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\rightarrow \infty $) 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.).

Log-concavity of a joint survival function is proposed as a model for bivariate increasing failure rate (BIFR) distributions. Its connections with or distinctness from other notions of BIFR are discussed. A necessary and sufficient condition for a bivariate survival function to be log-concave (BIFR-LCC) is given that elucidates the impact of dependence between lifetimes on ageing. Illustrative examples are provided to explain BIFR-LCC for both positive and negative dependence.

Goldman (2010) proved that the distribution of a stationary determinantal point process (DPP) Φ can be coupled with its reduced Palm version Φ0,! such that there exists a point process η where Φ=Φ0,!∪η in distribution and Φ0,!∩η=∅. The points of η characterize the repulsive nature of a typical point of Φ. In this paper we use the first-moment measure of η to study the repulsive behavior of DPPs in high dimensions. We show that many families of DPPs have the property that the total number of points in η converges in probability to 0 as the space dimension n→∞. We also prove that for some DPPs, there exists an R∗ such that the decay of the first-moment measure of η is slowest in a small annulus around the sphere of radius √nR∗. This R∗ can be interpreted as the asymptotic reach of repulsion of the DPP. Examples of classes of DPP models exhibiting this behavior are presented and an application to high-dimensional Boolean models is given.

Customers arrive sequentially at times x 1 < x 2 < · · · < x n and stay for independent random times Z 1, …, Z n > 0. The Z-variables all have the same distribution Q. We are interested in situations where the data are incomplete in the sense that only the order statistics associated with the departure times x i + Z i are known, or that the only available information is the order in which the customers arrive and depart. In the former case we explore possibilities for the reconstruction of the correct matching of arrival and departure times. In the latter case we propose a test for exponentiality.