Trimmed sums of long range dependent moving averages

Statistics and Probability Letters 78 (2008) 2536–2542

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Trimmed sums of long range dependent moving averages Rafał Kulik a,∗ , Mohamedou Ould Haye b a Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Av., K1N 6N5 Ottawa, ON, Canada b School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, K1S 5B6 Ottawa, ON, Canada

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a b s t r a c t

Article history: Received 31 October 2006 Received in revised form 7 May 2007 Accepted 14 February 2008 Available online 15 March 2008

In this paper we establish asymptotic normality of trimmed sums for long range dependent moving averages. Our results extend those of Ho and Hsing [Ho, H.-C., Hsing, T. 1996. On the asymptotic expansion of the empirical process of long-memory moving averages. Ann. Statist. 24, 992–1024] and Wu [Wu, W.B., 2005. On the Bahadur representation of sample quantiles for dependent sequences. Ann. Statist. 33 1934–1963]. © 2008 Elsevier B.V. All rights reserved.

1. Introduction Let {i , i ≥ 1} be a centered sequence of i.i.d. random variables. Consider the class of stationary linear processes Xi =

∞ X

ck i−k ,

i ≥ 1.

(1)

k=0

We assume that the sequence ck , k ≥ 0, is regularly varying with index −β, β ∈ (1/2, 1). This means that ck ∼ k−β L0 (k) as k → ∞, where L0 is a slowly varying function at infinity. We shall refer to all such models as long range dependent (LRD) linear processes. In particular, if the variance exists (which is assumed throughout the whole paper), then the covariances ρk := EX0 Xk decay at the hyperbolic rate, ρk = k−(2β−1) L(k), where limk→∞ L(k)/L20 (k) = B(2β − 1, 1 − β) and B(·, ·) is the beta-function. Consequently, the covariances are not summable (cf. Giraitis and Surgailis (2002)). Assume that X1 has a continuous distribution function F . For y ∈ (0, 1) define Q (y) = inf {x : F (x) ≥ y}, the P corresponding quantile function. Given the ordered sample X1:n ≤ · · · ≤ Xn:n of X1 , . . . , Xn , let Fn (x) = n−1 ni=1 1{Xi ≤x} be the empirical distribution function and Qn (·) be the corresponding left-continuous sample quantile function, i.e. Qn (y) = Xk:n for P k−1 < y ≤ nk . Define Ui = F (Xi ) and En (x) = n−1 ni=1 1{Ui ≤x} . Denote by Un (·) the corresponding uniform sample quantile n function. Let r be an integer and define Yn,r =

n X

X

r Y

i=1 1≤j1 <···≤jr s=1

so that Yn,0 = n, and Yn,1 =

cjs i−js ,

Pn

i=1

n ≥ 1,

Xi . If p

< (2β − 1)−1 ; then

σn2,p := Var(Yn,p ) ∼ n2−p(2β−1) L20p (n).

∗ Corresponding author. Tel.: +1 613 562 5800; fax: +1 613 562 5776. E-mail addresses: [email protected] (R. Kulik), [email protected] (M. Ould Haye). 0167-7152/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2008.02.033

R. Kulik, M. Ould Haye / Statistics and Probability Letters 78 (2008) 2536–2542

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Define now the general empirical, the uniform empirical, the general quantile and the uniform quantile processes respectively as follows:

βn (x) = σn−,11 n(Fn (x) − F (x)), αn (y) = σn,1 n(En (y) − y), −1

y ∈ (0, 1),

qn (y) =

σn,1 n(Q (y) − Qn (y)),

un (y) =

σn,1 n(y − Un (y)),

−1

−1

x ∈ R, y ∈ (0, 1),

y ∈ (0, 1).

The aim of this paper is to study the asymptotic behavior of trimmed sums from the long range dependent sequence defined by (1). P Let Tn (m, k) = ni=−mk +1 Xi:n and note that (see below for a convention concerning integrals) Tn (m, k) = n

Z

1−k/n m/n

Qn (y)dy.

