In order to obtain an empirical counterpart of the second-order conditionU ∈ 2RVγ,ρ and therefore be able to quantify the convergence rate in the empirical MDA-condition, it is necessary to investigate the limiting behavior of intermediate order statisticsXn−k,n. From the fact that Xn,n/U(n)→d Hγ if and only if F is heavy-tailed, one might expect that scaling Xn−k,n by its theoretical counterpart F←(1−k/n) =U(n/k) results in some nondegenerate limit distribution. However, recall that in Corollary 3.7 it was shown that Xn−k,n/U(n/k) →P 1 provided k→ ∞, butn/k →0 asn → ∞. Moreover, we obtain the following limit theorem for √
k(Xn−k,n/U(n/k)−1), which is a modified version of Theorem 2.4.1 in de Haan and Ferreira (2006).
Theorem 3.9. Let X1,n ≤ X2,n ≤ · · · ≤ Xn,n be the nth order statistics of an i.i.d. sample with common distribution F satisfying 1−F ∈ RV−1/γ. Suppose that the corresponding tail quantile function U is in 2RVγ,ρ with γ >0, ρ≤0and auxiliary function A. Then,
√ k
Xn−k,n
U(n/k)−1 d
→N(0, γ2), provided that k(n)→ ∞, k/n→0 and limn→∞
√kA(n/k) exists and is finite.
The proof is essentially based on de Haan and Ferreira (2006), p. 41-42. We slightly modify their arguments in order to meet our assumptions. First, we establish the following auxiliary result going back to Smirnov (1949).
Lemma 3.2 (Lemma 2.2.3, de Haan and Ferreira (2006) ).
Let U1,n ≤U2,n ≤ · · · ≤Un,n be the nth order statistics from a standard uniform distribution. Then, as n→ ∞, k→ ∞, n−k → ∞
Uk,n−bn an
→d N(0,1), where
bn:= k−1 n−1, an:=
r
bn(1−bn) 1 n−1.
Proof. We closely follow de Haan and Ferreira (2006), p.41.
The density of Uk,n is given by
fUk,n(x) = n! Considering the first factor yields
anbk−1n (1−bn)n−k = Further, applying Stirling’s formula forn!, we obtain
n! Note that due to
k−1 bn
=n−1 = n−k 1−bn
the highest-order terms cancel. The coefficient of x2/2 is given by (k−1)
Since the other terms are of smaller order, we have pointwise convergence of fZk,n(x) → φ(x), where φ is the density of the standard normal distribution.
The weak convergence of the probability distributions follows from Scheff´e’s Theorem.
The next corollary is a consequence of Smirnov’s Lemma.
Corollary 3.11 (Corollary 2.2.2, de Haan and Ferreira (2006)).
Let Y1,n ≤ Y2,n ≤ · · · ≤ Yn,n be the nth order statistics from a standard Pareto distribution given by Fpa,1(y) = 1−1/y, y ≥ 1. Then, provided k → ∞, but
k/n→0 as n→ ∞,
√ k
k
nYn−k,n−1
→d N(0,1).
Proof. From Smirnov’s Lemma we deduce that
√ k
n
kUk+1,n−1 d
→N(0,1), which implies n/k Uk+1,n
→P 1. Thus,
√ k
k
nUk+1,n −1
→d N(0,1).
Now, sinceYn−k,n
=d FY←(1−Uk+1,n) andFY←(1−y) = y−1 we obtain Yn−k,n
= (Ud k+1,n)−1. The assertion follows.
Proof of Theorem 3.9.
Observe that Xn−k,n
=d U(Yn−k,n), where Yi,n is the i-th order statistic from a standard Pareto distribution. Hence, we have
√ k
Xn−k,n
U(n/k)−1 d
=√
k U nknkYn−k,n
U(n/k) −1
! .
According to (3.46), a possible auxiliary function A0 for U ∈2RVγ,ρ is given by
A0(t) :=
ρ(1−lims→∞s−γU(s)/(t−γU(t))), ρ <0, 1−R1
0 s−γU(s)ds/(t1−γU(t)), ρ= 0.
