Four strategies for bias correction of the maximum likelihood estimator of the parameters in the Type I generalized logistic distribution are studied. First, we consider an analytic bias-corrected estimator, which is obtained by deriving an analytic expression for the bias to order n ⁻¹; second, a method based on modifying the likelihood equations; third, we consider the jackknife bias-corrected estimator; and fourth, we consider two bootstrap bias-corrected estimators. All bias correction estimators are compared by simulation. Finally, an example with a real data set is also presented.

With the emergence of the COVID-19 pandemic, all existing health protocols were tested under the worst health crisis humanity has experienced since the Black Death in the 14th century. Countries in Latin America have been the epicenter of the COVID-19 pandemic, with more than 1.5 million people killed. Worldwide health measures have included quarantines, border closures, social distancing, and mask use, among others. In particular, Chile implemented total or partial quarantine measures depending on the number of infections in each region of the country. Therefore, it is necessary to study the effectiveness of these quarantines in relation to the public health measures implemented by government entities at the national level. The main objective of this study is to analyze the effectiveness of national- and region-level quarantines in Chile during the pandemic based on information published by the Chilean Ministry of Health, and answers to the following question are sought: Were quarantine measures in Chile effective during the COVID-19 pandemic? The causal effect between the rates of COVID-19 infections and the population rates in Phase 1 and Phase 2 quarantines in the period from March 2020 to March 2021 in different regions of Chile were evaluated using intervention analyses obtained through Bayesian structural time series models. In addition, the Kendall correlation coefficient obtained through the copula approach was used to evaluate the comovement between these rates. In 75% of the Chilean regions under study (12 regions out of a total of 16), an effective Phase 1 quarantine, which was implemented to control and reduce the number of cases of COVID-19 infection, was observed. The main regions that experienced a decrease in cases were those located in the north and center of Chile. Regarding Phase 2, the COVID-19 pandemic was effectively managed in 31% (5 out of 16) of the regions. In the south-central and extreme southern regions of Chile, the effectiveness of these phases was null. The findings indicate that in the northern and central regions of Chile, the Phase 1 quarantine application period was an effective strategy to prevent an increase in COVID-19 infections. The same observation was made with respect to Phase 2, which was effective in five regions of northern Chile; in the rest of the regions, the effectiveness of these phases was weak or null.

A generalized gamma (GG) distribution of four parameters was first introduced by Amoroso 1925, and since then, different distributions emerged as subclasses of this model. This model is commonly used to model lifetime data or data with a right skewed unimodal density function. In this article, we use a reparameterization of the GG distribution that is compared with other usual two-parameter distributions, Weibull, generalized exponential (Gupta and Kundu 1999), and gamma, using a real data set with a high coefficient of asymmetry and kurtosis (Valenzuela M. 2009). Akaike's information criterion and Bayesian information criterion indicates that our reparametrization of the gamma distribution is better. Besides a Monte Carlo simulation study, shows the behavior of five estimation methods: least squared, weighted least squared, moments, probability weighted moments and maximum likelihood methods.