Estimation of the parameter controlling the speed of convergence in ...

Estimation of the parameter controlling the speed of convergence in extreme value theory M. I. Fraga Alves∗

D.E.I.O. and C.E.A.U.L., University of Lisbon

Laurens de Haan∗

Tao Lin

Erasmus University Rotterdam

January 11, 2002

Abstract. We present, in a semi-parametric context, estimators of the second

ρ, a parameter controlling the speed of convergence in extreme value theory. Under a third order extended regular variation condition on the tail of the underlying distribution function F , asymptotical normality is proven for these estimators, provided one employs many more upper order statistics than in the case of estimation of the first order parameter γ. An exhaustive simulation study has been done for practical validation of the asymptotic results for finite samples. Illustration will be presented for samples from underlying Cauchy and Arcsin models. Also an application to mortality data was considered to have a global picture of the exact performance of the estimators. order parameter

1

Introduction.

X ,X ,

F

Let 1 2 ¢ ¢ ¢ be i.i.d. random variables with distribution function . The distribution is in the domain of attraction of an extreme value distribution if and only if ( ) := (1 (1 ¡ )) satisfies an extended regular variation property:

Ut

F ←

/

U (tx) U (t) = x lim →∞ a(t) γ ¡

t

γ

¡

1




=

1

x

yγ −1dy,

(1.1)

for all x > 0 with γ a real-valued parameter and a a suitable positive function. The limit function should be interpreted as log x for γ = 0. The speed of convergence of the partial maxima towards the limit distribution and also ∗

Research partially partially supported by FCT / POCTI / FEDER.

1

the asymptotic normality of estimators of the parameter γ (and other quantities) are controlled by a second order extended regular variation property:

−

log U (tx)

log U (t)

a0 (t)

→∞

lim

¡

γ−

x

−1

γ−

A(t)

t

= 1 ρ

=

 γ− +ρ  x −1 ¡ xγ− −1

x 1

γ− +ρ γ− γ − 1  y ρ−1 − y u dudy,

(1.2)

1

for all x > 0 with ρ the non-positive second order parameter, a0 > 0 and A a suitable positive or negative function. As before the limit function is defined by continuity for γ = 0 and/or ρ = 0 . The function j j , which is regularly varying of order and tends to zero, represents the speed of convergence. Hence large values of j j correspond to rapid convergence, whereas for example = 0 means very slow convergence (logarithmic or even worse). Estimators of e.g. are generally functions of a number, say , of top order statistics. In order to get asymptotic normality (or even consistency) for those estimators one needs = ( ) ! 1 as ( ) ! 0 , as the sample size tends to infinity, but this still leaves much freedom. Adaptive optimal choices for are known. For those choices one needs to estimate , the second order parameter. We present an estimator for , under the assumption that 0, which converges at a polynomial rate. The idea behind the estimator (or in fact a class of estimators) is as follows. Most estimators of have the following type of expansion:

A

ρ

ρ

ρ

γ

k

k kn

n

k n /n

k

ρ

ρ

ρ
0, with ρ — 0, η — 0, as t ! 1, with functions a1 and a2 having constant sign. Let X1, X2 , ¢ ¢ ¢ be i.i.d. with distribution function F . Define for α > 0

Mn



1 k −1

α k i=0 (log Xn−i,n log Xn−k,n ) , where X1,n X2,n Xn,n are the n th order statistics. Next define for α > 0 (α)

:= —

¡

— ¢¢¢ —



¡



(α) (1) α +1 M n Mn gα := α 1 ¡ Mn(α+1) .

It follows as in Dekkers et al. [3] that

gα ¡!

1

p

1

¡ γ−

with γ− := min(γ, 0), as

n

! 1, for all α > 0.

Next we introduce

T n :=

g1 ¡ g2 g2 ¡  g3 3

and for α > 0

¡

(α + 2) gα+2 + αgα 2(α + 1) gα+1 Tn(α) := . (α + 3) gα+3 + (α + 1) gα+1 2(α + 2)gα+2

¡

We are now able to introduce the estimators for ρ:

ρ  := 3

¡ 8ˆγ− + 6 ¡ 12ˆγ− , Tn ¡ 3

(2.2)

and

ρα := 1 ¡ (α + 4)ˆγ− ¡ Tn(α) where

γˆ− := 1 ¡ 12



1

¡ (α + 2)ˆγ−



,

(2.3)

Mn(2)  . Mn(2) ¡ Mn(1) 2

Theorem 2.1. Suppose

(2.4)

that the second order condition (1.2) holds with sequences of integers = ( ) satisfying ( ) = ( ) and j ! 1, as ! 1, we have

ρ < 0. For k A(n/k) p ρ¡!ρ p

j

k

n

kn

kn

on

and p ρα¡!ρ for all α > 0. Theorem 2.2.

