It may seems a very silly question for you but it is still quite unclear to me:
What's the meaning of the parameters 'a' and 'b' presented in the Heffernan & Tawn 2004 multivariate model http://onlinelibrary.wiley.com/doi/10.1111/j.1467-9868.2004.02050.x/abstract ?
What they do represent? Are they the location 'a' and scale 'b' parameters?
thank you very much
The quantities $\mathbf{a}_{|i}$ and $\mathbf{b}_{|i}$ in the article both represent a collection of $d-1$ functions $a_{j|i}$ and $b_{j|i}$, for $j \neq i$ in relation with a random vector $\mathbf{Y} =[Y_i]_i$ with length $d$. The functions $a_{j|i}(y_i)$ and $b_{j|i}(y_i)$ have as their argument a possible value $y_i$ for the component $Y_i$. The functions act somewhat as a regression function and a variance function in a general (non-linear) regression with response $Y_j$ and regressor $Y_i$, but the regression form holds only conditional on a large value of the regressor $Y_i$.
As done in some other later articles by the authors and their co-authors, it helps to consider the case $d=2$ and change the notations so that the random vector of interest is $[X,\,Y]$. Then under ``suitable conditions'' sketched later, the following holds: Conditional on a large value $x$ of $X$, the distribution of $Y$ tends to depend on $x$ only through a location parameter $a(x)$ and a scale parameter $b(x) > 0$. In other words the r.v. $Z := [Y - a(X)] / b(X)$ is independent of $X$ conditional on $X$ when $X$ is large, and therefore has a distribution $G(z)$ no longer depending on $X$. To a certain extend, $Z$ can be compared to the unobservable error term in a regression-like context, although it does not necessarily have mean zero and although its distribution is unknown and unspecified.
One possible ``suitable condition'' is that $X$ and $Y$ both have an exponential tail, or equivalently a Gumbel tail. The magical thing is that for a large number of dependence models used in practice, only one special form of functions $a(x)$ and $b(x)$ exists, depending on a small number of parameters. For the usual non-negative tail dependence, a general form is $$ a(x) = \alpha x; \quad b(x) = x^\beta $$ with $\alpha \leqslant 1$ and $\beta < 1$. In practice we can first transform $X$ and $Y$ so that they have an exponential tail, using a univariate tail estimation. By using a first guess for the distribution $G(z)$ e.g. normal, we can find estimates of the parameters $\alpha$ and $\beta$ which control the tail dependence, and then guess $Z$ and find a non-parametric estimate of $G(z)$.
