Sociedad de Estad´ıstica e Investigaci´ on Operativa Test (2006) Vol. 15, No. 2, pp. 375–396
Bias Reduction in Risk Modelling: Semi–Parametric Quantile Estimation M. Ivette Gomes∗ DEIO and CEAUL, Faculty of Science University of Lisbon, Portugal
Fernanda Figueiredo CEAUL, Faculty of Economics University of OPorto, Portugal
Abstract In Statistics of Extremes we are mainly interested in the estimation of quantities related to extreme events. In many areas of application, like for instance Insurance Mathematics, Finance and Statistical Quality Control, a typical requirement is to find a value, high enough, so that the chance of an exceedance of that value is small. We are then interested in the estimation of a high quantile χp , a value which is overpassed with a small probability p. In this paper we deal with the semiparametric estimation of χp for heavy tails. Since the classical semi-parametric estimators exhibit a reasonably high bias for low thresholds, we shall deal with bias reduction techniques, trying to improve their performance.
Key Words: Heavy tails, high quantiles, semi-parametric estimation, bias reduction, statistics of extremes. AMS subject classification: Primary 62G32, 62E20; Secondary 65C05.
1
Introduction
In the most diversified areas of research we are often interested in finding a value, high (or low) enough, so that the chance of an exceedance (or of a non-exceedance) of that value is small. For instance, in Insurance ∗
Correspondence to: M. Ivette Gomes. DEIO, Faculdade de Ciˆencias de Lisboa, Bloco C6, Piso 4, Campo Grande, 1749-016 Lisboa, Portugal. E-mail:
[email protected] Received: November 2003;
Accepted: August 2004
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Mathematics and Finance, a typical requirement is the estimation of a level χp = V aRp , the Value at Risk at the level p, a value which is overpassed with a small probability p = pn → 0, as n → ∞. Also in Statistical Quality Control, the on-line region depends strongly on the estimation of quantiles χp , with p quite close either to 0 or to 1. And in the design of dams, a typical requirement is that the dam height must be high enough so that the chance of a flood is no more than once in ten thousand years. We are thus in the field of Statistics of Extremes, a field in which we are mainly interested in the estimation of quantities related to extreme or even rare events, being scarce the information from previous experiments. Indeed, one of the main goals of extreme value theory is to do inference in the far tail of a probability distribution function (d.f.). For this it is necessary to extend the empirical d.f. beyond the range of the available data, estimating the tail parameter as accurately as possible. We shall assume that the d.f. F satisfies a regular variation condition, i.e., 1 − F (tx) = x−1/γ t→∞ 1 − F (t) lim
(1.1)
for all x > 0, where γ (> 0) is the tail parameter (see, for instance Galambos (1987), for details on Extreme Value Theory). Equivalently, if we consider the quantile function U (t) = F ← (1 − 1/t), t ≥ 1, with F ← denoting the generalized inverse function of F , i.e., F ← (t) = inf{x : F (x) ≥ t}, (1.1) is equivalent to say that lim {ln U (tx) − ln U (t)} = γ ln x
t→∞
(1.2)
for all x > 0. Several semi-parametric estimators of the tail parameter γ are available in the literature. In order to get the asymptotic non-degenerate behaviour of those estimators, we need more than the first order condition (1.1) (or equivalently (1.2)). A convenient condition to achieve asymptotic normality is the second order condition ln U (tx) − ln U (t) − γ ln x xρ − 1 = t→∞ A(t) ρ lim
(1.3)
for all x > 0, where A is a suitably chosen function of constant sign near infinity (positive or negative), and ρ ≤ 0 is the second order parameter. The limit function in (1.3) is necessarily of this given form, and |A| ∈ RVρ
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(see Geluk and de Haan, 1987). The notation RVβ stands for the class of regularly varying functions at infinity with index of regular variation equal to β, i.e., positive measurable functions g such that lim g(tx)/g(t) = xβ , t→∞ for all x > 0. We want to estimate χp ≡ V aRp : 1 − F (χp ) = p,
p small,
i.e., 1 χp := U , p
p = pn → 0,
n pn → K, as n → ∞, 0 ≤ K ≤ 1, (1.4)
and we shall assume to be working inside Hall’s class of models in Hall and Welsh (1985), where ρ < 0 and U (t) ∼ C tγ . More specifically, we shall assume that U (t) = Ctγ (1 + Dtρ + o(tρ )) ,
ρ < 0,
as t → ∞.