In particular, note thatA0(t)∼A(t) and therefore limn→∞
√kA0(n/k) exists and
is finite, as well. Applying uniform approximations of Corollary 3.8 yields Then, according to Corollary 3.11, we get
√
kA0(n/k) is bounded by assumption, the other terms go to zero in probability and the result follows.
Remark. We offer some intuition for the fact that √
k(Xn−k,n/U(n/k)−1)tends to a nondegenerate limit by same heuristics as was used earlier to explain the limiting behavior of Xn,n/U(n). Therefore assume for the moment that F is absolute continuous with density f and that the first two moments ofXn−k,n exist.
The expected value ofXn−k,ncan be approximated byF−1((n−k)/(n+1))∼U(n/k).
Furthermore, according to Rice (1995), the variance ofXn−k,n can be approximated by the von Mises condition (Corollary 3.5), we have for F ∈M DA(Hγ) and under the additional assumption that f is ultimately monotone
t→∞lim
U(t)−1 tf(U(t)) =γ.
Thus, Since k(n) is intermediate, we have
n→∞lim
Hence, the variance of Xn−k,n increases too slowly with n such that the variance of Xn−k,n/U(n/k) collapses as n → ∞. This results in Xn−k,n/U(n/k)→P1.
Moreover, assume that there is some rate function a(n), which stabilizes the variance of Xn−k,n/U(n/k)−1 such that
This implies in particular
a2(n) suggesting a(n) =√
k. Then, the corresponding variance is approximately given by where we again make use of the von Mises condition.
The result of the Theorem 3.9 can be enlarged in the following way.
Corollary 3.12. Let X1,n≤X2,n ≤ · · · ≤Xn,n be the order statistics of an i.i.d.
sample with common distribution F ∈M DA(Hγ). Suppose that the corresponding tail quantile function U is in 2RVγ,ρ with γ >0, ρ≤0 and auxiliary function A.
Then, for any fixed p∈[0,1),
Proof. Observe that according to Theorem 3.9 we have
√ k
Xn−k,n
U(n/k)−1
→d N(0, γ2),
for any intermediate sequence k = k(n). Setting k0(n) = (1−p)k(n), where p∈[0,1), we getk0 → ∞, n/k0 →0 as n→ ∞. Thus
√ k0
Xn−[k0],n
U(n/k0) −1
→d N(0, γ2). (3.54) This corresponds the assertion.
As an immediate consequence we obtain the following result.
Corollary 3.13. Let X1,n≤X2,n ≤ · · · ≤Xn,n be the order statistics of an i.i.d.
sample with common distribution F ∈M DA(Hγ). Suppose that the corresponding tail quantile function U is in 2RVγ,ρ with γ > 0, ρ ≤ 0 and auxiliary function
A. Then, provided limn→∞
√kA(n/k) = λ ∈ R\ {0}, we obtain for any fixed p∈[0,1),
√ k
Xn−[k(1−p)],n
U(n/k) −(1−p)−γ
→d N
λ(1−p)−γ(1−p)−ρ−1
ρ , γ2(1−p)−2γ−1
. as n→ ∞, k→ ∞ but k/n→0.
Proof. First decompose
√ k
Xn−[k(1−p)],n
U(n/k) −(1−p)−γ
=√ k
Xn−[k(1−p)],n
U(n/k) − U(n/(k(1−p))) U(n/k)
+√
k
U(n/(k(1−p)))
U(n/k) −(1−p)−γ
=:E1+E2.
In fact, E1 can be considered as a normalized estimation error, since it de-scribes the limiting behavior of the difference between Xn−[k(1−p)],n/U(n/k) and U(n/(k(1−p)))/U(n/k). Recall that U(n/(k(1−p)))/U(n/k) corresponds to the p-quantile of relative excesses above U(n/k) = F←(1−k/n). Further, the deterministic term E2 characterizes the limiting behavior of the approximation errorU(n/(k(1−p)))/U(n/k)−(1−p)−γ scaled by k−1/2.
