Abstract

In this paper, the authors introduce a new three-parameter lifetime probability distribution known as the Marshall-Olkin-generated log-logistic (LL) distribution. They thoroughly examine and describe this distribution, providing insights into its characteristics and its suitability for various applications. This newly constructed distribution's density function exhibits characteristics of both symmetry and right-skewness, providing modelling flexibility across a range of datasets. Because of its skewness coefficient, which can take both positive and negative values, a wide range of data asymmetries can be represented. The Marshall-Olkin generated LL distribution's corresponding hazard rate displays a variety of characteristics, including monotonic increase, increasing-constant, constant, upside-down, and monotonic drop. Because of its adaptability, the distribution can successfully capture various risk or failure rate patterns across time. Using a number of techniques, the researchers expand this distribution to the bivariate domain. Its utility in modelling multivariate lifetime data and inter-variable relationships is improved by these extensions. The researchers use the maximum likelihood method to estimate the parameters of the distribution, which ensures reliable and effective parameter estimation from observed data. They carry out an extensive simulation research to analyse biases and mean squared errors in a range of scenarios and sample sizes in order to evaluate the finite behaviour of the maximum likelihood estimators. In real-life and reliability applications, this meticulous methodology aids in evaluating the estimators' precision and dependability. Because it may offer a comprehensive and nuanced knowledge of high financial risks, the Peaks Over a Random Threshold Value-at-Risk (PORT-VaR) study is crucial for evaluating Norwegian fire insurance claims. This financial analysis is given extra consideration.