(2)

Ho and Hsing observed in Ho and Hsing (1996) that, under appropriate conditions on F , n X −1 Xi = oP (1), sup qn (y) + σn,1 y∈[y0 ,y1 ]

(3)

i=1

where 0 < y0 < y1 < 1. Eq. (3), together with (2), yields asymptotic normality in the case of heavy trimming m = mn = [δ1 n], k = kn = [δ2 n], where 0 < δ1 < δ2 < 1 and [·] is the integer part; see Ho and Hsing (1996, Corollary 5.2). This agrees with the i.i.d. situation; see Stigler (1973). We note in passing that the asymptotic behavior of heavy trimmed sums, under weaker regularity conditions such as those of Ho and Hsing, can also be concluded from Lemma 16 in Wu (2005). However, the representation (2) requires some additional assumptions on F . In order to avoid them, we may study R asymptotics for the trimmed sums via the integrals of the form αn (y)dQ (y). This approach was initiated in two papers by Csörgő et al. (1986a,b) Then, Csörgő, Haeusler, Horváth and Mason took this route to provide the full description of the weak asymptotic behavior of the trimmed sums in the i.i.d. case. We refer the reader to Csörgő et al. (1991) for an extensive up-to-date discussion and a survey of results. In the LRD case, instead of using the Brownian bridge approximation, we can use the reduction principle for the general empirical processes as studied in Giraitis and Surgailis (2002) and Ho and Hsing (1996) or Wu (2003); see Lemma 2. We can use then an approach similar to that of the above mentioned authors to establish the asymptotic normality in the case of moderate trimming, complementing the Ho and Hsing result. In what follows C will denote a generic constant which may be different in each of its appearances. Also, for any sequences an and bn , we write an ∼ bn if limn→∞ an /bn = 1. Further, let `(n) be a slowly varying function, possibly different in each place at which it appears. On the other hand, L(·), L0 (·), L1 (·), etc., are slowly varying functions, fixed from the time they appear. Moreover, g(k) denotes the kth-order derivative of a function g and Z is a standard normal random variable. For any stationary sequence {Vi , i ≥ 1}, we will denote by V the random variable with the same distribution as V1 . 2. Statement of results and discussion Let F be the marginal distribution function of the centered i.i.d. sequence {i , i ≥ 1}. Also, for a given integer p, the derivatives F(1) , . . . , F(p+3) of F are assumed to be bounded and integrable. Note that these properties are inherited by the distribution F as well (cf. Ho and Hsing (1996) or Wu (2003)). Furthermore, assume that E41 < ∞. These conditions are needed to establish the reduction principle for the empirical process and will be assumed throughout the paper. We need to impose some conditions on F . The first assumption is that the right tail of the distribution F satisfies the following von Mises condition: lim

x→∞

xf (x)

1 − F (x)

= α > 0.

(4)

The condition (4) together with its counterpart for the left tail will be referred to as X ∈ MDA(Φα ), since, in particular, (4) implies that X belongs to the maximal domain of attraction of the Fréchet distribution with index α. Then Q (1 − y) = y−1/α L1 (y−1 )

as y → 0

(5)

and the density-quantile function f Q (y) = f (Q (y)) satisfies f Q (1 − y) = y1+1/α L2 (y−1 ) as y → 0, where L2 (u) = α(L1 (u))−1 . The second type of assumption is that F belongs to the maximal domain of attraction of the double-exponential Gumbel distribution. Then the corresponding von Mises condition implies

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lim

f Q (1 − y)

R1

1−y

(1 − u)/f Q (u)du y2

y→0

= 1.

 −1 R1 Thus, with L3 (y−1 ) = y−1 1−y (1 − u)/f Q (u)du one has f Q (1 − y) = yL3 (y−1 ) and L3 is slowly varying at infinity. The

above assumption, together with its left-tailed counterpart, will be referred to as X ∈ MDA(Λ). P Recall that Qn (y) = inf {x : Fn (x) ≥ y} = Xk:n if k−n 1 < y ≤ nk . Let Tn (m, k) = in=−mk +1 Xi:n and Z 1−k/n µn (m, k) = n Q (y)dy. m/n

The main result of this paper is the following theorem. Theorem 1 (Moderate Trimming). Let p be the smallest positive integer such that (p + 1)(2β − 1) > 1 and assume that for R1 r = 1, . . . , p, 0 F (r) (Q (y))dQ (y) < ∞. Let kn = nξ , ξ ∈ (0, 1). Assume that either F ∈ MDA(Λ) or F ∈ MDA(Φα ) for some α < ∞ such that  1−ξ 3   if β ≥ , α > 1−β  4 (6) 1 3   α ≥ 4 . if β ∈ , 2 4 d