Suppose that the third order condition (2.1) holds with

ρ < 0, η < 0. For sequences of integers k = k(n) satisfying k(n) = o(n) and

→∞

lim

n

→∞

lim

n

→∞

lim

n

p

kjA(n/k)j

p

2

kA

→∞

= 1,

(n/k ) =

p

lim inf

n

kjB (n/k)j >

0,

λ1, f inite

p

kA(n/k)B(n/k)

=

λ2 , f inite

f

g

where A(t) := a1 (t)/a0(t) and B(t) := a2 (t)/ a0 (t)a1 (t) , we have that

p

f ¡ ρg

kA(n/k) ρ 

and

p

f

kA(n/k) ρ α

¡ ρg

have asymptotically a normal distribution, as

4

n ! 1.

Remark 2.1. (IMPORTANT REMARK)

The estimator

ρ, for example, is a function of γˆ− and T n . Now the proof T n , a function of the upper k order statistics, converges to

will show that a constant depending on γ and ρ and the most efficient choice of k is the one given in Theorem 2.2. But the most efficient choice of the number of upper order statistics on which γ ˆ− is based, is much smaller (cf. Remark 2.4 below). Hence, in the application of the estimator ρ  (and also ρα), we choose ˆ− , according to different numbers of upper order statistics for T n and for γ Remark 2.4.

Remark 2.2. In the proof we shall sketch how to obtain the mean and variance of the limiting normal distribution.

Remark 2.3. We have thus shown that, at least in principle, the estimation of

ρ converges at a polynomial rate.

Remark 2.4. Another conclusion is that for estimating

ρ

efficiently one needs to take into account many more order statistics than for estimating the first order parameter ( i.e., ( )( ( )+ ( )) [0 ) as opposed to ( ) ( ) ). Hence any procedure to estimate the two using the same number of order statistics seems less efficient.

p kA n/k A n/k B n/k ! λ 2 , 1 kA n/k ! λ 2 ¡1, 1

p

γ

Remark 2.5. The optimal rate of convergence for p

/pk

γˆ

is 1 1 with j 1j 1 and the optimal rate of conver1 ( 1) ! p 1 with 0 p 2( gence for  is 1 2j ( 2)j with 2 2 ) ! 2 2 (0 1). Hence the optimal rate for  is always less than the one for ˆ (i.e., order ρ/(1−4ρ) as opposed to order nρ/(1−2ρ) ).

k A n/k ρ

3

λ < λ < / k A n/k k A n/k λ ρ γ

,

n

Proofs

We just give the proof of Theorem 2.2. The proof of Theorem 2.1 is similar but simpler. The proof of Theorem 2.2 consists of a series of Lemmas, each one of which is only used in the proof of the next one. First we sketch how to prove Potter type inequalities related to (2.1) (Lemma 3.1). This will lead to an asymptotic expansion for Mn(α) (Lemma 3.2). In turn this implies an asymptotic expansion for gα (Lemma 3.3). This leads to the asymptotic normality of T n and Tn(α) and finally to the desired asymptotic normality for ρ and ρα . The first Lemma concerns the extension in third order of the results of de Haan and Stadtm¨uller [6] and Drees [4] . 5

Lemma 3.1. Write γ and Dγ (x) := 1x yγ −1 dy = x γ−1   x γ −1  y ρ−1 1 xγ +ρ −1 Hγ,ρ (x) := 1 y u dudy = 1 ρ γ +ρ

¡ xγγ−1



.

Suppose f is a measurable function and there exist functions a0 (positive) and a1 and a2 (positive or negative) such that for all x > 0

→∞ff

lim t

(tx)

¡ f (t) ¡ a0(t)Dγ (x) ¡ a1 (t)Hγ,ρ (x)g/a2 (t) =: R(x),

(3.1)

!1

exists (t ), then for a judicious choice of a0 , a1 and a2 and given that D, H and R are not linearly dependent, the limit function R will be of the form




Rγ,ρ,η (x) :=

with ρ

—

,η

0

→∞

lim t

→∞

lim t

→∞

lim t

—

1

x

yγ −1




y

1

uρ−1




u

1

sη−1 dsdudy,

(3.2)