ConsideringE1 we have E1 =√
k(1−p)−γU(n/(k(1−p))) (1−p)−γU(n/k)
Xn−[k(1−p)],n
U(n/(k(1−p))) −1
.
Due to U ∈RVγ, we have
n→∞lim
U(n/(k(1−p))) (1−p)−γU(n/k) →1.
Together with Corollary 3.12, this yields E1 = (1d −p)−γ−1/2p
k(1−p)
Xn−[k(1−p)],n
U(n/(k(1−p))) −1
→d N(0, γ2(1−p)−2γ−1).
For the second term we obtain E2 =√
kA(n/k) 1 A(n/k)
U(n/(k(1−p)))
U(n/k) −(1−p)−γ
, where A is an auxiliary function forU ∈2RVγ,ρ. Since by assumption
n→∞lim
√
kA(n/k) = λ∈R, we have
n→∞lim E2 =λ(1−p)−γ(1−p)−ρ−1
ρ .
for any fixedp∈[0,1). Hence
√ k
Xn−[k(1−p)],n
U(n/k) −(1−p)−γ
→d N
λ(1−p)−γ(1−p)−ρ−1
ρ , γ2(1−p)−2γ−1
.
Remark. Observe that the conditionlimn→∞
√kA(n/k) =λ∈R ensures that the normalized approximation error converges to a finite limit. In fact, this conditions bounds the rate of k tending to infinity.
In order to obtain a functional version of Corollary 3.13 we define the so-called tail quantile process
en,k(p) :=
√ k
Xn−[k(1−p)],n
U(n/k) −(1−p)−γ
for anyp∈[0,1). It turns out that, ifk is intermediate, en,k(p) can be approxi-mated by some sequences of Brownian motions. The following theorem is due to Drees (1998). We state the version of de Haan and Ferreira (2006). Note that for notational convenience we consider ˜en,k(s) :=en,k(1−s) for any s∈(0,1].
Theorem 3.10 (Theorem 2.4.8, de Haan and Ferreira (2006)).
Suppose X1, . . . , Xn are i.i.d. random variables with distribution function F ∈ M DA(Hγ) (γ > 0). Moreover, assume that U ∈ 2RVγ,ρ with ρ ≤ 0. Then, we can define a sequence of Brownian motions {Wn(s)}s≥0 such that forA0, given in Theorem 2.10, and ε >0 sufficiently small,
sup Remark. It should be clear that one has to tight down the process e˜n,k(s) for s→0in order to obtain uniform convergence. This is connected with the different limiting behavior of Xn−k,n and Xn,n.
Following the proof of Theorem 3.10, stated in de Haan and Ferreira (2006), we require the following auxiliary result, which originally stems from Cs¨org˝o and Horv´ath (1993).
Proposition 3.6 (Proposition 2.4.9, de Haan and Ferreira 2006).
LetX1, X2, . . . be i.i.d. random variables with distribution function F with support (x∗, xF) and inverse function Q(t) := F←(t). Further, assume
In order to prove Theorem 3.10 we divide the range of possibles-values in (3.55) in 0< s < k−1 and k−1 ≤s≤1 and start with the latter case. With the help of Proposition 3.6 we can establish the following Lemma.
Lemma 3.3 (de Haan and Ferreira (2006), Lemma 2.4.10).
LetY1, . . . , Yn be i.i.d. random variables with distribution function 1−1/y, y ≥1, and Y1,n ≤ · · · ≤Yn,n the corresponding order statistics. For each γ ∈R we can define a sequence of Brownian motions {Wn(s)}s≥0 such that for each ε >0
sup
Proof of Lemma 3.3 (see de Haan and Ferreira (2006), p.53).
First note that the distribution function ofZi = (Yiγ−1)/γ is given by
Now, exploiting the fact that for γ ∈R k
and
Proof of Theorem 3.10.