(1.5)
Remark 1.1. Notice that since χp = U (1/p) and U (t) ∼ C tγ , an b p−bγ , with γ obvious estimator of χp is χ bp = C b any consistent estimator of γ. Also, with Yi:n , 1 ≤ i ≤ n, denoting the set of ascending order statistics (o.s.) associated to a standard independent identically distributed (i.i.d.) Pareto sample from the model FY (y) = 1 − y −1 , y ≥ 1, n γ p γ Xn−k:n = U (Yn−k:n ) ∼ C Yn−k:n ∼ C . k b = k γb Xn−k:n , and, Consequently, an obvious estimator of C is C n γb k χ bp (k) ≡ χ bp (k; γb) = Xn−k:n (1.6) np is the obvious quantile estimator, to be considered later on, in Section 3 of this paper.
We are going to base inference on the largest k top o.s., and as usual in semi-parametric estimation of parameters of rare events, we shall assume that k is an intermediate sequence of integers, i.e., k = kn → ∞,
k = o(n)
as n → ∞.
(1.7)
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Our aim is essentially to present new reduced bias estimators for χp in (1.4) and to prove their consistency and asymptotic normality under appropriate conditions. The classical semi-parametric quantile estimators are provided by plugging in χ bp (k; γb(k)), in (1.6), any classical tail index estimator, γ b(k). For heavy-tailed underlying models, the most proeminent classical tail index estimator is the Hill estimator (see Hill, 1975), with a functional form γ bnH (k) :=
k 1 X {ln Xn−i+1:n − ln Xn−k:n } , k
(1.8)
i=1
where Xi:n , 1 ≤ i ≤ n, is the sample of ascending o.s. associated to our original i.i.d. sample, (X1 , X2 , . . . , Xn ). The Hill estimator is weakly consistent for the estimation of the tail index γ, whenever (1.7) holds and we are under the first order framework√in (1.1) (equivalently, (1.2)). Under the second or der framework in (1.3), k γ bnH (k) − γ is asymptotically normal with null √ mean value whenever k A(n/k) → 0, i.e., for very high thresholds Xn−k:n , but there appears a non-null asymptotic bias, given by λ/(1 − ρ), whenever √ k A(n/k) → λ 6= 0, finite. In this paper we are going to base our quantile estimation on reduced bias estimators of the tail index γ, to be specified in Section 2, getting thus reduced bias estimators of any quantile. For a √ (•) (•) reduced bias tail index estimator, b γn (k) say, k γ bn (k) − γ is going √ to be asymptotically normal with null mean, even when k A(n/k) → λ, finite, non-necessarily null, i.e., the dominant component of a classical tail index estimator’s bias, which is of the order of A(n/k), is going to be of a smaller order when we consider these reduced bias estimators. Then, the optimal level, i.e., the level k0 at which the mean squared error of our estimator (as a function of k) is minimum, is going to be larger than the optimal level associated to any classical quantile estimator. Consequently, one of the main features of these new reduced bias quantile estimators is that for estimating χp more upper order statistics should be used than in the classical estimation of a quantile. We shall work with the statistics k 1X := [ln Xn−i+1:n − ln Xn−k:n ]α , α ∈ R+ , (1.9) k i=1 h i (1) which generalize the Hill estimator in (1.8) Mn ≡ γ bnH . These statistics were introduced, and studied under a second order framework, in Dekkers
Mn(α) (k)
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et al. (1989). For more details on these statistics, and the way they may be used to build alternatives to the Hill estimator, see Gomes and Martins (2001) and also Caeiro and Gomes (2002a,b). The starting point is a wellknown expansion for Mnα (k), for any α > 0: Mnα (k)
=
γ α µ(1) α +
1 γ α σα(1) √
k
(α) Pk
+
αγ α−1 µ(2) α (ρ)A(n/k)
(1+op (1))
(α)
where Pk is a standard normal random variable (r.v.), and, with W denoting a unit exponential r.v., α µ(1) α := E[W ] = Γ(α + 1),
σα(1) := and µ(2) α (ρ)
p p Var[W α ] = Γ(2α + 1) − Γ2 (α + 1)
ρW Γ(α) 1 − (1 − ρ)α −1 α−1 e := E W = . ρ ρ (1 − ρ)α
In Section 2 we shall review some of the reduced bias estimators of the tail index γ, already available in the literature. Those estimators depend on an external estimation of the second order parameter ρ < 0 in (1.5), also addressed in this same section. In Section 3 we shall discuss the estimation of a high quantile. Finally, in Section 4, we present the simulated behaviour of our χp -estimators for parent distributions in Hall’s class, like the Fr´echet and the Burr.