. Moreover,

0

a2 (tx)

γ + ρ+η

, = x a2 (t) a1 (tx) ¡ a1 (t)xγ +ρ a2(t)

a0 (tx) ¡ a0 (t)xγ

¡

=

γ +ρ

x

a1(t)xγ

¢ ¢

xη

¡1

η

−

xρ 1

ρ

a2 (t)

,

(3.3)

γ =x

¢

Hγ,ρ (x)

and for each ǫ > 0 there exists t0 such that for t ¸ t0, tx   f (tx) f (t) a0 (t)Dγ (x) a1 (t)Hγ,ρ (x) a2 (t)

e−ǫ| log x| e−γ −ρ−η 

−

−

−

t0

¸

¡R



x) — ǫ.

γ,ρ,η (

(3.4)

Proof

The first part of the proof is similar to that in de Haan and Stadtm¨ uller [6] but there are some differences. We sketch how to proceed. Similar to (2.4) of [6] we get for the limit function R

R(xy) ¡ R(x) =

→∞

lim

t

 γ +ρ R(y)(1 + o(1)) aa22((txt)) + Hγ ,ρ (y) a1 (tx)−aa2 1(t()t)·x +Dγ (y)

−a0 (t)xγ −a1 (t)xγ · xρρ−1

a0 (tx)

a2 (t)



. (3.5)

We have required that there do not exist constants c2 and c3 such that

R(y) ´ c2 Hγ,ρ (y) + c3 Dγ (y). Consequently, the set of vectors

f(R(y), H (y), D(y))gy>0 6

is not contained in a plane, hence there are y1, y2 , y3 such that the matrix 3

(R(yi ), H (yi ), D(yi ))i=1 has rank 3. Then also the transposed matrix has rank 3. So there are no z1 , z2 , z3 such that



zi R(yi ) =



i



zi H (yi ) =



i



zi D(yi ) = 0.

i

Now take z1, z2, z3 such that

zi H (yi ) =

i



zi D(yi ) = 0,

i

then we must have

6

zi R(yi ) = 0.

i

Now multiply the two sides of (3.5) by zi , i = 1, 2, 3, and add the three equations. The resulting equation shows that limt a1 (tx)/a1 (t) exists for x > 0. The limit could be zero (but that is not possible here) or else xγ +ρ+η (this is the definition of η). Similarly

→∞

→∞

lim

t

a1 (tx) ¡ a1(t) ¢ xγ +ρ

(3.6)

a2 (t)

and

→∞

lim

a0 (tx) ¡ a0(t) ¢ xγ

t

¡

a1(t) ¢ xγ

¢

xρ

−1

ρ

(3.7)

a2 (t)

must exist. The limit of (3.7) can be taken of the form xγ Hγ,ρ (x) and then η the limit of (3.6) must be xγ+ρ ¢ x η−1 . Now (3.5) leads to the functional equation (x, y > 0) γ + ρ+η

x

γ

R(y) = R(xy) ¡ R(x) ¡ x

¢

Hρ,η (x)

yγ

¡

γ

1

¡

γ +ρ

x

¢

Hγ,ρ (y) ¢

xη

¡

η

1

.

The solution is R(x) = Rγ,ρ,η (x) + c ¢

xγ +ρ+η

¡

1

γ +ρ+η

.

The second part disappears upon changing the auxiliary functions a0, a1 and a2 a little. In order to prove (3.4) we distinguish various cases. For γ = ρ = η = 0 the proof is similar to the one in Omey and Willekens [7]: 7

t

¡

t

define h(t) := f (t) ¡ 1t 0 f (u)du and k(t) := h(t) 1t 0 h(u)du. Then k is in the class Π with auxiliary function a2 . Next we can write f as a functional of k. The result follows when using Potter type inequalities for k. In all other cases we can use earlier results since the limit relation can be simplified: 1. η < 0, ρ < 0: check the limit relation for + ¡ ρt (γ +¡ρ1) . γ

f0 (t) := f (t)

ρ

6 −1 γ f0 (t) := f (t) ¡ γ t log t.

2. η < 0, ρ = 0, γ = 0: take

3. η < 0, ρ = γ = 0: take 2

f0 (t) := f (t) ¡ (log t) /2.

4. η = 0, ρ < 0: take f0 (t) := f (t) ¡

tγ

¡1 γ

.

