A framework for Cyber risk has been provided by the IMF including quantitative modeling in the work proposed by Bouveret (2018). The table below summarizes the modeling results from a similar application applied to a medium-sized bank using a range of distributions.

 

a) State the null hypotheses tests used to evaluate whether a Poisson or a negative binomial fit the data and explain the results (parameter estimates) reported in the table above. 

b) What are the properties of the Poisson and negative binomial distributions 

c)  The table does not provide results for the goodness of fit test for the log-normal distribution. Propose a test that you may use and discuss its properties in detail. Consider LFHS events in answering the question.

Maximum likelihood Estimation Distribution Poisson Negative Binomial Log-Normal parameter A n p H O p-values 0.28 Lower bound 0.04 8.33 5.88 0.47 The goodness of Fit - Pearson's Chi-Squared Test Distribution Poisson Negative Binomial 9.96 2.44 MLE 9.25 6.09 0.48 11.01 3.08 Upper bound 12.60 6.12 0.48 12.63 3.72