2
The tail index estimation
Among the reduced bias estimators of the tail index already introduced and studied at sub-optimal levels, which for these type of estimators are levels √ k such that k A(n/k) → λ, finite, we shall select here the two ones with smaller asymptotic variances. Notice that these estimators have not yet been studied at their optimal levels, i.e., at the levels k0 which minimize their mean squared errors, and this turns out to be an interesting topic of open research. These estimators require the estimation of the second
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order parameter ρ. Such an estimation will be briefly discussed before the discussion of the tail index estimators to be incorporated in the estimation of high quantiles. 2.1
Estimation of the second order parameter
We shall consider here particular members of the class of estimators of the second order parameter ρ proposed in Fraga Alves et al. (2003). Under adequate general conditions, they are semi-parametric asymptotically normal estimators of ρ, which show highly stable sample paths as functions of k, the number of top o.s. used, for a wide range of large k-values. Such a class of estimators is parameterized in a tuning parameter τ and depends on the statistics τ τ /2 (1) (2) Mn (k) − Mn (k)/2 if τ > 0 τ /2 τ /3 (3) − Mn (k)/6 Mn(2) (k)/2 Tn(τ ) (k) := (2) (1) ln Mn (k) −12 ln M n (k)/2 1 ln Mn(2) (k)/2 − 1 ln Mn(3) (k)/6 if τ = 0, 2
3
which converge towards 3(1 − ρ)/(3 − ρ), independently of τ , whenever the second order condition (1.3) holds and k is such that k → ∞, k = o(n) and √ k A(n/k) → ∞, as n → ∞. The estimators are thus given by ! (τ ) 3(T (k) − 1) n ) ρb(τ . (2.1) n (k) := min 0, (τ ) Tn (k) − 3 We shall formalize, without proofs, the main distributional results of the estimators in (2.1). Proofs may be found in Fraga Alves et al. (2003).
Theorem 2.1. If the second order condition (1.3) holds, with ρ < 0, k is a sequence of intermediate integers, i.e., (1.7) holds, and √ lim k A(n/k) = ∞, (2.2) n→∞
(τ )
then ρbn (k) in (2.1) converges in probability towards ρ, as n → ∞.
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Theorem 2.2. If we further assume that a third order condition holds, which we here write as lim
ln U (tx)−ln U (t)−γ ln x A(t)
B(t)
t→∞
−
xρ −1 ρ
=
x2ρ − 1 , 2ρ
(2.3)
then necessarily with |B| ∈ RVρ , ρ < 0, together with (2.2), and also the validity of the condition √ lim k A2 (n/k) = λ1 , finite, n→∞
together with lim
n→∞
√
k A(n/k) B(n/k) = λ2 , finite, (τ )
we may guaranteee asymptotic normality of the estimators ρbn (k) in (2.1). (τ ) 1 Moreover, ρbn (k) − ρ = Op √kA(n/k) .
Remark 2.1. The theoretical and simulated results in Fraga Alves et al. (2003), together with the use of these estimators in the Generalized Jackknife statistics of Gomes et al. (2000) led us to advise in practice the consideration of the level k1 = min(n − 1, [2n/ ln ln n])
(2.4)
(not chosen in an optimal way), and of the tuning parameters τ = 0 for the region ρ ∈ [−1, 0) and τ = 1 for the region ρ ∈ (−∞, −1). In the simulations of Section 4 we have always used τ = 1 in (2.1) and the level k1 in (2.4). Anyway, we advise practitioners not to choose blindly the (τ ) value of τ in (2.1). It is sensible to draw a few sample paths of ρbn (k), for instance for τ = 0, 0.5, 1, as functions of k, electing the value of τ which provides higher stability for large k. We may choose any adequate (τ ) stability criterion for ρbn (k), 1 ≤ k < n, like the one herewith described: let us define ) χin ,jn (τ ) := χ b1/2 ρb(τ n (k), in ≤ k ≤ jn and
jn n o2 X (τ ) Σin ,jn (τ ) := ρbn (k) − χin ,jn (τ ) , k=in
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where χ b1/2 (uk , in ≤ k ≤ jn ) denotes the sample median of the sample (uk , in ≤ k ≤ jn ). We may then take τ0 := arg min Σin ,jn (τ ), τ
ρb0 := χin ,jn (τ0 ),
for suitably chosen values of in and jn , like for instance in = n0.95 and jn = n0.99 . The choice of these values, provided they are large enough, is not quite important, due to the robustness of the method regarding such a choice.
