You have a data set to which you want to fit the Gumbel distribution, which is an extreme value

distribution. The Gumbel distribution Gumbel $(,)$ is a two$-$parameter family with location

parameter $$ and scale parameter $.$ Its cdf is given by

$F(x$;$,)$:$=e^{-e^{-x}},$ where $,z=\frac{x-}{}$

$4$

Furthermore, this distribution has mean $$ and variance ${}^2$ given by

$=+,{}^2=\frac{{}^2}{6}{}^2$

where $$~~$0\mathrm{.}57721756$ is the Euler$-$Mascheroni constant.

$($a$)$ Compute a formula for the pdf $f(x$;$,)$ of this distribution. Hence write a function

Gumbel$\_$pdf $(x,$ alpha, beta$)$ which computes this density at a vector of values $x.$ Use this

function to write a function Gumbel$\_$LL$(x,$ alpha, beta$)$ which computes the log$-$likelihood

of the data $x$ as a function of the parameters alpha, beta.

$($b$)$ Download the file QM$\_$Exam$\_$Gumbel$\_$data.csv$,$ which contains $400$ draws from a certain dis$-$

tribution. You suspect this distribution may be modelled by the Gumbel $(,)-$distribution.

Write a matlab script Q$3.$m which does the following:

$($i$)$ Computes and displays method$-$of$-$moments estimates alpha$\_$MM$,$ beta$\_$MM of $,.$

$($ii$)$ Uses maximum likelihood estimation to compute and display a set of point estimates

alpha$\_$MLE, beta$\_$MLE of alpha, beta.

Here, employ the matlab function fminsearch to perform the optimization, with the

point $[$alpha$\_$MM $,$ beta$\_$MM$]$ as starting point.

$($iii$)$ Computes and displays $95\%-$confidence intervals $[$L$\_$alpha, U$\_$alpha$]$ and $[$L$\_$beta,

U$\_$beta$]$ using the pivotal bootstrap method. Here, use B $=500$ bootstrap simula$-$

tions.

$($iv$)$ Plots a Q$-$Q plot of the data $x$ against a vector $Y$ consisting of $1000$ Gumbel$($alpha$\_$MLE,

beta$\_$MLE$)-$distributed random numbers.

$[$Hint: In order to generate $Y,$ recall that if $F$ is a cdf and $U$ is a uniformly distributed

random number on $0,1,$ then the random number $F^{-1}(U)$ is distributed according to

$F.]$