5. η = 0, ρ = 0, γ 6 = 0: x−γ f[f (tx) ¡ f (t) ¡ a0 (t)Dγ (x) ¡ a1 (t)Hγ,0 (x)] /a2 (t) ¡ Rγ,0,0 (x)g

=



a0 (tx) (tx)γ

+



¡

a1 (tx) (tx)γ

a0 (t) tγ

¡

¡

a1 (t) tγ

a1 (t) tγ





log x

/ (γa2(t)t−γ )



/ γ 2a2 (t)t−γ



¡ γ−1Hγ,0(x)

¡ γ−2 log x

+ f0 (atx2 ()t−)xfγ0 (t) . For the first two terms we can use earlier results. For the last one we use

→∞

lim

t

f0(tx)

¡ f0(t) = 0.

a2 (t)

8

Using either Bingham, Goldie and Teugels ([2], Th.3.1.7.c) or Geluk and t0 , de Haan ([5], Cor.1.15) we find for each ǫ > 0 a t0 such that for t tx

¸

¸ t0

  f0 (tx)  

¡ f0(t)  < ǫxγ+ǫ . 

a2 (t)

The result follows.

Lemma 3.2. Suppose that the third order conditions hold for log U with p ρ < 0, η < 0 and kjA(n/k )j ! 1. We then use a simplified version γ − +ρ x valid in that case: Hγ− ,ρ (x) can be taken to be γ +ρ−1 upon modifying −

=

the auxiliary functions a little. In particular, we replace A by A∗ A/ρ. Let Y1 , Y2, ¢ ¢ ¢ be i.i.d. with distribution function ¡ /x, x ¸ , and let Y1,n — Y2,n — ¢ ¢ ¢ — Yn,n be the n ¡ th order statistics. For any α > 0, n ! 1, k = k(n) ! 1, k(n)/n ! 0 we have

1 1

−α

1

a0 (Yn−k,n )

=

1

k

k−1 

M

i=0

with for x >

0

Mt (x) :=

− k 1

k i=0

1

flog U (Yn−i,n ) ¡ log U (Yn−k,n )gα

Y

Yn−k,n ( n−i,n

/Yn−k,n



α

)

Dγ (x) + A∗ (t)Hγ

(x) − ,ρ

−



op A∗ nk B nk

+

(

)

(



(3.8)

)



+ A∗ (t)B (t)Rγ− ,ρ,η (x)

_ 0.

The right hand side of (3.8) has the same distribution as

M

1 k

k

Yn−k,n ( i

i=1

Y′

Y

′ α )





+

op A∗ nk B nk (

)

(

(3.9)

)

where 1 , Y2 , ¢ ¢ ¢ are i.i.d. with distribution function 1 ¡ 1/x, independent of Yn−k,n . Next (3.9) equals ′

x

ki  Y − −  αA n ki  Y − −  − Hγ ,ρ Yi k k k − −   − −  − α α− A nk k Hγ ,ρ Yi op −  − − αA nk B nk k ki Y − R Y o γ

1

+ +

=1

(

2

1)

∗( )

i

γ

1

2( ) 1 ∗

( )1

α

+

k i=1

1 ∗( ) γ

Yi

γ

γ

=1

i

γ−

1

γ

=1 α 2 α 1

1

i

2

−

γ

(

1

1, and

α 1

)(1 +

−

(

(1))

γ− ,ρ,η ( i )(1 + p (1))

9

¸

) (3.10)

where Y1 , Y2 , ¢ ¢ ¢ be i.i.d. with distribution function 1 ¡ 1/x, in turn can be written as

  k i=1

1 k

− −1 γ−

γ

Yi

α

A∗

+

A2∗ nk b(1) A∗ (α,γ− ,ρ)

(

op A2∗

n) k

+

(

+

(

)

+

(

o A∗

n )) + ( p k

with

bα



:=

(

=

1, which

b

B nk b(2) (α,γ− ,ρ,η) (

B

(

(3.11)

)

n )) k

 −γ −αΓ(α+1)  Γ(1− − ) − ¡  α − γ +ρ Γ(α+1− − ) −  Γ( + 1) (1−ρ)−α−1 − 1 ρ

γ

)

1 ρ

γ

α

¸

n) k (α,γ− ,ρ)

n) k

(

b(α,γ− ,ρ)

(

x

α

− γ1− ) Γ(α− γ1 ) − Γ(

,

ρ

γ
0.

(3.13)

−

Define γ + ρ+η + ǫ Qt,ǫ (x) := ǫjA∗ (t)B (t)jx −

.