2.2
The reduced bias estimators of the tail index
We shall make first a review of some explicit reduced bias tail index esti(•) mators, γ bn (k), so far proposed in the literature. Notice that they are all based on an external estimation of the second order parameter ρ. More than that: an asymptotic distributional representation of the type
(•)
σ• (•) d γ bn(•) (k) = γ + √ Qk + op (A(n/k)), k
(2.5)
with Qk an asymptotic standard normal r.v., holds true for any of the esti(•) mators γn (k) herewith considered, i.e., as mentioned before, b √ (•) k γ bn (k) − γ is asymptotically normal with null mean value not only √ √ when k A(n/k) → 0 (as usual), but also when k A(n/k) → λ, finite and (τ ) non-null. Let us denote ρb = ρbn (k1 ) any of the estimators in (2.1), with k1 in (2.4). 1. In Gomes and Martins (2001): If we consider the reduced bias estimator proposed in this paper, obtained through the external estimation of the second order parameter ρ, we get the estimator (b α)
γ bn(1) ≡ γ bn(bα,1) (k) :=
where α b is such that:
Mn (k) h iαb −1 , (1) Γ(b α + 1) Mn (k)
(1 − ρb)αb −1 [1 + ρb(b α − 2)] = 1.
(2.6)
(2.7)
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Bias Reduction in Risk Modelling
The asymptotic variance of the estimator in (2.6), i.e., the value σ•2 in (2.5) associated to this estimator, is ! (1) Γ(2αρ + 1) (1) 2 2 2 2 σ1 ≡ ϕ1 (ρ) = γ , − αρ (1) Γ2 (αρ + 1) with α(1) ρ :
(1)
(1 − ρ)αρ
−1
[1 + ρ(α(1) ρ − 2)] = 1.
2. In Caeiro and Gomes (2002b): The estimator herewith considered is γ bn(2)
with
≡
γ bn(bα,2) (k) α b=−
:=
(2b α)
Γ(b α) (b α−1)
Mn
(k)
Mn (k) Γ(2b α + 1)
h i p ln 1 − ρb − (1 − ρb)2 − 1 ln(1 − ρb)
!1/2
.
,
(2.8)
(2.9)
The asymptotic variance σ22 of the estimator in (2.8) is now ruled by ( (2) (2) 2 γ Γ(4αρ ) 4 Γ(2αρ − 1) ϕ22 (ρ) = + (2) (2) 2 4 α(2) Γ2 (αρ ) ρ Γ (2αρ ) ) (2) 2 Γ(3αρ ) −1 , − (2) (2) αρ Γ(αρ )Γ(αρ ) with (2)
α(2) ρ = α0 (ρ) = −
i h p ln 1 − ρ − (1 − ρ)2 − 1 ln(1 − ρ)
.
