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Burr distribution is Burr Type XII distribution which is one among the twelve types of the continuous distributions in Burr system. It has two positive shape parameters, namely k and c. It is implied from the probability density function which can be either decreasing or unimodal, and the hazard rate function which can be either decreasing or upside-down bathtub-shaped. The other distributional properties and the moment properties of Burr distribution will be discussed in more detail. By considering these properties, we will study its tail behaviour. To estimate the parameters k and c, the maximum likelihood method will be considered. Based on the properties of the data representing the remission time of bladder cancer patients, we infer that Burr distribution is suitable to model the data. The goodness-of-fit using the Kolmogorov–Smirnov test shows that Burr distribution fits well to the data.

This paper aims to compare the safe-haven roles of gold and Bitcoin for energy commodities, including oils and petroleum, during COVID-19. Specifically, we examine the presence of reduction in downside risk after mixing gold/Bitcoin with such energy commodities. To do this, we account for dependence among energy commodities and gold/Bitcoin returns by applying a (vine) copula. The findings show that gold substantially reduces the downside risk of a portfolio containing any allocation to gold and energy commodities, indicating its safe-haven ability. In contrast, Bitcoin’s safe-haven functionality is inconsistent since the downside risk reduction is achieved for Bitcoin’s small allocation only.

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Risk in finance may come from (negative) asset returns whilst payment loss is a typical risk in insurance. It is often that we encounter several risks, in practice, instead of single risk. In this paper, we construct a dependence modeling for financial risks and form a portfolio risk of cryptocurrencies. The marginal risk model is assumed to follow a heteroscedastic process of GARCH(1,1) model. The dependence structure is presented through vine copula. We carry out numerical analysis of cryptocurrencies returns and compute Value-at-Risk (VaR) forecast along with its accuracy assessed through different backtesting methods. It is found that the VaR forecast of returns, by considering vine copula-based dependence among different returns, has higher forecast accuracy than that of returns under prefect dependence assumption as benchmark. In addition, through vine copula, the aggregate VaR forecast has not only lower value but also higher accuracy than the simple sum of individual VaR forecasts. This shows that vine copula-based forecasting procedure not only performs better but also provides a well-diversified portfolio.