3.10 Tail empirical process
4.2.2 Hill estimator
The most popular tail index estimator was proposed by Hill (1975) and can be motivated in various ways. We start with the so-called MLE approach stated in Embrechts et al. (1997) corresponding to the original arguments used by Hill (1975). Subsequently, we demonstrate that the Hill estimator can be constructed
fromp-Quantile Tail Index Estimators introduced in the last section.
LetX be a Pareto distributed random variable with distribution function Fpa,γ for some γ >0. Then, Y := logX is exponential distributed with parameter 1/γ.
Hence, the MLE of γ is given by
ˆ
A slight generalization of the underlying distribution F concerns F(x) = Cx−1/γ, x≥u >0,
with u known, which means thatF exhibits a power tail behavior above a certain known thresholdu. In absence of precise parametric informationC =u1/γ one merely assumes that the underlying distribution behaves like a Pareto distribution above a certain known threshold u. In such cases the exact MLE ofγ is non-applicable since one is faced with the problem of a simultaneous estimation ofγ and C. Conditional MLE provides a possible solution for this problem.
LetK be the number of order statistics which lie above a certain known threshold u, i.e.
K =|{i:Xi,n> u, i = 1, . . . , n}|.
The Hill estimator can be derived as a conditional MLE of γ conditioned on the event {K = k} by maximizing the joint likelihood of (Xn−k+1,n, . . . , Xn,n). with u < xk <· · ·< x1. The conditional log-likelihood function is then given by
l(γ, C;x1, ..., xk) = log
with first partial derivatives
This yields the conditional MLEs ˆ
A comparison of the Hill estimator with ˆγM L,n,u in (4.8) shows that conditioning on the event{k =K} results in a replacement of the deterministic thresholdu by the random thresholdXn−k+1,n. Thus, the Hill estimator is often referred to as an approximated maximum likelihood estimator in the literature.
Remark. The Hill estimator is usually defined by
ˆ Note, that the slight deviation from the conditional MLE of γ is asymptotically negligible.
We provide a different approach representing the Hill estimator as a weighted average ofp-Quantile Tail Index Estimators. Therefore, define for any fixedk ∈N a sequence of probabilities pi =i/(k+ 1) fori= 1, . . . , k as well as a sequence of
SincePk
i=1log(k+1−i) =Pk
i=1log(k) = log(k!) we have, due to Stirlings formula, Pk
i=1log(k+ 1−i)∼klog(k)−k as k → ∞. This yields
k→∞lim
k
X
i=1
wk(pi) = lim
k→∞
1
k log(k+ 1)k−log(k!)
= lim
k→∞
1
k log(1 + 1/k)k+k
= lim
k→∞
1 +k k
= 1,
which justifies the use of the Hill estimator as an estimator ofγ. Note, in particular, that the number ofp-Quantile Tail Index Estimators involved in the construction of the Hill estimator tends to infinity ask → ∞. This considerably improves the asymptotic properties of the underlying estimator.
Observe that for a sequence of i.i.d. random variables Y1, ..., Yl with common Pareto distribution,Fpa,γ, the Pareto MLE is given by
ˆ
γM L,k = 1 k
k
X
i=1
log(Yi). (4.10)
Considering an i.i.d. sequence, X1, . . . , Xn, with common distribution function F ∈M DA(Hγ) and replacing the original observations (Y1, ..., Yk) in (4.10) by k relative excesses above Xn−k,n, given by
Xn−k+1,n Xn−k,n
, . . . , Xn,n Xn−k,n
, yields
ˆ
γn,k(H) = 1 k
k
X
i=1
log
Xn−k+i,n Xn−k,n
.
Hence, the Hill estimator can be considered as a tail index estimator based on relative excesses in the sense of Definition 4.4.
Theorem 4.7 (Consistency of the Hill estimator, Resnick (2007)).