3. In Gomes and Martins (2002): Among the estimators considered in this paper, we have selected here the following two: the Generalized Jackknife estimator, (2)
γ bn(3) ≡ γnGJ(bρ) (k) :=
based on
(3)
−(2 − ρb)γn (k) + 2γn (k) , ρb
γn(2) (k) := Mn(2) (k)/(2Mn(1) (k))
(2.10)
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and γn(3) (k)
:=
q
(2)
Mn (k)/2;
and the “Maximum Likelihood” (M L) estimator bn(4) ≡ γnML(bρ) (k) := γ −
k 1 X −bρ i Ui k i=1
k 1X Ui k i=1 k k k P −bρ P P −bρ ! i Ui − k i Ui
i=1
k P
i=1
i−bρ
based on the scaled log-spacings
i=1
k P
i=1
i−bρ Ui
−k
i=1 k P
i=1
i−2bρ Ui
, (2.11)
Ui := i [ln Xn−i+1:n − ln Xn−i:n ] . The asymptotic variances of the estimators in (2.10) and in (2.11) are ruled by γ 2 (2ρ2 − 2ρ + 1) ϕ23 (ρ) = ρ2 and γ 2 (1 − ρ)2 ϕ24 (ρ) = = ϕ23 (ρ) − γ 2 , ρ2 respectively. (j)
The quantile estimators associated to the tail index estimators b γn given (j) (3) (4) before will be denoted χ bn , j = 1, 2, 3, 4. The estimators χ bn and χ bn L will also be denoted χ bGJ and χ bM n n , respectively. The classical quantile estimator, i.e., the one associated to the Hill estimator, will be denoted (0) either χ bn or χ bH n . The standard deviations ϕj (ρ), j = 1, 2, 3, 4, are drawn in Figure 1. The comparison of those standard deviations enable us to draw the following conclusions: 1. The so-called M L estimator, first introduced in Beirlant et al. (1999) and Feuerverger and Hall (1999), studied with a misspecification of ρ = −1 in Gomes and Martins (2004), and further studied under an external estimation of ρ in Gomes and Martins (2002), is, among these four reduced bias estimators considered, the one which provides the smallest asymptotic variance for all values of ρ.
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ϕ j (ρ) 3
j =1
2
j=2
j=3
j=4
ρ
1 0
1
2
3
4
5
Figure 1: Asymptotic standard deviations of the tail index estimators under study, up to a factor γ
2. The one with the largest asymptotic variance is the one in Caeiro and Gomes (2002b). This class of estimators was first considered in Caeiro and Gomes (2002a), where the tuning parameter α was chosen through an adequate stability criterion of the sample paths. It was then such a choice of α that would give rise to a possible estimate of ρ, useless for applications, due to its high variance.
3
How to estimate a high quantile
In a semi-parametric context, and for heavy tails, the most usual estimator of χp is, as previously mentioned, χ bp (k) := Xn−k:n
k np
bγ (k)
=: Xn−k:n aγnb (k) ,
an =
k , npn
(3.1)
where γ b(k) is any semi-parametric estimator of the tail index γ. The estimator in (3.1) is of the type of the one introduced in Weissman (1978). Details on semi-parametric estimation of extremely high quantiles for a
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general tail index γ ∈ R may be found in de Haan and Rootz´en (1993) and more recently in Ferreira et al. (2003). Approaches associated to bias reduction techniques, similar to the one herewith presented, may be found in Matthys and Beirlant (2003). We state, without proof the following lemma. Lemma 3.1. Under the second order framework in (1.3) and for intermediate k, i.e., k such that (1.7) holds, we may write ! (1) γ Pk A(n/k) d (1) √ + (3.2) (1 + op (1)), Mn (k) = γ + 1−ρ k (1)
where Pk
is a standard normal r.v. Also, γ Bk d Xn−k:n = U (n/k) 1 + √ + op (A(n/k)) , k
(3.3)
(1)
with Bk standard normal, independent of Pk . We may further state: Theorem 3.1. Under the conditions of Lemma 3.1, in Hall’s class of models, where (1.5) holds, and whenever √ ln npn = o k , (3.4)
√ let us consider a tail index estimator γ bn (k) such that k (b γn (k) − γ) has 2 an √ asymptotic variance equal to σ and a null asymptotic bias, even when k A(n/k) → λ, non-null and finite, i.e., d
γ b(k) − γ =
σ Qk √ + op (A(n/k)), k
with Qk an asymptotically standard normal r.v. Then, √ √ k (b χp (k) − χp ) d = Qk + op ( k A(n/k)). γ σ an ln an U (n/k)
(3.5)
(3.6)
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We may also write √ √ χ bp (k) k d − 1 = Qk + op ( k A(n/k)). σ ln an χp
(3.7)
We √ thus have asymptotic normality with a null√bias not only when k A(n/k) → 0, as usually happens, but also when k A(n/k) → λ, nonnull, but finite. Proof. The use of the δ-method, together with the fact that ln an = o enables us to write
√ k ,
d
aγnb(k) = aγn + aγn ln an (b γ (k) − γ) (1 + op (1)).
