Modeling the vulnerabilities lifecycle and exploitation frequency are at the core of security of networks evaluation. Pareto, Weibull, and log-normal models have been widely used to model the exploit and patch availability dates, the time to compromise a system, the time between compromises, and the exploitation volumes. Random samples (systematic and simple random sampling) of the time from publication to update of cybervulnerabilities disclosed in 2021 and in 2022 are analyzed to evaluate the goodness-of-fit of the traditional Pareto and log-normal laws. As censoring and thinning almost surely occur, other heavy-tailed distributions in the domain of attraction of extreme value or geo-extreme value laws are investigated as suitable alternatives. Goodness-of-fit tests, the Akaike information criterion (AIC), and the Vuong test, support the statistical choice of log-logistic, a geo-max stable law in the domain of attraction of the Fréchet model of maxima, with hyperexponential and general extreme value fittings as runners-up. Evidence that the data come from a mixture of differently stretched populations affects vulnerabilities scoring systems, specifically the common vulnerabilities scoring system (CVSS).

The importance of tourism in the Azores’ economy has been increasing over the years. In order to respond to higher tourist demand, new accommodation establishments and restaurants started activity in the region, creating new jobs and wealth. This trend of economic expansion has been slowed down, possibly reversed, since the COVID-19 pandemic and the various strategies adopted by the government to decrease the infection rate, which led to the arrival of fewer tourists. To assess the pandemic impact after the first lockdown on the economy of São Miguel, the biggest and most visited island of the Azores, surveys were designed for the tourism accommodation and restaurant sectors, both influenced by tourism activity. The main aim was to estimate the mean income drop in the 2020 tourism high season, in percentage, compared to the homologous pre-pandemic 2019 period. The results highlight an adverse impact on those sectors, with the greatest mean income drop being estimated for the local accommodation sector ($78.7\pm 3.6\%$), followed by the traditional hotel sector ($74.7\pm 4.6\%$) and the restaurant sector ($58.5\pm 6.5\%$). Moreover, an almost 60% drop in the mean occupancy rate during the 2020 tourism high season, compared to 2019, was estimated for the tourism accommodation sector.

Inferences on the location parameter $\lambda$ in location-scale families can be carried out using Studentized statistics, i.e., considering estimators $\tilde{\lambda }$ of $\lambda$ and $\tilde{\delta }$ of the nuisance scale parameter $\delta$, in a statistic $T=g(\tilde{\lambda },\tilde{\delta })$ with a sampling distribution that does not depend on $(\lambda ,\delta )$. If both estimators are independent, then T is an externally Studentized statistic; otherwise, it is an internally Studentized statistic. For the Gaussian and for the exponential location-scale families, there are externally Studentized statistics with sampling distributions that are easy to obtain: in the Gaussian case, Student’s classic t statistic, since the sample mean $\tilde{\lambda }=\overline{X}$ and the sample standard deviation $\tilde{\delta }=S$ are independent; in the exponential case, the sample minimum $\tilde{\lambda }=X_{1:n}$ and the sample range $\tilde{\delta }=X_{n:n}-X_{1:n}$, where the latter is a dispersion estimator, which are independent due to the independence of spacings. However, obtaining the exact distribution of Student’s statistic in non-Gaussian populations is hard, but the consequences of assuming symmetry for the parent distribution to obtain approximations allow us to determine if Student’s statistic is conservative or liberal. Moreover, examples of external and internal Studentizations in the asymmetric exponential population are given, and an ANalysis Of Spacings (ANOSp) similar to an ANOVA in Gaussian populations is also presented.

Discussiones Mathematicae Probability and Statistics