The Convergence Rate for the Normal Approximation of ...yqi/mydownload/yqi08.pdf · The Convergence Rate for the Normal Approximation of Extreme Sums ... WCNA 2008, Orlando, July

The Convergence Rate for the Normal Approximation of Extreme Sums

The Convergence Rate for the NormalApproximation of Extreme Sums

Yongcheng Qi

University of Minnesota Duluth

WCNA 2008, Orlando, July 2-9, 2008


This talk is based on

a joint work with Professor Shihong Cheng


Outline

Outline

IntroductionMain ResultsSketch of Proofs


Outline

Outline



Outline

Outline



Introduction

Extreme Sum

Let {X ,X1,X2, · · · } be a sequence of independent andidentically distributed random variables with distribution FFor each n ≥ 1 let Xn,1 ≤ · · · ≤ Xn,n denote the orderstatistics of X1, · · · ,Xn

Define Sn,k =∑k

j=1 Xn,n−j−1 as an extreme sum, wherek = kn satisfies

limn→∞

kn =∞ and limn→∞

kn

n= 0 (1)


Introduction

Extreme Sum


Define Sn,k =∑k


limn→∞


kn

n= 0 (1)


Introduction

Extreme Sum


Define Sn,k =∑k


limn→∞


kn

n= 0 (1)


Introduction

Motivation

Why extreme sums?Influence of extremes sums in the partial sums

∑nj=1 Xj

–Trimmed sums or extreme sums

Estimation of the tail of a distribution in extreme-valuestatistics–Hill estimator and moment estimator for tail-index


Introduction

Motivation


∑nj=1 Xj




Introduction

Motivation


∑nj=1 Xj




Introduction

Domain of Attraction of An EV Distribution

Assume that F belongs to the domain of attraction of anextreme value distribution, i.e. there exist constants cn > 0and dn ∈ R such that

Xn,n − dn

cn

d→ exp{−(1 + γx)−1/γ} =: Gγ(x) (2)

for some γ ∈ R.Gγ : extreme-value distributionγ > 0: Frechetγ = 0: Gumbelγ < 0: Weibull


Introduction

Domain of Attraction of An EV Distribution

Assume that F belongs to the domain of attraction of anextreme value distribution, i.e. there exist constants cn > 0and dn ∈ R such that

Xn,n − dn

cn

d→ exp{−(1 + γx)−1/γ} =: Gγ(x) (2)

for some γ ∈ R.Gγ : extreme-value distributionγ > 0: Frechetγ = 0: Gumbelγ < 0: Weibull


Introduction

Some Distributions

Heavy-tailed distributions (γ > 0)

1− F (x) = x−1/γL(x), x > 0

where L(x) is a slowly varying function at infinity;Normal and exponential (γ = 0)Distributions with a finite right-endpoint ωF (γ < 0)

1− F (x) ∼ c(ωF − x)−1/γ as x ↑ ωF .


Introduction

Some Distributions


1− F (x) = x−1/γL(x), x > 0


1− F (x) ∼ c(ωF − x)−1/γ as x ↑ ωF .


Introduction

Some Distributions


1− F (x) = x−1/γL(x), x > 0


1− F (x) ∼ c(ωF − x)−1/γ as x ↑ ωF .


Introduction

Limiting Distribution of Sn,kn under (2)

Csorgo, Horvath and Mason (1986),Csorgo and Mason (1986),Lo (1989),Haeusler and Mason (1987), andCsorgo, Haeusler and Mason (1991).In particular, Csorgo, Haeusler and Mason (1991) provedthat

the limiting distribution of Sn,kn is normalif and only if γ ≤ 1/2.


Introduction

Limiting Distribution of Sn,kn under (2)

Csorgo, Horvath and Mason (1986),Csorgo and Mason (1986),Lo (1989),Haeusler and Mason (1987), andCsorgo, Haeusler and Mason (1991).In particular, Csorgo, Haeusler and Mason (1991) provedthat

the limiting distribution of Sn,kn is normalif and only if γ ≤ 1/2.


