3.10 Tail empirical process
4.1.1 Influence Function of the Huberized Tail Index Estimator . 127
Hampel (1968) introduced the Influence Function in order to measure the quanti-tative robustness of estimators which can be written as functionals with respect to the empirical distribution function.
Definition 4.3 (Influence function).
The influence function of a functional T at the distribution F is defined as IF(x;T, F) := lim
ε→0
T((1−ε)F +εδx)−T(F)
ε , (4.6)
where δx is the Dirac measure. It describes the influence of an infinitesimal contamination of F by a point mass at x on T. Observe that IF(x;T, F) is the Gˆateaux derivative of T at F in the direction of δx.
The Influence Function can be used to approximate the bias causes by a contami-nation of the model distributionF by a fractionε ∈(0,0.5) of some distribution
F1. The first order approximation of T((1−ε)F +εF1) is then given by T((1−ε)F +εF1)≈T(F) +ε
Z
IF(x;F, T)dF1(x) provided R
IF(x;F, T)dF1(x) exists and the corresponding remainder term is negligible.
Moreover, the Influence Function allows to describe qualitative robustness prop-erties of an estimator by some characteristic quantities. For instance, Hampel (1968) defined the so-called gross error sensitivity by
γ∗ = sup
x
|IF(x;F, T)|.
For a bounded Influence Function we haveγ∗ <∞and the underlying estimator is called B-robust, which stands for bias-robust. That means that the linear approxi-mation of the bias caused by an arbitrary (infinitesimal) contamination is bounded.
The Influence Function of an M-functional has the following striking property.
Corollary 4.2 (Hampel 1968).
Let Tψ be some M-functional (associated with ψ) and set λF(t) =R
ψ(x, t)dF(x).
Then the Influence Function of Tψ is given by
IF(x;F, Tψ) =−ψ(x, t)|t=Tψ(F) λ0F(t)|t=Tψ(F) provided λ0F(t)|t=Tψ(F)6= 0.
Note that in case of an M-functional theIF is proportional to ψ. Hence, the prop-erties ofψ translate directly to the Influence Function determining the qualitative robustness of the corresponding M-estimator. This allows to design estimators with desired robustness properties by choosing a suitable ψ-function.
In case of tail index estimation we are not primarily interested in B-robust estimators but mainly in estimators which are less sensitive to deviations from the ideal model at low quantiles, since heavy-tailed distributions typically exhibit deviations from Pareto distribution in this area. The Influence Function of the Pareto MLE is given by
IF(x;Fpa,γ0, TM L) = log(x)−1/γ,
1 2 3 4 5 6 7
Figure 4.2: Influence Function of THub(v,∞) for v ∈ {−1,−0.5,0,0.25} together with Pareto density(left)and a contaminated Pareto density (right). The corresponding huberized region is highlighted.
while the Influence Function of the Huberized Tail Index Estimator is given by IF(x;Fpa,γ0, THub) = [γ0−1log(x)−1]uv −(v+e−(v+1)−e−(u+1))
γ0−1((v+ 2)e−(v+1)−(u+ 2)e−(u+1)) .
In order to reduce the sensitivity ofTM L at low quantiles we choose u=∞ and v > −1. It should be mentioned that bounding the ψ-function of the MLE in this way has two additional effects on the resulting Influence Function. First, the correction term which makes sure that the corresponding estimator is a well-defined parametric M-estimator of the Pareto family {Fpa,γ, γ > 0}, causes in general a horizontal shift of ψM L. Note that for u=∞ and v >−1 this shift will be positive. The second effect is the change of the slope in the area whereψM L is not huberized. It is caused by the change of ∂γ∂ λ(v,u)F
for anyv >−1. Thus,−∂γ∂ λ(v,∞)F
pa,γ0(γ) is decreasing in v, meaning that the slope of the corresponding Influence Function is increasing inv. Therefore, Huberization at low quantiles leads in fact to an increase of weights for extremely large observa-tions, since the Influence Function withv >−1 will outrun the Influence Function of Pareto MLE. The behavior of influence functions foru= ∞and different lower thresholds v is illustrated by Figure 4.2.
Deviations from the model distribution at lower quantiles get a smaller weight in theψv,∞-function than in the ψM L-function and therefore have a smaller effect on the resulting estimate. To illustrate the last point, we consider the gross error model
Fε,γ0,γ1(x) = (1−ε)Fpa,γ0(x) +εFpa,γ1(x), x≥1, γ0 > γ1.
