I'm working through a Power-Point presentation about extreme value theory with application to finance. My question is about a technique to calculate the confidence interval of a $k$ $n$-block return level $R_{n,k}$.

Suppose we have divided our data in $m$ blocks, each block of size $n$. We denote with $M^i_n$ the maximum of the $i$-th block. The Fisher-Tippett theorem tells us that we should model the distribution function of such a $M^i_n$ with a GEV distribution $H_{\xi,\mu,\sigma}(x)$. Suppose this is done using the likelihood approach, i.e. we get $H_{\hat{\xi},\hat{\mu},\hat{\sigma}}(x)$. The $R_{n,k}$ is defined to be $P(M_n>R_{n,k})=\frac{1}{k}$. As an approximation we use:

$$R_{n,k}\approx H^{-1}_{\hat{\xi},\hat{\mu},\hat{\sigma}}(1-\frac{1}{k})$$

Now in one example we use some data of the S&P 500 with $n=260$ and $k=40$ with the result

$$R_{260,40}\approx H^{-1}_{\hat{\xi},\hat{\mu},\hat{\sigma}}(1-\frac{1}{40})=6.8$$

which is a estimate of $R_{260,40}$. On the slides there is a sentence, I quote: "It is important to construct confidence intervals for such statistics. We use asymptotic likelihood ratio ideas to construct asymmetric intervals – i.e. the profile likelihood method."

How exactly is this confidence interval for $R_{260,40}$ obtained?


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