1. The generalised Pareto distribution is useful in modelling extreme value problems or rare events. For example, it can be used to model geophysical phenomena such as floods or extreme windstorms. A random variable X is said to have a generalised Pareto distribution GP(8), where # >0, if its cumulative distribution function is Fx(x)=1-(1+1), :>0. Let X1. .... X, be a random sample on GP(0) and I1, ...; I, be the corresponding ob- servations. (a) Obtain the distribution of Y, := = In(1+ X,) and show that Em(1+X,) is x (2n). 1-1 (b) Obtain the maximum likelihood estimator, call it 7, and deduce a 100(1 - 0)% two-sided confidence interval for & based on the maximum likelihood estimator. You may need to use the result in (a). (c) The sample minimum is the smallest observation in the sample. Denote it as m and let M be the corresponding random variable, i.e. M = min(X1, ..., X,). Obtain the cumulative distribution function of M and deduce a 100(1 - o)% two-sided confidence interval for # based on the sample minimum. (d) A sample of 7 observations on GP() gave: 6.30260 0.53457 0.97925 5.57901 2.08050 1.08441 54.2865 Compute the two 95% confidence intervals described above (in (b) and (c)) for this data set. Comment on the relative sizes of the two confidence intervals. (e) Compute () := E[M] for d do at a significance level of a based on T. (g) Construct a test of hypothesis for Ho : # = do against the alternative H1 : d > do at a significance level of a based on M - observe that A tends to increase as o increases