Observe the distributional equality ofXn−[ks],n andU(Yn−[ks],n), where{Yi,n}i=1,...,n are order statistics of an i.i.d. sequence of random variables from the standard Pareto distribution FP a,1(x) = 1− 1/x with x ≥ 1. Under the assumption U ∈2RVγ,ρ, we obtain by the uniform inequalities in Corollary 3.8 and by setting t=n/k as well as x= nkYn−[ks],n Considering the product of (3.60) and (3.61) we get
k
Again by Lemma 3.3 we obtain for ε <−ρ+ 1/2, k
nYn−[ks],n
γ+ρ+ε
=Op(s−γ−1/2−ε). (3.63) Combining (3.59), (3.60), (3.62) and (3.63) yields
sup range oft values available in Proposition 3.6. Thus, by takingt =n/(n+ 1) in (3.57) and using similar arguments as in the proof for s-values k−1 ≤s ≤1, we
obtain
Moreover, since
For a proof of the second statement in Theorem 3.10 note first that ifU ∈RVγ, we have where A is some auxiliary function. Altogether this yields
t→∞lim
Moreover, using Theorem 2.5, we can prove that for each ε >0 there exists some t0 such that with A0 from Corollary 3.8. The rest of the proof will be established by analogous arguments as above. By the uniform inequality (3.70) we get for any k−1 < s≤1
log U nknkYn−[sk],n
where Y1, . . . , Yn are i.i.d. random variables from a standard Pareto distribution and Y1,n ≤ · · · ≤ Yn,n are the corresponding order statistics. Note, that since limγ→0(x−γ−1)/γ = log (x−1) for any x >0 we obtain by Lemma 3.3
Again, exploiting the distributional equivalence,U(Yn−[ks],n)=d Xn−[sk],n, we obtain sup stan-dard Pareto distributed, then log(Y) is standard exponential distributed. Thus, flog(Y)(y) =e−y, for y >0 and Q(t) = −log(1−t). This yields f(Q(t)) = 1−t.
Since the exponential distribution satisfies the conditions of Theorem 3.6 we obtain fort =n/(n+ 1)
Since
Now, sinceε >0, the term inside the absolute value is op(1). Moreover, due to
Since assertion follows due to (3.71)-(3.75).
Alternative proof for 0< s < k−1.
Deviating from the arguments stated in de Haan and Ferreira (2006), we provide a more simple proof of (3.56) for the range 0< s < k−1. Observe that the main part of the proof consists of establishing
k−ε
Moreover, sinceU ∈RVγ and U is monotone increasing, we obtain by Potter’s bounds that for any δ1, δ2 >0 and k, n/k large enough
fork, n/k large enough, which implies
Asymptotic approximations of the tail quantile process are of limited use for our main purpose, the construction of tail index estimators. A major problem is the presence of the unknown tail quantile function U. Replacing U(n/k) by its empirical counterpart Xn−k,n allows to approximate the limiting behavior of relative excesses without knowledge ofU.
Corollary 3.14. Suppose X1, . . . , Xn are i.i.d. random variables with distribution function F ∈ M DA(Hγ). Moreover, assume that U ∈ 2RVγ,ρ, where a possible auxiliary function A0 is given by Corollary 3.8. Then, we can define a sequence of Brownian motions {Wn(s)}s≥0 such that for ε >0 sufficiently small,
sup
Note that the random part in both uniform approximations results from a Brownian bridgeB(s) :=W(s)−sW(1) for s∈[0,1].
Proof. First, consider the decomposition
√ Hence, due to (3.55), it suffices to prove
sup
Note that Additionally, we have, again due to (3.55),
For the second statement, observe that evaluating (3.56) ats = 1 yields the assertion follows, due to (3.79) and (3.56).
An immediate consequence is stated by the next corollary.
Corollary 3.15. Let X1, . . . , Xn be a sequence of i.i.d. random variables with common distribution function F ∈M DA(Hγ) with γ >0 and assume U ∈2RVγ,ρ with auxiliary function A0, defined by Corollary 3.8. Then, for any intermediate sequence of order statistics, i.e. k → ∞, but n/k → 0 for n → ∞, satisfying
According to Theorem 3.10 we have
√
Now, consideringII, we directly obtain
II =γs−γ−1Wn(s) +λs−γs−ρ−1
ρ +op(1)s−γ−1/2−ε. Putting this together yields
√
Note that for any fixeds∈(0,1] the Brownian bridge B(s) :=W(s)−sW(1) is a normal distributed random variable with mean 0 and variances(1−s). Thus, we have For the second statement, note that due (3.77)
√ Substitutings by 1−p proves the second assertion.
