile estimators, L-moments, ordinary and weighted least squares, maximum product of spacings, maximum goodness-of-fit estimators and compare them with respect to mean relative estimates, mean squared errors and the 95% coverage probability of the asymptotic confidence intervals using extensive numerical simulations. Next, we consider the Bayesian inference approach using reference priors. The Metropolis-Hasting algorithm is used to draw Markov Chain Monte Carlo samples, and they have in turn been used to compute the Bayes estimates and also to construct the corresponding credible intervals. Five real data sets related to the minimum flow of water on Piracicaba river in Brazil are used to illustrate the applicability of the discussed procedures.

The Generalized gamma (GG) distribution plays an important role in statistical analysis. For this distribution, we derive non-informative priors using formal rules, such as Jeffreys prior, maximal data information prior and reference priors. We have shown that these most popular formal rules with natural ordering of parameters, lead to priors with improper posteriors. This problem is overcome by considering a prior averaging approach discussed in Berger et al. [Overall objective priors. Bayesian Analysis. 2015;10(1):189-221]. The obtained hybrid Jeffreys-reference prior is invariant under one-to-one transformations and yields a proper posterior distribution. We obtained good frequentist properties of the proposed prior using a detailed simulation study. Finally, an analysis of the maximum annual discharge of the river Rhine at Lobith is presented.

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In this paper we propose to make Bayesian inferences for the parameters of the Lomax distribution using non-informative priors, namely the Jeffreys prior and the reference prior. We assess Bayesian estimation through a Monte Carlo study with 500 simulated data sets. To evaluate the possible impact of prior specification on estimation, two criteria were considered: the bias and square root of the mean square error. The developed procedures are illustrated on a real data set.

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