Then by Lemma 3.1 for x ¸ 1 j

(log U (tx) ¡ logU (t)) /a0(t) ¡ Mt (x)j — Qt,ǫ (x),

hence (Mt (x) ¡ Qt,ǫ (x)) _ 0 — (log U (tx) ¡ logU (t)) /a0(t) — Mt (x) + Qt,ǫ (x) 11

Bα

and, since Mt (x) > 0, j

(log U (tx) ¡ logU (t))α /fa0 (t)gα ¡ Mtα (x)j —

—

(Mt (x) + Qt,ǫ (x))

α

For α

Hγ− ,ρ (x) —

(log x)2

(Mt (x) ¡ Qt,ǫ (x)) _ 0 .

¡f

Qt,ǫ (x)

1 note, since

¸

(3.14)

α g

and Rγ− ,ρ,η (x) —

—

ǫjA∗ (t)B(t)jxǫ , Dγ− (x)

(log x)3 ,

— log x,

(Mt (x) + Qt,ǫ (x))α ¡ f(Mt (x) ¡ Qt,ǫ (x)) _ 0gα =α —




Mt (x)+ Qt,ǫ (x)

sα−1 ds — 2αQt,ǫ (x) (Mt (x) + Qt,ǫ (x))α−1

−Qt,ǫ (x))∨0

(Mt (x)

2αǫjA∗ (t)B (t)jxǫ



log x + jA∗ (t)j(log x)2

+jA∗ (t)B (t)j(log x)3 + ǫjA∗ (t)B (t)jxǫ

— 2αǫ A∗ (t)B(t) xǫ 4α−1 j

j

α−1



(log x)α−1 + jA∗ (t)jα−1 (log x)2α−2

 +jA∗ (t)B (t)jα−1(log x)3α−3 + jA∗ (t)B(t)jα−1 xǫ(α−1) . p

Yn−k,n ¢ nk ¡!1, n ! 1, we can assume

Since

A∗ (Yn−k,n )B(Yn−k,n )A−∗ 1( nk )B−1( nk )

—

,

2

A∗ (Yn−k,n ) 1, B(Yn−k,n ) 1. Hence, with Zi,k := Yn−i,n /Yn−k,n , (i = 0, 1, 2, —

n −1 ∗ ( k )j j

A

j

B

—

  1 f (n ) j  k 

−

a0 (Yn−k,n )g−α 1 k

¡ k1



k−1 i=0

flog U (Yn−i,n ) ¡ log U (Yn−k,n )gα

M k −1

f

i=0

¢ ¢ ¢ ),

  α k,n )g  

Yn−k,n (Yn−i,n /Yn−

1 n −1 ( n )¢ — 2αǫA∗ (Yn−k,n )B (Yn−k,n )A− ∗ ( k )B k α−1 ¢4

¢

1 k

Z k

i=1

ǫ i,k

 (log Zi,k )α−1 + (log Zi,k )2α−2 + (log Zi,k )3α−3 + (Zi,k )ǫ(α−1)

12

—

4α ¢ α ¢ ǫ ¢

k

−1

1 k



α−1 2α−2 3α−3 ǫ(α−1) (log Zi,k ) + (log Zi,k ) + (log Zi,k ) + (Zi,k )

ǫ Zi,k

i=0

(3.15) Now for any

k

i=1

ϕ k

d ϕ(Zi−1,k )=

i=1

ϕ(Yi ) ′

with Y1 , Y2 , ¢ ¢ ¢ i.i.d. with distribution function 1 ¡ 1/x, x ¸ 1 and independent of Yn−k,n , hence by the law of large numbers the above expression converges (n ! 1) to ′

′

4α ¢ α ¢ ǫ ¢ E

   eǫZ Z α−1 + Z 2α−2 + Z 3α−3 + eǫ(α−1)Z

with Z a standard exponential random variable. We have proved (3.8) for α 1.

¸

For 0 < α < 1, note first that for ρ < 0, η < 0, each t and γ Qt,ǫ (x)

→∞

x↓1

¡2

¡1

decreases monotonically,

γ−

Qt,ǫ (x) ¡

lim x

lim

γ−

1x

Qt,ǫ (x) ¡

1 2

1 2

γ x−

¢

−

1

γ

γ−

¢

¡

x

¡

1

γ−

=

1

−

2γ

< 0,

= ǫjA∗ (t)B (t)j > 0,

hence there exists 1 — q(t) such that

  Q  Q Q

− x) ¡ 12 ¢ x γ −1 γ

t,ǫ (

t,ǫ (q (t)) ¡

¢

γ−

−

Clearly q(t) »

#

> 0, γ−

(q(t))

−

γ

x) ¡ 12 ¢ x γ −1

t,ǫ (

q(t) ¡ 1

1 2

−

−1

for 1 < x < q(t) = 0,

< 0,

(3.16) for x > q(t).