Then σn−,11 (Tn (kn , kn ) − µn (kn , kn )) → Z . 2.1. Remarks Remark 2.1. Let us discuss condition (6) of Theorem 1. Recall that the main assumption is E41 < ∞ which implies EX14 < ∞ and thus α ≥ 4, in particular. If β is close to 43 then there is no additional restriction on moments. However, the restriction on α is very sensitive for β close to 12 or 1. The condition E41 < ∞ appeared since we used the reduction principle from Wu (2003). This moment restriction can be weakened to E|1 |2+δ < ∞, δ > 0, as indicated in Giraitis and Surgailis (2002). However, in this case the rates in the reduction principle are not as good as those in Wu (2003), which is crucial in our method. R1 Remark 2.2. The conditions 1/2 F (r) (Q (y))dQ (y) < ∞, r = 1, . . . , p, are not restrictive at all, since they are fulfilled for most distributions with a regularly varying density-quantile function f (Q (1 − y)), for which we refer the reader to Parzen (1979). Consider for example the Pareto case, Q (y) = C (1 − y)−1/α for all y such that 0 < y1 < y < 1, where y1 is fixed. Then for r ≥ 1 and all x exceeding some value x0 , F (r) (x) = Cx−(α+r) . Thus, we have Z 1 Z 1 F (r) (Q (y))dQ (y) = (1 − y)(r−1)/α dy < ∞. 1/2∨x0

1/2∨x0

If, additionally, we impose inf x∈(a,b) f (x) > 0, where a = sup{x : F (x) = 0}, b = inf {x : F (x), x = 1}, −∞ ≤ a < b ≤ ∞, then R1 in view of the assumed boundedness of derivatives F (r) (·), the integral 1/2 F (r) (Q (y))dQ (y) is finite. The condition is trivially fulfilled for the exponential right tail. Consider the standard normal distribution. Then F (r) (x) = f (x)W(r−1) (x), where W(r−1) (·) is a polynomial of order r − 1. Consequently, the finiteness of the integral is equivalent to R1 r−1 (y)dy < ∞, which is the case. 1/2 Q Remark 2.3. As noted by a referee, in some applications one knows the properties of f , rather than those of f . Assume that F ∈ MDA(Φα ). Then also F ∈ MDA(Φα ) since limx→∞ P(X1 > x)/P(|| > x) = const. ∈ (0, ∞). For α > 2 the P 2 result is valid as long as ∞ j=0 cj < ∞, in particular, in the case of long range dependence (see Mikosch and Samorodnitsky (2000) for details). If 1 is normally distributed, then X is too; thus in this special case both F and F belong to MDA(Λ). Pn Remark 2.4. In a similar vein to Theorem 1, partial sums i=1 G(Xi ), where G is a measurable function, may be considered. Assuming for example that Xi , i ≥ 1, is a Gaussian sequence and that G has the Hermite rank τ , one can state the corresponding results, replacing σn,1 and Z by σn,τ and Zτ respectively, where Zτ is a (possibly non-Gaussian) random variable defined as the τ -fold integral with respect to a Brownian motion; see Dobrushin and Major (1979). In the context of Theorem 1, however, one needs to assume that G(X1 ) is in the appropriate domain of attraction. For example, if G(x) = log(x+ )α and X ∈ MDA(Φα ), then G(X ) ∈ MDA(Λ). The result of Theorem 1 is still valid, since the Hermite rank of G is 1. On the other hand, if G(x) = x2 − 1 or G(x) = |x| − E|X1 |, then the Hermite rank is 2 and the domain of attraction for G(X ) can be easily characterized by that of X .

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3. Proofs Let p be a positive integer and let Sn,p (x) =

n X

(1{Xi ≤x} − F (x)) +

i=1

=:

n X

p X

(−1)r−1 F (r) (x)Yn,r

r=1

(1{Xi ≤x} − F (x)) + Vn,p (x),

i=1

where F (r) is the rth-order derivative of F . Setting Ui = F (Xi ) and x = Q (y) in the definition of Sn (·) we arrive at its uniform version, S˜ n,p (y) =

n X

(1{Ui ≤y} − y) +

i=1

=:

n X

p X

(−1)r−1 F (r) (Q (y))Yn,r

r=1

(1{Ui ≤y} − y) + V˜ n,p (y).

i=1

Define ( dn,p =

1 5/2 n−(1−β) L− (log log n)3/4 , 0 (n)(log n)

n−p(β− 2 ) L0 (n)(log n)1/2 (log log n)3/4 , 1

p

(p + 1)(2β − 1) ≥ 1 (p + 1)(2β − 1) < 1.

We shall need the following lemma. Lemma 2 (Wu, 2003). Let p be a positive integer. Then, as n → ∞, 2 p n X X r−1 (r) E sup (1{Xi ≤x} − F (x)) + (−1) F (x)Yn,r = O(Ξn + n(log n)2 ), x∈R i=1

r=1

where ( O(n),

Ξn =

2(p+1)

O(n2−(p+1)(2β−1) L0

(n)),

(p + 1)(2β − 1) > 1 (p + 1)(2β − 1) < 1.

