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b/notebooks/tex_source/sections/deriving_theorem_two.tex index 91439b0..dc32d8f 100644 --- a/notebooks/tex_source/sections/deriving_theorem_two.tex +++ b/notebooks/tex_source/sections/deriving_theorem_two.tex @@ -1,4 +1,4 @@ -Use the approximation $\ln(1+x) \approx 1 + x$ for $|x| \ll 1$ and $F(z) \approx +Use the approximation $\ln(1+x) \approx x$ for $|x| \ll 1$ and $F(z) \approx 1$ for large enough $z$ to derive. diff --git a/notebooks/tex_source/sections/peaks_over_thresholds.tex b/notebooks/tex_source/sections/peaks_over_thresholds.tex index dfa4577..f842c13 100644 --- a/notebooks/tex_source/sections/peaks_over_thresholds.tex +++ b/notebooks/tex_source/sections/peaks_over_thresholds.tex @@ -24,9 +24,10 @@ \textbf{Q:} What should $u$ and the data set fulfill in order for the above approximation to be accurate? -\textbf{A:} It should be small enough such that many data points are larger than it. -Then the approximation in $P(X>u) \approx \frac{N_u}{N}$ holds (the estimator is -not too biased). +\textbf{A:} +% It should be small enough such that many data points are larger than it. +% Then the approximation in $P(X>u) \approx \frac{N_u}{N}$ holds (the estimator is +% not too biased). \hrulefill\\* @@ -67,11 +68,12 @@ \textbf{Q:} Intuitively, what does $u$ need to fulfill for both approximations to hold? -\textbf{A:} $u$ should be small enough such that the approximation $P(X>u) \approx -\frac{N_u}{N}$ holds and sufficiently large such that the generalized pareto -distribution is a good estimate of the tail of the distribution for values -larger than $u$. Intuitively, it should be at the *beginning of the tail*, where -for values larger than $u$ only the tail behavior plays a role - i.e. no more -local extrema or other specifics of the underlying distribution of the data. +\textbf{A:} +% $u$ should be small enough such that the approximation $P(X>u) \approx +% \frac{N_u}{N}$ holds and sufficiently large such that the generalized pareto +% distribution is a good estimate of the tail of the distribution for values +% larger than $u$. Intuitively, it should be at the *beginning of the tail*, where +% for values larger than $u$ only the tail behavior plays a role - i.e. no more +% local extrema or other specifics of the underlying distribution of the data. \hrulefill\\* diff --git a/notebooks/tex_source/sections/taking_a_look_back.tex b/notebooks/tex_source/sections/taking_a_look_back.tex index 5426809..f185613 100644 --- a/notebooks/tex_source/sections/taking_a_look_back.tex +++ b/notebooks/tex_source/sections/taking_a_look_back.tex @@ -3,7 +3,15 @@ us about the quality of the fit? -Well, the truth is, not too many. First notice the following exact equality: +Well, the truth is, not too many. + +Remember that the block-maximum is defined as + +\begin{equation} + M_n \coloneqq \max\{ X_1, X_2, \dots, X_n \} +\end{equation} + +First notice the following exact equality: \begin{equation}