1 as t ! 1 and

ǫjA∗ (t)B (t)j.

(3.17)

a) For x ¸ q(t) by (3.16) and (3.13)
Mt(x)+Qt,ǫ (x) sα−1ds — α ¢ 2Qt,ǫ (x) ¢ (Mt(x) ¡ Qt,ǫ (x))α−1 α Mt (x)−Qt,ǫ (x)  α−1 1−α ¢ 2 ¢ Q (x) ¢ D (x) — α ¢4 . t,ǫ γ− 13



.

Hence as before

A∗ ( nk ) −1 B( nk ) −1

j

j

j

j

¢

   ¢ f 

a0 (Yn−k,n )g−α k1

— 2α ǫ ¢

—

1−α ¢4

1 ¢2 k

2α ¢ ǫ ¢ 41−α ¢ 2 k1

− k 1 i=0

flog U (Yn−i,n ) ¡ log U (Yn−k,n )gα 1{Zi,k ≥q(Yn−k,n )}

− Z  Z

−  Z k 1

i=0

1 k

γ−

ǫ i,k

i,k

ǫ Zi,k

i,k

i=0

k 1

−

¡ M

¡1

α−1

¡1

i=0

   i,k )1{Zi,k ≥q(Yn−k,n )}  

Z

α

Yn−k,n (

1{Zi,k ≥q(Yn−k,n )}

γ−

γ−

k 1

α−1

γ−



α−1 ¢ 2E eǫZ eγ− Z −1 ¡! 2α ¢ ǫ ¢ 4 γ−

α−1 

.

b) For 1 < x — q(t) by Lemma 3.1, (3.16) and (3.13) jA∗

( nk )j−1 jB ( nk )j−1 jfa0 (Yn−k,n )g−αj ¢ ¢

 −  1 k

k 1 i=0

A∗ ( nk ) −1 B( nk ) −1

— j

j

j

j

¢

1

k

flog U (Yn−i,n ) ¡ log U (Yn−k,n

   Zi,k q(Yn−k,n )} 

1{

≤

¢

k−1 

i=0

)gα

M

Z

Yn−k,n ( i,k ) +

α

QYn−k,n (Zi,k )

1{Zi,k ≤q(Yn−k,n )} (3.18)

A∗ ( nk ) −1 B( nk ) −1

— j

j

j

−  j

¢

1

k

¢

k 1

c¢

i=0

γ−

Zi,k

¡1

γ−

14

α

+

QYn−k,n (Zi,k )

1{Zi,k ≤q(Yn−k,n )}

A∗ ( nk ) −1 B( nk ) −1

— j

j

j

j

¢

1

k

k−1 

j

i=0

j

j

1 ¢

k

with

k−1

Z

c0 ¢ QY

= A∗ ( nk ) −1 B( nk ) −1 j

¢

α

( i,k ) n−k,n

1{Zi,k ≤q(Yn−k,n )}

¢



β c0 ǫ A∗ (Yn−k,n )B(Yn−k,n) Zi,k j

i=0

β := γ− + ρ + η + ǫ. Now 1 k

k

i=1

α

j

Yiαβ ¢ 1{Yi ≤q(Yn−k,n )} — (q(Yn−k,n ))αβ

1

1{Zi,k ≤q(Yn−k,n )}

{ k

k i=1

1 Yi ≤q(Yn−k,n )} .

Using characteristic functions it is easy to see that (q(Yn−k,n ) provided



¡ 1)−1 k1

k q( nk ) ¢

¡

k

i=1

1{Yi ≤q(Yn−k,n )} ¡!1 p

 1 ! 1, i.e.,



k A∗ ¢

n k

B

 n   k

! 1.