Then, since χp = U (1/p) = U (nan /k), and (3.3) holds,
U (nan /k) U (n/k) χ bp (k) − χp = Xn−k:n anbγ (k) − U (n/k) Xn−k:n γ Bk γ (k) − γ)(1 + op (1)). = U (n/k) 1 + √ + op (A(n/k)) aγn ln an (b k
Consequently, if we choose a tail index estimator with a null dominant component of asymptotic bias, i.e., if (3.5) holds, (3.6) follows, as well as the remaining of the theorem. Remark 3.1. The result in Theorem 3.1 may be easily generalized to a real tail index γ provided we are able to get reduced bias estimators of γ ∈ R, and this is still, as far as we know, an open question. p
Remark 3.2. In √ order to have χ bp − χp → 0 we need to guarantee that aγn ln an U (n/k) / k → 0. Consequently, √ even if K in (1.4) is non-null (and then an = O(k), we need to have ln k/ k = o(n−γ ), and this happens for k = n , > 0 only if γ < 1/2. √ p Remark 3.3. However, χ bp /χp → 1, as n → ∞ whenever ln an = o k , √ and this happens whenever ln npn = o k , one of the conditions in Theorem 3.1.
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Remark 3.4. The simulation results obtained for a reasonable large class of heavy-tailed models, and partially presented in Section 4, enable us to advance that the high stability of sample paths achieved with the reduced bias estimates of the tail index is no longer achieved for these reduced bias high quantiles’ estimates. Anyway, the proposed estimators perform better than the classical one, i.e., the one where Hill’s tail index estimator is used, and for most of the simulated models exhibit a much more stable mean value pattern, as a function of k, the number of top o.s. used.
4
Simulated properties of the VaRp -estimator
For the estimation of the second order parameter ρ we have here used the value τ = 1 in (2.1), together with the level k1 in (2.4). In Figures 2 and 3 we show the simulated patterns of mean value, E[·], and root mean (j) (j) bp (k)/χp , for p = 1/n and j = 0, squared error, RM SE[·], of Qp (k) := χ 3 and 4, based thus on the Hill (H), the Generalized Jackknife (GJ) and the “Maximum Likelihood” (M L), respectively. The sample size used in these simulations is N = 5000. Results for the other two estimators are not a long way from the ones got for the GJ-estimator and are not pictured. The models underlying the original data are the Fr´echet model with γ = 0.25 and the model with (γ, ρ) = (0.25, −1). The Fr´echet(γ) d.f. Burr is F (x) = exp −x−1/γ , x ≥ 0, γ > 0, (for which ρ = −1), and the Burr 1/ρ d.f. is F (x) = 1 − 1 + x−ρ/γ , x ≥ 0, γ > 0, ρ < 0. As may be seen from these figures we do not gain a lot regarding mean squared error at the optimal level, but we gain in mean value stability along a wide range of k-values. This gives an obvious indication about the sample paths’ stability of these new estimators; note that, under a different scale, the sample paths are identical to the corresponding mean values. For large k-values, the classical quantile estimator based on Hill’s tail index estimator exhibits without doubt a much higher bias, due to the existence of that dominant term of asymptotic bias, of the order of A(n/k), with A the function in (1.3). Figures 4 and 5 are equivalent to Figures 2 and 3, but now for Burr models with (γ, ρ) = (0.25, −0.5) and (γ, ρ) = (0.25, −2), respectively.
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E[.]
RMSE[.]
1.5
0.4
H
H ML 0.2
1
GJ
ML
GJ
0
200
400
600
800
k
0
k
0.5
0
1000
200
400
600
800
1000
Figure 2: Fr´echet parent with γ = 0.25
E[.]
RMSE[.]
1.5
0.4
H H ML 1
0.2
GJ
ML
GJ 0.5
k 0
200
400
600
800
1000
k
0 0
200
400
600
800
1000
Figure 3: Burr parent with (γ, ρ) = (0.25, −1)
The behaviour exhibited by the reduced bias estimators of high quantiles, in the region of ρ values close to 0, is quite different from the one we have got for the reduced bias tail index estimators. Indeed, in this region of ρ-values, a region where the reduced bias estimators of the tail index exhibit a stable sample path comparatively to the classical estimators, the associated estimators of high quantiles reveal a mean value pattern similar to that of the classical one, although with a slightly better performance at the optimal level. The reason for this is now under current research: most probably this is due to the fact that here the rate of convergence is no
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E[.]