Remark. We would like to stress on the fact that replacing the deterministic threshold U(n/k) by some random threshold Xn−k,n yields to a less volatile tail quantile process. To see this, observe that setting p= 1−s in Theorem 3.10 yields
√
Now, replacingU(n/k) by its empirical counterpart Xn−k,n yields Since 0< p <1, we obtain the remarkable fact that
Avar Indeed, one would expect an increase in the variance if deterministic quantities are replaced by random one. However, since Xn−[(1−p)k)],n and Xn−k,n are correlated, we obtain a variance reduction in the empirical tail quantile process.
Next, we clarify the role of the condition limn→∞
√kA(n/k) =O(1) considering the process
for some fixeds ∈(0,1] and some suitable scaling sequence wn,k. The following result is based on arguments provided in de Haan and Ferreira (2006), p. 77.
Corollary 3.16. Let X1, . . . , Xn be a sequence of i.i.d. random variables with common distribution function F ∈ M DA(Hγ) with γ > 0 and X1,n ≤ X2,n ≤
· · · ≤Xn,n the corresponding order statistics. Moreover, assume that U ∈2RVγ,ρ with some γ >0, ρ <0 and A0 given by Corollary 3.8.
(iii) If √ Note that (3.85) and (3.87) easily follows from (3.84) and (3.86) by the delta-method. Starting with I1 we have
I1 = 1
SinceXn−[ks],n/U(n/k) is bounded in probability for any fixeds ∈(0,1] and
√
Considering the second term, we get I2 = 1
The proof of the second assertion in (i) as well as for the remaining statements (ii) and (iii) follows by similar arguments.
Remark. Assume that the second order parameter ρ <0 as well as the auxiliary function A are known. Moreover note that since |A| ∈ RVρ there exist some L ∈ RV0 such that |A(t)| = tρL(t). Assume, for simplicity that limt→∞L(t) <
∞. Choosing an intermediate sequence k such that √
k|A0(n/k)| → ∞ leads to n−2ρ/(1−2ρ) = o(k) and therefore nρ/(1−2ρ) = o(A0(n/k)). That means the convergence rate A0(n/k) in (3.82) is slower than nρ/(1−2ρ). Moreover, if we choose k such that √
kA0(n/k) → 0, we get k(n) = o(n−2ρ/(1−2ρ)) which means that the convergence rate k−1/2 in (3.86) is again slower than nρ/(1−2ρ). However, if √
kA0(n/k) =λ∈R\{0}, we get k−1/2 = O(n−ρ/(1−2ρ)). That means we obtain the best rate of convergence in the second case of Corollary 3.16.
Note that applying similar arguments as in the proof of Theorem 3.10 more general versions of Corollary 3.16 can be established.
For the sake of completeness, we state a weighted approximation of the tail quantile processXn−[ks],n, assuming that F ∈M DA(G˜γ) with ˜γ ∈R. This result originates from Drees (1998). We state the version of de Haan and Ferreira (2006).
Proposition 3.7 (Theorem 2.4.2, de Haan and Ferreira (2006)).
Suppose X1, . . . , Xn are i.i.d. random variables with common distribution function F ∈M DA(G˜γ) (˜γ ∈R). Moreover, assume that U ∈2ERVγ,ρ with ρ≤0. Then, we can define a sequence of Brownian motions {Wn(s)}s≥0 such that for suitable
functions a0, ˜a0 and Ψ, defined in Proposition 2.4, and ε sufficiently small, sup
k−1≤s≤1
sγ+1/2+ε
√ k
Xn−[ks],n−U(n/k)
a0(n/k) − s−γ−1 γ
−s−γ−1Wn(s)−√
k˜a0(n/k)Ψγ,ρ s−1
→P 0, (3.88) as n→ ∞, provided k → ∞, k/n→0 and √
k˜a0(n/k) =O(1).