LetX1, . . . , Xn be a sequence of i.i.d. random variables with common distribution function F ∈ M DA(Hγ) with γ > 0. Then as n → ∞, k = k(n) → ∞ but k/n→0,
ˆ
γn,k(H)→P γ.
Consistency of the Hill estimator was first established by Mason (1982) (see Em-brechts et al. 1997). We follow the approach stated in Resnick (2007), using the tail empirical measure and the concept of weak convergence of random measures.
Proof. Recall that 1−F ∈RV−1/γ implies ˆνn,k ⇒νγ. Following Resnick (2007) we consider the functional
T(ν) = Z ∞
1
ν(x,∞]dx x ,
defined on M+(0,∞]. In order to obtain weak convergence of T(ˆνn,k) we require the following auxiliary result:
Theorem 4.8 (Billingsley (1968), Theorem 4.2).
Suppose that {XM n, XM, Yn, X;n ≥1, M ≥1} are random elements of the metric space (S,S) and are defined on a common domain. Suppose that, for each M, as n→ ∞,
XM n ⇒XM, and as M → ∞
XM ⇒X.
Suppose further that for all ε >0,
Mlim→∞lim sup
n→∞
P(d(XM n, Yn)> ε) = 0.
Then as n→ ∞, we have
Yn⇒X.
In order to apply this theorem, define:
XM n :=
Z M 1
ˆ
νn,k(x,∞]x−1dx and
XM :=
Z M 1
νγ(x,∞]x−1dx.
Since the integration is over a finite region, we can apply the continuous mapping theorem in this case. It follows
Z M 1
ˆ
νn,k(x,∞]x−1dx⇒ Z M
1
νγ(x,∞]x−1dx= Z M
1
x−1/γ−1dx=γ(1−M−1/γ),
asn → ∞due to ˆνn,k⇒νγ. Moreover,
→P γ follows. In order to prove the mentioned condition, we will consider the event {(Xn−k,n/U(n/k))∈(1−η,1 +η)}for some η >0 and decompose the above probability as follows
P The second probability is negligible, due to
P
The upper bound can in turn be bounded using Markov’s inequality
Hence, the above probability converges to zero asM → ∞.
So far, we have proved that Yn := done by rewritingYn as
Yn= and using the identity
Z ∞
This yields Therefore, we can deduce
1
Asymptotic normality of the Hill estimator was derived by many authors (see e.g. Hall 1982, Davis and Resnick 1984, Cs¨org˝o and Mason 1985, Cs¨org˝o et al.
1985, H¨ausler and Teugels 1985, Cs¨org˝o and Viharos 1997, de Haan and Resnick 1998, de Haan and Peng 1998, among others). The following theorem corresponds Theorem 3.2.5. in de Haan and Ferreira (2006).
Theorem 4.9 (Asymptotic normality of the Hill estimator).
Suppose that U ∈ 2RVγ,ρ with γ > 0, ρ ≤ 0 and auxiliary function A with ultimately constant sign satisfying|A| ∈RVρandlimn→∞A(t) = 0. Then, provided
k(n)→ ∞, k/n→0 asn → ∞ and limn→∞
Proof. We follow the arguments of de Haan and Ferreira (2006), p.76, exploiting the second assertion in Theorem 3.10. Recall that fork → ∞,k/n→0 asn → ∞, local uniformly for 0< s≤1. Hence,
Z 1
and therefore
Second proof of asymptotic normality of the Hill estimator.
We follow the arguments in de Haan and Ferreira (2006), p. 162.
After rewriting the Hill estimator, we will use the uniform approximation of the tail empirical process stated in Proposition 3.8 to prove the asymptotic normality.
Similar to the consistency proof we obtain by integration by parts ˆ
where for the last equality we substituted s by tU(n/k). Equipped with this
We start with I. According to Theorem 3.8, we have
√
SinceXn−k,n/U(n/k)→P 1 and the supremum above is bounded, we have
√
Considering II, we can use the uniform convergence in Theorem 3.8 in order to