パレト分布

更新　The definition I used for Value at Risk was slightly inaccurate. The quantile return is the return such that the probability of a strictly worse return is equal to the probability theta, not the probability of a return of the same value or worse.

まだVaRの論文を読んでいますけど二つの点について書きます。
Firstly, I am not sure of the precise definition of the quantile function x(p), where x(p) is a return value and p a probability. Is x(p) the maximum loss (minimum return) such that the probability of a return less than x(p) is p, or is x(p) the maximum loss such that the probability of a return less than or equal to x(p) is p?
Secondly, I commented before about the use of the Generalized Pareto Distribution in the exposition of the Extreme Value Theory(EVT). However, I was unclear about how the parameters beta and xi were chosen via maximum likelihood. The derivation involved a log-likelihood computation. If the cumulative distribution function:
1-(1+(xi*x)/beta)^(-1/xi) were used, the maximum likelihood formula in the article seemed impossible to derive.
However, if one takes ln of the probability density function:
(1/beta)(1+(xi*x)/beta)^(-1/xi-1), the formula matches. It seems to make sense that one need only maximize the probability generated by the pdf. The pdf is actually derived from the cumulative distribution via a derivative, I believe.
The Pareto Distribution basically is the right half of a spike function. In the limiting case, it seems to be the right half of the Dirac delta function. It can be used to express the length distribution of jobs assigned to a supercomputer (a few long jobs take up most of the time).
f(x) = alpha*(min^alpha/x^(alpha+1)) if x>=min or 0 (x