![]() Hill‘s estimator is based on the assumption that the denominator above is almost 1 (which means that, as ), i.e. The estimator of the slope is (considering only the largest observations) since a natural estimator for is the order statistic, the slope of the straight line is the opposite of tail index. Equivalently, the exists a slowly varying function such that. , so that, where is some slowly varying function. Consider some heavy tailed distribution, i.e. The slope here is somehow related to the tail index of the distribution. Residual standard error: 0.1299 on 99 degrees of freedom ![]() The slope can be obtained using a linear regression, > B= ame(X= log(Xs),Y= log((n:1)/(n+1))) (which is close to the relation we derived using a GPD model). if we invert that function, we derive an estimator for the quantile function Since is a slowly varying function, it seem natural to assume that this ratio is almost 1 (which is true asymptotically). Assume here that, where is a slowly varying function. + u-beta/xi+beta/xi/(1-xi)*(100/2.5*(1-p))^(-xi)Īn alternative is to use Hill’s approach (used to derive Hill’s estimator). If we substitute estimators to unknown quantities on that expression, we get Thus, assuming again that above (and therefore above that high quantile) a GPD will fit, we can write So that we recognize the mean excess function (discussed earlier). Note that it is also possible to derive an estimator for another population risk measure (the quantile is simply the so-called Value-at-Risk), the expected shortfall (or Tail Value-at-Risk), i.e. This is similar with the following outputs, with the return period of a yearly event (one observation out of 250 trading days) > gpd.q(tailplot(gpd(X,quantile(X.975))), 1-1/250, ci.type = large values of the opposite of log returns, plotted below) The code we wrote last week was the following (here based on log-returns of the SP500 index, and we focus on large losses, i.e. we keep the largest observations to fit a GPD, then this estimator can be written We have seen last week that, if we assume that above a threshold, a Generalized Pareto Distribution will fit nicely, then we can use it to derive an estimator of the quantile function (for percentages such that the quantile is larger than the threshold) This week, we conclude the part on extremes with an application of extreme value theory to risk measures. So that the (empirical) mean of those estimator isīias computer Extremes Hill MAT8595 normality R-english regular variation second order Again, consider thousands of samples, and let us see how Hill’s estimator is behaving, Here, with > n=500īut it’s based on one sample, only. First, let us consider a Pareto survival function, and the associated quantile function > alpha=1.5 In order to illustrate this point, consider the following code. Further, under additional technical conditions ![]() (under additional assumptions on the rate of convergence, it is possible to prove that ). Then we did say that satisfies some consistency in the sense that if, but not too fast, i.e. To be more specific, we did see see that if, with, then Hill estimators for are given byįor. In the MAT8595 course, we’ve seen yesterday Hill estimator of the tail index.
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