Remark(Motivation)
  1. Use \(F\) for extreme value distribution \[ F(u)= \exp\{-\exp(-u)\}. \] Two-parameter extreme value distribution (Gumbel distribution) is \[\begin{eqnarray*} &&F(x;a,b)=\exp\left\{-\exp\left(-\frac{x-a}{b}\right)\right\}, \mbox{ }\mbox{ } -\infty<a<\infty,\\ &&E(X)=a+b(0.577), \mbox{ } \sigma(X)=\frac{\pi b}{\sqrt{6}}. \end{eqnarray*}\]

The binary GLM has form

\[\begin{eqnarray*} &&\pi_i=\exp\{-\exp(\sum_j \beta_j x_{ij})\}\\ &\iff& \log(-\log \pi_i) =\sum_j \beta_j x_{ij} \mbox{ : log-log link,}\\ &\mbox{ }& \log(-\log (1-\pi_i))= \sum_j \beta_j x_{ij} \mbox{ : complementary log-log link.} \end{eqnarray*}\]

  

Remark(Interpretation)￩

For the two complementary log-log model, we have \(\log\left\{-\log (1-\pi_i)\right\}\)-\(\log\left\{-\log (1-\pi_h)\right\}=\beta'(x_i-x_h)\). So \[ \frac{\log(1-\pi_i)}{\log(1-\pi_h)}=\exp\{ \beta'(x_i-x_h)\}\\ \iff 1-\pi_i= (1-\pi_h)^{\exp\{\beta'(x_i-x_h)\}}. \] So, \(P(\text{failure})\) at \(x_i\) is power \(\exp\{\beta'(x_i-x_h)\}\) of \(P(\text{failure})\) at \(x_h\).

  

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