Using Lemma 2 we obtain

σn−,1p sup |Sn (x)| =

(

Oa.s (n−( 2 −p(β− 2 )) L0 1

−(β− 21 )

Oa.s (n

x∈R

(n)(log n)5/2 (log log n)3/4 ), L0 (n)(log n)1/2 (log log n)3/4 ), 1

−p

(p + 1)(2β − 1) > 1 (p + 1)(2β − 1) < 1.

Since σn,p /σn,1 ∼ n−(β− 2 )(p−1) L0 (n), we obtain 1

p−1

n X σn,p sup |βn (x) + σn,1 Vn,p (x)| = sup σn−,1p (1{Xi ≤x} − F (x)) + σn−,1p Vn,p (x) = oa.s (dn,p ). σn,1 x∈R x∈R i=1





−1

Consequently, via {αn (y), y ∈ (0, 1)} = {βn (Q (y)), y ∈ (0, 1)}, sup |αn (y) + σn−,11 V˜ n,p (y)| = Oa.s (dn,p ).

y∈(0,1)

(7)

By (7), the next lemma is obvious. Lemma 3. Let p be a positive integer. Assume that for r = 1, . . . , p, an → ∞, an = o(n),

σn−,11

Z

1−an /n

an /n

d

V˜ n,p (y)dQ (y) → Z .

R 1 (r ) 0 F (Q (y))dQ (y) < ∞. Then for any 0 < an such that

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3.1. Integral functionals of the empirical process Let ψµ (y) = (y(1 − y))µ , y ∈ [0, 1], µ > 0. Let εn = n−κ , κ > 0 and consider the following class of functions: ) ( Z Ω0 = Ω0 (µ) =

K : K : [0, 1] → R, nondecreasing,

1

0

ψµ (y)dK (y) < ∞ .

Lemma 4. Fix κ > 0, p ∈ N. Let µ > 0 be such that  1−β   , if (p + 1)(2β − 1) ≥ 1, µ <

κ

p(β − 12 )   µ < ,

(8)

if (p + 1)(2β − 1) < 1.

κ

Then sup

sup

K ∈Ω0 εn
Z t   αn (y) + σn−,11 V˜ n,p (y) dK (y) = oa.s (1).

(9)

s

The most interesting case is K = Q . Corollary 5. Assume that either E|X |α < ∞ for all α > 0 or (5) holds with some α < ∞ such that  3 κ   if β ≥ , α > 1−β 4   1 3   α ≥ 4 if β ∈ , . 2 4 Let p be the smallest positive integer such that (p + 1)(2β − 1) > 1. Then Z t   −1 ˜ = oa.s (1). V ( y ) d Q ( y ) sup ( y ) + σ α n , p n n,1 1/nκ
(10)

s

Proof of Lemma 4. From (7) and the choice of µ, sup

|αn (y) + σn−,11 V˜ n,p (y)|

y∈(n ,1−n )

ψµ (y)

= Oa.s (1).

We have for all εn < s < t < 1 − εn and all K ∈ Ω0 , Z t  Z t  sup αn (y) − σn−,11 V˜ n,p (y) dK (y) ≤ s

s

|αn (y) − σn−,11 V˜ n,p (y)|

y∈(εn ,1−εn )

ψµ (y)

ψµ (y)dK (y) = oa.s (1). 

Proof of Corollary 5. If β ≥ 34 , set p = 1 in (8). If F ∈ MDA(Φα ), then E|X |α−δ < ∞ for all δ ∈ (0, α). Since we assumed α > κ/(1 − β), choose δ ∈ (0, α) such that α − δ > κ/(1 − β) still holds. Set µ = (α − δ)−1 < (1 − β)/κ. Then we have R1 E|X |1/µ+δ/2 < ∞. Thus 0 ψµ (y)dQ (y) < ∞. Consequently, Q ∈ Ω0 and (9) applies. 3 If β < 4 and F ∈ MDA(Φα ), then take in (8) the smallest integer p such that (p + 1)(2β − 1) > 1. Now, (1 − β)−1 < 4. Further, since E41 < ∞, we have EX14 < ∞ and thus α ≥ 4. Consequently, α > (1 − β)−1 . We may choose δ ∈ (0, α) such that α − δ > (1 − β)−1 and continue as in the previous case. Likewise, if E|X |α < ∞ for all α > 0 then choose α big enough that α > (1 − β)−1 . Consequently, we may continue as in the case of Q being regularly varying. Therefore, (10) has been proved.  3.2. Proof of Theorem 1 Integration by parts yields

σn,1 −1

nX −kn i=kn +1

!