RMSE[.] 0.4
H
ML
H
ML
1
0.2
GJ GJ
0.5
0
k 0
200
400
600
800
1000
k 0
200
400
600
800
1000
Figure 4: Burr parent with (γ, ρ) = (0.25, −0.5) 1.5
E[.]
0.4
RMSE[.]
H
H
1
0.2
GJ ML
ML 0.5
k 0
200
400
600
800
1000
GJ
0
k 0
200
400
600
800
1000
Figure 5: Burr parent with (γ, ρ) = (0.25, −2)
√ √ longer 1/ k as for the tail index estimation, but ln an / k, with an → ∞ as n → ∞, being the bias highly relevant for values of ρ close to 0. For small values of ρ, here illustrated with ρ = −2, the patterns of mean values of the new estimators exhibit clearly wider stability regions, but the gain in mean squared error, at the optimal level, is not at all significant. We shall also present here, for finite values of n, more specifically for n = 100, 200, 500, 1000 and 2000, the simulated mean values and mean (j) squared errors of Qp , j = 0, 3 and 4, at their optimal levels, i.e. of
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(j) (j) (j) (j) (j) Qp0 := χ bp (b k0 (n))/χp , b k0 (n) := arg mink M SE[b χn (k)], j = 0, 3, 4, (1)
(2)
p = 1/n. As said before, the distributional behaviour of Qp and Qp is (3) quite similar to that of Qp ≡ QGJ p , and has been omitted from the tables. The peculiar behaviour of the mean squared error pattern, exhibited in Figure 4 by the M L-estimator and in Figure 5 by the GJ-estimator, with two minima, led us to perform the search over a region of k-values smaller than a value kmax < n − 1, in order to obtain the relevant minimum value, i.e., the first one, which we think to be the one underlying the developed theory. The size N × r of the multi-sample simulation used, where N denotes the number of runs in each of r replicates of the experiment, is related to the computer time and not to the precision of the estimates. For samples of size n ≤ 500 we have used a 5000× 10 simulation, for n = 1000 a 5000× 5 simulation and for n = 2000 only a 5000 × 2 simulation was performed. For any details on multi-sample simulation refer to Gomes and Oliveira (2001). As mentioned before, Figures from 2 till 5 have been based on the first replicate of 5000 runs. Among the estimators considered, and for every n, the one providing the smallest bias and the smallest mean squared error is underlined. Tables 1 and 2 are related to underlying Fr´echet and Burr parents, respectively. In summary we may draw the following conclusions: 1. The new reduced bias quantile estimators have in general reasonably stable sample paths, which make less troublesome the choice of the optimal level k. 2. For Fr´echet models, and among the estimators considered, the M Lestimator is the best one, but not a long way from the other reduced bias estimators. The same happens for a Burr model with ρ = −1, provided n is large. For small n (n < 1000) the GJ estimator overpasses the M L estimator, when both are computed at their optimal levels.
M SE QML p0
M SE QGJ p0
M SE QH p0
E QML p0
E QGJ p0
E QH p0
n
500
0.9719 ±0.0027
0.9299 ±0.0018
1.0602 ±0.0047
0.9897 ±0.0036
0.9539 ±0.0008
1.0537 ±0.0028
0.0363 ±0.0007
0.0405 ±0.0012
0.0361 ±0.0010
0.0242 ±0.0007
0.0268 ±0.0005
0.0272 ±0.0007
0.0130 ±0.0002
0.0145 ±0.0003
0.0184 ±0.0003
1000
0.9957 ±0.0018
0.9697 ±0.0015
1.0480 ±0.0030
Mean values at the optimal level
200
0.0075 ±0.0002
0.0083 ±0.0001
0.0135 ±0.0002
Mean squared errors at the optimal level
0.9508 ±0.0044
0.9058 ±0.0043
1.0636 ±0.0049
100
0.0059 ±0.0005
0.0067 ±0.0002
0.0097 ±0.0001
0.9991 ±0.0060
0.9774 ±0.0016
1.0402 ±0.0034
2000
GJ ML Table 1: Simulated distributional behaviour of QH echet parent with p0 (j = 0), Qp0 (j = 3) and Qp0 (j = 4), for a Fr´ γ = .25.
