Bounds of Shannon entropy and extropy and their application in exploring the extreme value behavior of a large set of data

We derive bounds for the Shannon entropy and the extropy of the largest order statistic and the associated normalized sequence in the extreme value theory setting. The largest order statistic comes from a set of n independent and identically distributed random variables of finite or infinite ( n right arrow normal infinity n → ∞ n→ ∞ ) number, governed by a log-concave distribution. A characterization of the exponential distribution is provided as the model that maximizes the Shannon entropy and the extropy of the largest order statistic and the associated normalized maximum value in a large sample size regime. This characterization is exploited to provide an alternative, immediate proof of the convergence of the Shannon entropy and the extropy of the normalized maxima to the respective measures for the Gumbel distribution, studied recently for Shannon entropy by Johnson ((2024). J. Appl. Probab. 61 , 244–254.).