Introduction

Distributional Convergence Rate

Mijnheer (1988) showed that

supx|P(n−1/αSn,kn ≤ x)−G(x ;α)| = O(max(k1−1/α

n ,n−1))

if1− F (x) = x−α + O(x−r ) x →∞

for some r > α, 0 < α < 1, where G(x ;α) is an asymmetricstable law.

The above condition implies (2) with γ = 1/α > 1.


Introduction

Distributional Convergence Rate

Mijnheer (1988) showed that

supx|P(n−1/αSn,kn ≤ x)−G(x ;α)| = O(max(k1−1/α

n ,n−1))

if1− F (x) = x−α + O(x−r ) x →∞

for some r > α, 0 < α < 1, where G(x ;α) is an asymmetricstable law.

The above condition implies (2) with γ = 1/α > 1.


Main Results

Our interest is

To estimate rates of convergence to normality for the extremesums under conditions:

(2) with γ < 1/2, and

Von-Mises condition


Main Results

Our interest is


(2) with γ < 1/2, and

Von-Mises condition


Main Results

Our interest is


(2) with γ < 1/2, and

Von-Mises condition


Main Results

Von-Mises Conditions

Denote ωF = sup{x : F (x) < 1}.

The following von-Mises conditions are sufficient for (2): Fhas the derivative function F ′ satisfying

γ > 0 : ωF =∞, limx→∞

xF ′(x)

1− F (x)=

1γ

;

γ < 0 : ωF <∞, limx→∞

(ωF − x)F ′(x)

1− F (x)= −1

γ;

γ = 0 : limx→ωF

F ′(x)

[1− F (x)]2

∫ ωF

x[1− F (u)]du = 1.


Main Results


Denote ωF = sup{x : F (x) < 1}.

The following von-Mises conditions are sufficient for (2): Fhas the derivative function F ′ satisfying

γ > 0 : ωF =∞, limx→∞

xF ′(x)

1− F (x)=

1γ

;

γ < 0 : ωF <∞, limx→∞

(ωF − x)F ′(x)

1− F (x)= −1

γ;

γ = 0 : limx→ωF

F ′(x)

[1− F (x)]2

∫ ωF

x[1− F (u)]du = 1.


Main Results


Let V (t) = inf{x : [1− F (x)]−1 ≥ t} for t ≥ 1

These von-Mises conditions are equivalent to theassumption that V has a derivative functionV ′ ∈ Rv(γ − 1), where Rv(γ − 1) denotes the class of allregularly varying functions with index γ − 1, i.e.

limt→∞

V ′(tx)

V ′(t)= xγ−1, ∀x > 0. (3)

See Cheng, de Haan and Huang (1997).


Main Results


Let V (t) = inf{x : [1− F (x)]−1 ≥ t} for t ≥ 1

These von-Mises conditions are equivalent to theassumption that V has a derivative functionV ′ ∈ Rv(γ − 1), where Rv(γ − 1) denotes the class of allregularly varying functions with index γ − 1, i.e.

limt→∞

V ′(tx)

V ′(t)= xγ−1, ∀x > 0. (3)

See Cheng, de Haan and Huang (1997).


Main Results

Notation

Φ, φ – the standard normal distribution function and itsdensityD2X – the variance of a random variable X ;Z – a random variable with distribution functionFZ (x) = 1− x−1, x ≥ 1.Set

an =n − kn + 1

n + 1; bn =

√an(1− an)

n + 1;

ln =1

1− an; τn =

√an(EV (lnZ )− V (ln)

)DV (lnZ )


Main Results

Notation


an =n − kn + 1

n + 1; bn =

√an(1− an)

n + 1;

ln =1

1− an; τn =


)DV (lnZ )


Main Results

Notation


an =n − kn + 1

n + 1; bn =

√an(1− an)

n + 1;

ln =1

1− an; τn =


)DV (lnZ )