ǫ

Hence the right hand side of (3.18) is less than a constant times α . Relation (3.8) follows by adding the results of a) and b). In order to establish (3.10) we use for (a + b)α ¡ aα ¡ α ¢ aα−1b

< with

α(α−1)(α−2) 3 b (a 3!

a > 0 , a+b > 0

¡ α(α2−1) aα−2b2

_ (a + b))α−3

a = Dγ (x) and b = A∗ (t)Hγ− ,ρ (x) + A∗ (t)B (t)Rγ− ,ρ,η (x).

many terms of the form

1 r s A∗ (Yn−k,n )B (Yn−k,n )

k

k

i=1

We then get

′

f (Yi ) ∞

1

. with r, s non-negative integers and 1 f (s) ds s2 < We keep the terms with r + s 2. By the law of large numbers the other terms are of lower order. Hence (3.10). In order to get (3.11) from (3.10), we replace all but the first random part by their means which we can do by the law of large numbers. The explicit expansion (3.12) is obtained by using the beta-integral.

—

15

Y

Proof of Corollary 3.1 Apply the central limit theorem to 1

k

γ

i

k i=1

−

¡1

α

.

γ−

Proof of Lemma 3.3 Combination of expansion (3.10) for various values of

α.

Proof of Theorem 2.2 Since       n (1 ¡ γ− )g  ¡ 1 /A∗ n ¡ (1 ¡ γ ) g  ¡ 1 /A 1 2 − ∗  k    k , T n =  (1 ¡ γ− )g 2 ¡ 1 /A∗ nk ¡ (1 ¡ γ− )g3 ¡ 1 /A∗ nk

Lemma 3.3 and the delta method imply, with Bα and Tα defined in Lemma 3.3, n 

p

kA∗ k

 B ¡ B B1 ¡ B2 (Γ ¡ Γ ) =: T. d Γ1 ¡ Γ2 1 2

T n ¡ B ¡ B ¡! ¡ B2 ¡ B3 (B2 ¡ B3)2 2 3 2 3

Elementary but tedious calculations show that in fact

B1 ¡ B2 = 3 ¡ 12γ− ¡ 3ρ =: τ B2 ¡ B3 3 ¡ 8γ− ¡ ρ

.

Now

¡ 8ˆγ− + 6 ¡ 12ˆγ− . Tn ¡ 3 We use γ ˆ− = 1 ¡ 1/g ˆ1 , but we can do that in two ways: ρ  := 3

² Firstly, we can use different numbers of upper order statistics for γ ˆ−

´

T ´

n as explained in Remark 2.1. So, we now write γˆ− γˆ− (k1) and n and T

n ( ), with the sequence as specified in the conditions of the Theorem and n λ1 with 0 j 1 j 1. Then from Lemma 3.3 1 satisfying 1 ∗(k )

T k k

pk A

and Remark 2.5 lim n→∞

1

!

ρ γ third order parameter, X =U (Y ) with: log U (Y ) = ¡Y γ ¡ Y γ+ρ /2 and Y a r.v. with distribution F (y) = 1 ¡ y−1, y > 1, (ρ = ¡0.5 = γ ). Notice that d

the scale for the simulated mean vlue is different from the previous graphics, revealing a better performance for this model.

ρ Y

Figure 3: Simulated mean value (up) and mean square error (down) of α in (2.3) ( = 1), for a sample of size = 50000 from a model with no third γ γ+ρ 2 and order parameter, namely = ( ) with: log ( ) = 1 a r.v. with distribution ( ) = 1 , 1, ( = 0 5 = ), considering for − the true value — 5000 replications.

α

γ

n X UY U Y ¡Y ¡ Y F y ¡ y− y > ρ ¡ . γ

21

/

In Figure 4 we consider from a Arcsin model a smaller sample size, which lead us to a pattern not much different from Figures 1 or 2, but with a smaller stability region in the lowest portion of the sample. n = 5000,

Figure 4: Simulated mean value (up) and mean square error (down) of ρ in (2.2), for a sample of size n = 5000 from a Arcsin model (ρ = ¡2 = γ ) and γ− estimated by the negative part of Moment estimator γˆ− ´ γˆ− (k) in (2.4) at the optimal number k = kopt of upper order statistics — 5000 replications. In this section we will not present all the pictures associated to each approach for γ− , once it was checked that there were no big differences between them; also, it would become too heavy to present all the possibilities. However, it was found interesting to present a comparison between the two estimators under the three approaches for two models, in Cauchy with γ positive (γ = 1) and in Arcsin model with γ negative (γ = ¡2) . 22

Figure 5: Simulated mean value of ρ in (2.2), for a sample of size n = 50000 from a Arcsin model (ρ = ¡2 = γ ), with γ− according to: i) true value (up); estimated by the negative part of Moment estimator γˆ− ´ γˆ− (k) in (2.4) ii) at the optimal number kopt of upper order statistics (center); iii) γˆ− ´ γˆ− (k) against k (down) — 5000 replications. 23