Considering an i.i.d. sampleX1, . . . , Xn with common distributionFε,γ0,γ1, the Hu-berized Tail Index Estimator ˆγHub,n(v,∞) estimates the quantityγ(v,∞) :=γ(v,∞)(γ0, γ1, ε), defined as an implicit solution of
λFε,γ0,γ1(γ) = Z
ψv,u(x, γ) dFε,γ0,γ1(x) = 0.
Forv =−1 the solution of λFε,γ
0,γ1(γ) = 0 is explicitly given by γ(−1,∞) = (1−ε)γ0+εγ1 =γ0+ε(γ1−γ0)< γ0,
providedε >0 and γ0 > γ1. Thus, the corresponding asymptotical bias is given by
γ(−1,∞)−γ0 =ε(γ1−γ0).
Unfortunately,γ(v,∞) withv >−1 can be derived only numerically. Nevertheless, Figure 4.3 illustrates the results for v = 0 and v = −1 (Pareto MLE) with γ0 = 2, ε∈ {0.05, 0.1, 0.2} and various values of γ1 ∈ [0.5,2]. Note that there is a considerable bias reduction using the Huberized Tail Index Estimator in comparison to the Pareto MLE. Moreover, Figure 4.3 contains the first order approximation of γ(v,∞) given by
ˆ
γ(v,∞)=γ0+ε Z
IF(x, Fpa,γ0, THub)dFpa,γ1(x) =γ0+εγ0
γ1
γ0e−
γ γ0(v+1)
−e−(v+1) (v+ 2)e−(v+1) . Note that
ˆ
γ(−1,∞) =γ0 +ε(γ1−γ0) =γ(−1,∞).
0.5 1.0 1.5 2.0
1.801.851901.952.00
γ(v,∞) for cont. Pareto distribution with γ0=2, ε= 0.05, 0.1, 0.2
γ1
γ(v,∞)
v= −1 v=0 γ
^(0,∞) ε=0.2 ε=0.1 ε=0.05
Figure 4.3: γ(v,∞) as a function ofγ1 forγ0 = 2 and three different contamination levelsε∈ {0.05,0.01,0.02}together with the first order approximation of γ(0,∞) given by ˆγ(0,∞) as a function of γ1.
Moreover, comparing γ(0,∞) with ˆγ(0,∞) reveals that the difference remains in certain bounds. Hence, in this case, the first order approximation of the bias via the Influence Function can be used to quantify the scale of the bias resulting from contaminations of the model distribution and to describe the bias reduction resulting from Huberization.
4.2 Tail index estimators based on relative ex-cesses
The main disadvantage of the Huberized Tail Index Estimator is its inconsistency ifF is not exactly Pareto. In addition, despite the parametric convergence rate, its bias in the central limit theorem is not controllable without further specification of the underlying distribution. The source of these problems is the slowly varying
function in the tail. In general the influence of this function remains present, even if we consider observations beyond some finite threshold in order to construct estimators ofγ. One way to get rid of the slowly varying function is to assume that the underlying distribution possesses exact Pareto behavior beyond some valuex0, i.e. P(X > x) = cx−1/γ for x > x0 and some c, γ >0. However, this restrictive assumption is satisfied only by a small subclass of heavy-tailed distributions.
The classical way to construct tail index estimators is to use parametric estimators of γ inFpa,γ(x) := 1−x−1/γ and to replace the original observations by relative excesses above some high threshold. This replacement is justified by the empirical MDA-conditions resulting from regular variation of the distributional tail.
Definition 4.4 (Tail index estimator based on relative excesses).
Let Y1, . . . , Yn be an i.i.d. sequence of random variables with common Pareto distribution Fpa,γ(x) = 1−x−1/γ, x ≥1 and suppose ˆγn := ˆγn(Y1, . . . , Yn) is an estimator of the Pareto parameter γ. Further, let X1, . . . , Xn be an i.i.d. sequence of random variables with common heavy-tailed distribution F ∈M DA(Hγ) and X1,n ≤ · · · ≤ Xn,n the corresponding order statistics. Then, for any 1≤ k < n, we call
ˆ
γn,k := ˆγk
Xn−k+1,n
Xn−k,n , . . . , Xn,n Xn−k,n
a tail index estimator based on relative excesses above Xn−k,n.
Typically, an intermediate number of relative excesses is used, i.e. k = k(n)→ ∞ and k/n → 0 as n → ∞. This allows to exploit the regular variation of the tail quantile function and to establish weak convergence results. We introduce some tail index estimators based on relative excesses and discuss their asymptotic properties in the following.