Main Results

We are going to estimate the convergence rate

∆n(x) = P( Sn,kn − knEV (lnZ )√

kn(1 + τ2n )D2V (lnZ )

≤ x)− Φ(x)


Main Results

Theorem 1

Theorem

Under (1), if (3) holds for some γ ∈ [1/3,1/2), then

supx∈R|∆n(x)| = o(k−εn ) (4)

holds for anyε ∈ (0, (2γ)−1 − 1). (5)


Main Results

Theorem 2

Theorem

Under (1), if (3) holds for γ < 1/3, then

∆n(x) =1√kn

2(1− 2γ)1/2

6(2(1− γ)

)3/2(1− 3γ)

((ξ1(γ) + ξ2(γ)x2

)φ(x)

+Φγ(x))

+ o( 1√

kn

)(6)

holds uniformly on x ∈ R, where

ξ1(γ) = 7− 8γ + 9γ2, ξ2(γ) = 1− 16γ − 15γ2 + 26γ3

Φγ(x) = 2(1− 2γ)(1− 3γ)

∫ x

−∞(3u − u3)φ(u)du.


Sketch of the proofs


Express the extreme sum as a sum of kn conditionally iidRVs

Estimate the moments of these RVs–by using the regularity of V and V ′

Find normal approximate error for the distribution of thesum of these conditionally iid RVs





















Some Notation

Let Z1,Z2, · · · be independent random variables having thesame distribution function as Z and let Zn,1 ≤ · · · ≤ Zn,n bethe order statistics of Z1, · · · ,Zn. Then

(Xn,1, · · · ,Xn,n)d= (V (Zn,1), · · · ,V (Zn,n)).

For every u ∈ R set

an(u) = {(an + bnu) ∧ 1} ∨ 0; ln(u) =(1− an(u)

)−1;

Sn,kn (u) =kn∑

j=1

V (ln(u)Zj); φn(u) = φ(u)(1− u3 − 3u

3√

kn

).



Some Notation

Let Z1,Z2, · · · be independent random variables having thesame distribution function as Z and let Zn,1 ≤ · · · ≤ Zn,n bethe order statistics of Z1, · · · ,Zn. Then

(Xn,1, · · · ,Xn,n)d= (V (Zn,1), · · · ,V (Zn,n)).

For every u ∈ R set

an(u) = {(an + bnu) ∧ 1} ∨ 0; ln(u) =(1− an(u)

)−1;

Sn,kn (u) =kn∑

j=1

V (ln(u)Zj); φn(u) = φ(u)(1− u3 − 3u

3√

kn

).



Fact 1:

Under (1), we have

supx∈R

∣∣P(Sn,kn ≤ x)−∫ ∞−∞

P(Sn,kn (u) ≤ x)φn(u)du∣∣ = o

( 1√kn

).

(7)



Fact 2:

If (3) holds for γ ∈ R, then

limt→∞

V (tx)− V (t)tV ′(t)

=xγ − 1γ

, ∀x > 0 (8)

with convention xγ−1γ

∣∣∣γ=0

= log x for x > 0. Furthermore, for

any δ > 0, there exists a tδ > 0 such that

(1− δ)xγ−δ − 1γ − δ

≤ V (tx)− V (t)tV ′(t)

≤ (1 + δ)xγ+δ − 1γ + δ

(9)

holds for all x ≥ 1 and t ≥ tδ.



Fact 3:

Under (1) and (3), we have

limt→∞

∫ ∞1

[V (tv)− V (t)tV ′(t)

]j dvv2 =

∫ ∞1

(vγ − 1)j

γ jv2 dv , ∀γ < j−1

(10)for all j = 1,2, · · · .



Some estimates

We conclude

limn→∞

EV (lnZ )− V (ln)

lnV ′(ln)=

∫ ∞1

vγ − 1γ

dvv2 =

11− γ

if γ < 1.