392 M. I. Gomes and F. Figueiredo
GJ
M SE M SE
M SE
E
E E
QH p0 QGJ p0 L QM p0
QH p0 QGJ p0 L QM p0
QH p0 QGJ p0 L QM p0
M SE M SE
M SE
E
E E
QH p0 QGJ p0 L QM p0
M SE Qp0 L M SE QM p0
M SE QH p0
E
E E
QH p0 QGJ p0 L QM p0
n
0.0205 ± 0.0004 0.0350 ± 0.0007
0.0284 ± 0.0004 0.0555 ± 0.0008
0.0211 ± 0.0005 0.0438 ± 0.0008
0.0544 ± 0.0011
1.0875 ± 0.0044
0.0127 ± 0.0002 0.0229 ± 0.0004
0.0072 ± 0.0001 0.0113 ± 0.0001
0.0243 ± 0.0004
0.9778 ± 0.0011
1.0566 ± 0.0037 1.0108 ± 0.0008
0.0211 ± 0.0004 0.0265 ± 0.0002 0.0215 ± 0.0003
0.0377 ± 0.0005 0.0314 ± 0.0005
0.9475 ± 0.0033
0.9418 ± 0.0025 0.02931 ± 0.0007
1.0463 ± 0.0027 0.9000 ± 0.0116
1.0503 ± 0.0025 0.8991 ± 0.0035
0.0165 ± 0.0002 0.0126 ± 0.0001
0.0131 ± 0.0003
0.9560 ± 0.0030
1.0388 ± 0.0030 0.9337 ± 0.0019
ρ = −2
0.0355 ± 0.0005
0.0307 ± 0.0004 0.0364 ± 0.0005
0.9503 ± 0.0027
0.9265 ± 0.0036 0.0464 ± 0.0007
1.0674 ± 0.0072 0.9688 ± 0.0054
1.0694 ± 0.0047 0.8987 ± 0.0059
ρ = −1
0.0657 ± 0.0019
0.9687 ± 0.0046
0.0833 ± 0.0018
0.9191 ± 0.0057
500 1.1312 ± 0.0025 1.0379 ± 0.0034
ρ = −0.5 1.0966 ± 0.0033 1.0138 ± 0.0030
200
1.0873 ± 0.0044 0.9347 ± 0.0039
100
0.0113 ± 0.0003 0.0083 ± 0.0001
0.0090 ± 0.0002
0.9647 ± 0.0017
1.0334 ± 0.0019 0.9455 ± 0.0013
0.0065 ± 0.0001 0.0064 ± 0.0001
0.0180 ± 0.0003
0.9906 ± 0.0019
1.0544 ± 0.0016 1.0203 ± 0.0011
0.0199 ± 0.0004 0.0363 ± 0.0011
0.0560 ± 0.0007
1.0833 ± 0.0124
1.1717 ± 0.0014 1.0477 ± 0.0023
1000
0.0080 ± 0.0001 0.0058 ± 0.0000
0.0062 ± 0.0000
0.9727 ± 0.0000
1.0259 ± 0.0000 0.9598 ± 0.0034
0.0049 ± 0.0000 0.0044 ± 0.0000
0.0133 ± 0.0000
0.9929 ± 0.0000
1.0521 ± 0.0000 1.0166 ± 0.0000
0.0163 ± 0.0000 0.0279 ± 0.0009
0.0654 ± 0.0000
1.0780 ± 0.0097
1.2162 ± 0.0001 1.0496 ± 0.0000
2000
GJ ML Table 2: Simulated distributional behaviour of QH p0 (j = 0), Qp0 (j = 3) and Qp0 (j = 4), for Burr parents with γ = .25.
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3. For values of ρ close to zero, here illustrated with ρ = −0.5 in a Burr parent, the new estimators may exhibit sample paths similar to the classical ones. This may be partially due to the estimation of the parameter C, in (1.5), which needs perhaps to be improved. Indeed, we think that the semi-parametric estimation of this scale first order parameter C deserves further attention, but this overpasses the scope of the present paper. Among the quantile estimators herewith considered, the GJ-estimator is the best one, in the sense of stability around the target value γ and of minimum mean squared error, at its optimal level. 4. For small values of ρ, here illustrated with ρ = −2 in a Burr model, and regarding mean squared errors at optimal levels, the M L-estimator is able to overpass slightly the H-estimator for large n. However, the classical H-estimator of a high quantile reveals a slightly smaller bias at optimal levels.
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