Figure 6: Simulated mean value of ρα in (2.3) (α = 1), for a sample of size n = 50000 from a Arcsin model (ρ = ¡2 = γ ), with γ− according to: i) true value (up); estimated by the negative part of Moment estimator γˆ− ´ γˆ− (k) in (2.4) ii) at the optimal number kopt of upper order statistics (center); iii) γˆ− ´ γˆ− (k) against k (down) — 5000 replications. 24

Figure 7: Simulated mean value of ρ in (2.2), for a sample of size n = 50000 from a Cauchy model (ρ = ¡2, γ = 1 and γ− = 0), with γ− according to: i) true value (up); estimated by the negative part of Moment estimator γˆ− ´ γˆ− (k) in (2.4) ii) at the optimal number kopt of upper order statistics (center); iii) γ ˆ− ´ γˆ− (k ) against k (down) — 5000 replications. 25

Figure 8: Simulated mean value of ρα in (2.3) (α = 1), for a sample of size n = 50000 from a Cauchy model (ρ = ¡2, γ = 1 and γ− = 0), with γ− according to: i) true value (up); estimated by the negative part of Moment estimator γˆ− ´ γˆ− (k) in (2.4) ii) at the optimal number kopt of upper order statistics (center); iii) γˆ− ´ γˆ− (k) against k (down) — 5000 replications. 26

In Figure 9, we illustrate the exact performance of the estimators ρ in (2.2) and ρα (α = 1) in (2.3), a sample path for a generated sample, of size n = 50000 from a Arcsin model (ρ = ¡2 = γ ). Notice the smoother pattern of ρα .

Figure 9: Sample paths of ρ in (2.2) (up) and of ρα (α = 1) in (2.3) (down) against k, for a sample of size n = 50000 from a Arcsin model (ρ = ¡2 = γ ), with γ− estimated by the negative part of Moment estimator γˆ− ´ γˆ− (k) in (2.4) at the optimal number k = kopt of upper order statistics .

27

Finally, Figure 10 is equivalent to Figure 9, but for a smaller sample size of n = 5000. Again the smoother pattern is achieved by ρα , but only for large values k both paths of the estimators against k, have some interpretation for estimating ρ.

Figure 10: Sample paths of ρ in (2.2) (up) and of ρα (α = 1) in (2.3) (down) against k, for a sample of size n = 5000 from a Arcsin model (ρ = ¡2 = γ ), with γ− estimated by the negative part of Moment estimator γˆ− ´ γˆ− (k) in (2.4) at the optimal number k = kopt of upper order statistics .

28

We have also carried out an application of the estimators to real data. The data consist of the total life span (in days), of all people born in the years 1877-1881, still alive on the 1st of January of 1971 and who died as residents of the Netherlands, who reached the age of 94 years (10391 sample size) — data from the Dutch Statistical Office (C.B.S.). These data has been studied by Aarssen and de Haan [1], on the extremal parameter γ estimation context. Here we studied the exact performance of both estimators, ρ and ρα, for a wide set of values for α. In Figure 11 we presente the sample paths of ρ and ρα (α = 0.2) against k; the graphics seam to suggest a value for the second order ρ parameter between -2 and -3.

Figure 11: Sample paths of ρ in (2.2) and of ρα (α = 0.2) in (2.3) against for mortality data from Netherlands — ages for men and women born in 1877-1881 (10391 sample size). k,

29

References [1] Aarssen, K. and de Haan, L. (1994). On the maximal life span of humans. Mathematical Population Studies, vol.4(4), 259-81. [2] Bingham, N., Goldie, C., and Teugels, J. (1987). Cambridge University Press, New York.

Regular Variation.

[3] Dekkers, A. L. M., J. H. J. Einmahl and de Haan, L. (1989). A moment estimator for the index of an extreme-value distribution. Ann. Statist. 17, 1833-1855. [4] Drees, H. (1998). On smooth statistical tail functionals. Scandinavian Journal of Statistics , 25, 187-210. [5] Geluk, J. and de Haan, L. (1987). Regular Variation, Extensions and Tauberian Theorems. CWI Tract 40, Center for Mathematics and Computer Science, Amsterdam, Netherlands. [6] de Haan, L. and U. Stadtm¨uller (1996). Generalized regular variation of second order. J. Austral. Math. Soc. (A) 61, 381-395. [7] Omey,E. and Willekens, E. (1988). π-variation with remainder. J. London Math. Soc.

37, 105-118.

30