(11)

limn→∞

DV (lnZ )

lnV ′(ln)=

1(1− γ)(1− 2γ)1/2 =: σ(γ) if γ < 1/2.

(12)



Some estimates

We conclude

limn→∞

EV (lnZ )− V (ln)

lnV ′(ln)=

∫ ∞1

vγ − 1γ

dvv2 =

11− γ

if γ < 1.

(11)

limn→∞

DV (lnZ )

lnV ′(ln)=

1(1− γ)(1− 2γ)1/2 =: σ(γ) if γ < 1/2.

(12)



Fact 4:

If γ < 1/2, then under (1) and (3), the following equations holduniformly on u ∈ [−k1/6

n , k1/6n ]:

V (ln(u))− V (ln) =lnV ′(ln)u[1 + o(1)]√

kn; (13)

EV (ln(u)Z )−EV (lnZ ) =uτnDV (lnZ )√

kn+

(1 + γ)lnV ′(ln)u2[1 + o(1)]

2kn(1− γ);

(14)DV (lnZ )

DV (ln(u)Z )− 1 = −u[γ + o(1)]√

kn. (15)



Fact 5:

Suppose that (1) and (3) hold. If γ < 1/2, then there exist twofunctions αn,1(u) and αn,2(u) such that

ηn(x ,u) :=DV (lnZ )x −

√kn[EV (ln(u)Z )− EV (lnZ )]

DV (ln(u)Z )(16)

= x − τnu − αn,1(u)xu − αn,2(u)u2

holds for all x ∈ R and u ∈ [−k1/6n , k1/6

n ], where

limn→∞

√knαn,1(u) = γ and lim

n→∞

√knαn,2(u) = [2σ(γ)]−1

uniformly on [−k1/6n , k1/6

n ].



For n = 1,2, · · · and x ,u ∈ R, denote

Zn,j(u) =V (ln(u)Zj)− EV (ln(u)Z )√

D2V (ln(u)Z ), j = 1, · · · , kn;

Sn,kn (u) =1√kn

kn∑j=1

Zn,j(u).



Proof of Theorem 1.

Denote

∆n,1 = supx∈R

∣∣∣ ∫|u|>k1/6

n

[P(Sn,kn (u) ≤ ηn(x ,u))− Φ(x − τnu)]φ(u)du∣∣∣;

∆n,2 = supx∈R

∣∣∣ ∫ k1/6n

−k1/6n

[P(Sn,kn (u) ≤ ηn(x ,u))− Φ(ηn(x ,u))]φ(u)du∣∣∣;

∆n,3 = supx∈R

∣∣∣ ∫ k1/6n

−k1/6n

[Φ(ηn(x ,u))− Φ(x − τnu)]φ(u)du∣∣∣.



SinceΦ( x√

1 + τ2n

)=

∫ ∞−∞

Φ(x ± τnu)φ(u)du, (17)

we have from (7) and (16) that

supx∈R|∆n(x)| = sup

x∈R

∣∣∣∆n

( x√1 + τ2

n

)∣∣∣≤

∣∣∣ supx∈R

∫ ∞−∞

[P(Sn,kn (u) ≤ ηn(x ,u))− Φ(x − τnu)]φ(u)du|

+C√kn

≤ ∆n,1 + ∆n,2 + ∆n,3 +C√kn.



∆n,1 ≤ P(|N| > k1/6n ) = O(k−1/2

n );

From the Berry-Esseen theorem we get for any ε ∈ (0,1/2]

∆n,2 ≤ sup|u|≤k1/6

n

supx∈R|P(Sn,kn (u) ≤ ηn(x ,u))− Φ(ηn(x ,u))|

≤ Ck−εn

∆n,3 = O(k−1/2n ).



∆n,1 ≤ P(|N| > k1/6n ) = O(k−1/2

n );


∆n,2 ≤ sup|u|≤k1/6

n


≤ Ck−εn

∆n,3 = O(k−1/2n ).