Xi − µn (kn , kn )

Z

=−

1−kn /n

kn /n

+ σn−,11 n

Z

αn (y)dQ (y) + σn−,11 n 1−kn /n

Un−kn :n

Z

Ukn :n

k n /n

(En (y) − kn /n)dQ (y)

(En (y) − (1 − kn /n))dQ (y) =: B1 + B2 + B3 .

Corollary 5 applied with κ = 1 − ξ, together with Lemma 3 yields the asymptotic normality for B1 . It suffices to show that

B3 = oP (1). The term B2 is treated in the same way.

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2541

p

Lemma 6. For any kn → ∞, kn = o(n), we have Un−kn :n /(1 − kn /n) → 1. Proof. In view of (7) one obtains supy∈(0,1) |un (y)| = supy∈(0,1) |αn (y)| = OP (1). Consequently, sup |y − Un (y)| = sup σn,1 n−1 |un (y)| = sup σn,1 n−1 |αn (y)| = OP (σn,1 n−1 ).

y∈(0,1)

y∈(0,1)

y∈(0,1)

Thus, the result follows on noting that Un (1 − kn /n) = Un−kn :n .



An easy consequence of (7) is the following result. Lemma 7. For any kn → 0, sup

y∈(1−kn /n,1)

|αn (y)| = Oa.s. (dn,p ) + OP (f (Q (1 − kn /n))).

To prove that B3 = oP (1), let y be in the interval with the endpoints Un−kn :n and 1 − kn /n. Then k 1 − En (y) − n ≤ |En (1 − kn /n) − (1 − kn /n)|. n

p

Case 1, Y ∈ MDA(Φα ): By Lemma 6 we have Q (1 − kn /n)/Q (Un−kn :n ) → 1. Thus, by Lemma 7 B3 ≤

σn−,11 Q (1 − kn /n)|αn (1 − kn /n)|

|Q (1 − kn /n) − Q (Un−kn :n )| Q (1 − kn /n)

= σn−,11 Q (1 − kn /n)αn (1 − kn /n)op (1) 







= op σn−,11 Q (1 − kn /n)f Q (1 − kn /n) + op σn−,11 Q (1 − kn /n)dn,p = oP (1). Case 2, X ∈ MDA(Λ): Let Tn (λ) =

σn−,11 |αn (1 − kn /n)| Q (rn+ (λ)) − Q (rn− (λ)) ,



where rn+ (λ) = 1 − λknn , rn− (λ) = 1 − λ knn and 1 < λ < ∞ is arbitrary. Applying the argument as in the proof of Theorem 1 in Csörgő et al. (1986) we have lim inf P(|B3 | < |Tn (λ)|) ≥ lim inf P(rn− (λ) ≤ Un−kn :n ≤ rn+ (λ)). n→∞

n→∞

In view of Lemma 6, the lower bound is 1. Thus, limn→∞ P(|B3 | < |Tn (λ)|) = 1. Further, by Lemma 4 in Lo (1989) lim (Q (rn+ (λ)) − Q (rn− (λ)))L3 (n/kn ) = − log λ.

n→∞

Thus, for large n Tn (λ) =

σn−,11 |αn (1 − kn /n)|(L3 (n/kn ))−1 |Q (rn+ (λ)) − Q (rn− (λ))|L3 (n/kn )

≤ C1

σn−,11 σn−,11 f Q (1 − kn /n)(log λ) + C2 dn,p log λ L3 (n/kn ) L3 (n/kn )

almost surely with some constants C1 , C2 . Then both terms, for arbitrary λ, converge to 0. Thus, we have for sufficiently large n, Tn (λ) ≤ C1 log λ almost surely. Thus, limn→∞ P (|Tn (λ)| ≤ C1 log λ) = 1. Consequently, lim P(|B3 | > C1 log λ) = lim P(|B3 | > C1 log λ, |Tn (λ)| ≤ C1 log λ)

n→∞

n→∞

+ lim P(|Tn (λ)| > C1 log λ) ≤ lim P(|B3 | > |Tn (λ)|) + 0 = 0 n→∞

and thus B3 = oP (1) taking λ → 1.

n→∞



Acknowledgements This work was done during the first author’s stay at Carleton University. RafałKulik is grateful to Professors Miklos Csörgő and Barbara Szyszkowicz for support.

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