∆n,1 ≤ P(|N| > k1/6n ) = O(k−1/2

n );


∆n,2 ≤ sup|u|≤k1/6

n


≤ Ck−εn

∆n,3 = O(k−1/2n ).



Proof of Theorem 2

For n = 1,2, · · · and t , x ,u ∈ R, let

ρ(γ) =2(1 + γ)

√1− 2γ

1− 3γ;

G(x) = Φ(x)− ρ(γ)(x2 − 1)φ(x)

3√

kn;

Fn(x ,u) = P(Sn,kn (u) ≤ x);

fn(t ,u) = E exp{it Zn,1(u)}(the characteristic function of Zn,1(u));

ψn(t ,u) = log fn( t√

kn,u)

+t2

2kn.



Fact 6:

Suppose (1) and (3) hold. Then

limn→∞

sup|u|≤k1/6

n

|EZ 3n,1(u)− ρ(γ)| = 0, (18)

and for any positive p with γp < 1− 3γ

lim supn→∞

sup|u|≤k1/6

n

E |Zn,1(u)|3+p <∞. (19)



Fact 7:

Under (1) and (3), we have

sup|u|≤k1/6

n

supx∈R|Fn(x ,u)−G(x)| = o

( 1√kn

). (20)



Proof of Theorem 2.

We get

∆n

( x√1 + τ2

n

)=

∫ k1/6n

−k1/6n

[Φ(ηn(x ,u))− Φ(x − τnu)])φn(u)du

− ρ(γ)

3√

kn

∫ k1/6n

−k1/6n

[η2n(x ,u))− 1]φ(ηn(x ,u))φn(u)du

+

∫ k1/6n

−k1/6n

Φ(x − τnu))[φn(u)− φ(u)]du + o( 1√

kn

)=: Kn,1(x) + Kn,2(x) + Kn,3(x) + o

( 1√kn

)holds uniformly on x ∈ R.



Thank you very much!


References

References:Bingham, N.H., Goldie, C.M., Teugels, J.L. (1987). RegularVariation. Cambridge University, New York.Cheng, S. (1997). Approximation to the expectation of afunction of order statistics and its applications. Acta Math. Appl.Sinica (English Ser.) 13, 71-86.Cheng, S., de Haan, L. and Huang, X. (1997). Rate ofconvergence of intermediate order statistics. J. Theoret.Probab. 10, 1-23.Csorgo, S., Horvath, L. and Mason, D. M (1986). What portionof the sample makes a partial sum asymptotically stable ornormal? Probab. Theory Rel. Fields 72 1-16.Csorgo, S. and Mason, D. M (1986). The asymptoticdistribution of sums of extreme values from a regularly varyingdistribution. Ann. Probab. 14, 974-983.Csorgo, S., Haeusler, E. and Mason, D. M (1991). Theasymptotic distribution of extreme sums. Ann. Probab. 19,


References

783-811.Geluk, J. L. and de Haan, L. (1970). Regular Variation,Extension and Tauberian Theotems. CWI Tracts 40, Centre forMathematics and Computer Science, Amsterdam.Haeusler, E. and Mason, D. M (1987). A law of the iteratedlogaritheorem for sums of extreme values from a distributionwith a regularly varying upper tail. Ann. Probab. 15 932-953.Lo, G. S. (1989). A note on the asymptotic normality of sums ofextreme values. J. Statist. Plann. Inference 22, 127-136.Petrov, V. V. (1995). Limit Theorems of Probability Theory:Sequences of Independent Random Variables. ClarendonPress, Oxford.Reiss, R.-D. (1989). Approximate Distribution of OrderStatistics with Applications to Nonparametric Statistics.Springer, New York.

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The Convergence Rate for the Normal Approximation of ...yqi/mydownload/yqi08.pdf · The Convergence Rate for the Normal Approximation of Extreme Sums ... WCNA 2008, Orlando, July