Estimating the dimension of the model

As in the case of η, the asymptotic variance is symmetric at η= 0.5 for fixed alpha and has a pole at η = 0 of the form

g(α, η)∼ 1 η^6+α

7056(8−5α²+ 12α)(1−α)(2−α) 2(8−α)(6−α)(1−α)₄ .

As can be seen in figure 10, not all components of the asymptotic covariance matrix are symmetric. For a complete numerical implementation, see section 6.

true knot 0.3

0.4 0.5

0.6 0.7

d

−0.2

−0.1 0.0

0.1 1e+05 0.2

2e+05 3e+05 4e+05 5e+05

Asymptotic variance of ηη^

true knot 0.3

0.4 0.5

0.6 0.7

d

−0.2

−0.1 0.0

0.1 0.2 2e+07

4e+07 6e+07

Asymptotic variance of a^₅

true knot 0.3

0.4 0.5

0.6 0.7

d

−0.2

−0.1 0.0

0.1 0.2 2e+07

4e+07 6e+07

Asymptotic variance of a^₄

true knot 0.3

0.4 0.5

0.6 0.7

d

−0.2

−0.1 0.0

0.1 0.2

−6e+07

−4e+07

−2e+07 0e+00

Asymptotic covariance of a^

4 and a^

5

true knot 0.3

0.4 0.5

0.6 0.7

d

−0.2

−0.1 0.0

0.1 0.2 5e+07

1e+08

Trace of asymptotic covariance matrix

true knot 0.3

0.4 0.5

0.6 0.7

d

−0.2

−0.1 0.0

0.1 0.2 5.0e+20

1.0e+21 1.5e+21 2.0e+21

Determinant of asymptotic covariance matrix

Figure 10 Top: Asymptotic variance of η,ˆ ˆa₅ andaˆ₄ in case of a cubic spline with one free knot and different values of d. In contrast to ηˆand ˆa₅, the variance ofaˆ₄ is not symmetric at η = 0.5.

Bottom: Similarly to the variance ofaˆ4, the covariance of ˆa4 and ˆa5 is not symmetric. Note that the trace is not symmetric whereas the determinant appears to be symmetric atη = 0.5.

5.3 Estimating the dimension of the model

5.3.1. Throughout sections 5.1 and 5.2 we assumed that our model is identifiable in the sense of definition 4.1.4. In practice, however, this assumption may not hold: the exact number of knots is typically unknown and therefore needs to be estimated.

There exist several methods to determine the number of knots. In the case of two nested models, say for instance zero knots vs. one knot, formal tests, e.g. maximum

likelihood ratio types of statistics, can be applied. It is well-known that the asymptotic of these ratio test statistics do not follow the usual χ² distribution but some extremal law, see e.g. [AHHK08, Dav02, AJ03, Jar98, Fed75a] for both continuous and noncontinuous change points. For the noncontinuous case, testing procedures based on bootstrap are for example discussed in [HP05]. Other testing procedure for the noncontinuous case are for example the CUSUM test [Kim94, BDE75] and an union intersection approach [Haw89].

We will not pursue these approaches further but turn to another alternative method:

the use of information criteria. Classical examples where information criteria are used to determine the number of parameters of the model are linear models and autoregressive time series models, see e.g. [Sha97, HQ79, BBO98]. Similarly, the number of knots can be consistently estimated by applying an adequate information criterion. Yao [Yao88]

showed consistency of the Schwarz criterion for piecewise constant piecewise polynomials and iid normal random variables. Liu et al. [LWZ97] showed that for locally exponentially bounded iid errors the number of knots can be estimated consistently by minimising a modified Schwarz criterion

M IC(l) = log [S(ˆη₁, . . . ,ηˆ_l)/(n−p^∗)] +p^∗c₀(log(n))^2+δ0

n .

Here, S(ˆη₁, . . . ,ηˆ_l) denotes the residuals sum of squares obtained under a model with l knots and p^∗ parameters in total,c₀ and δ₀ denote strictly positive constants. By using a law of the iterated logarithm, we wish to generalise the latter result to fractional processes.

To this end, we need to fix some additional notation beforehand.

5.3.2. In what follows, let

x(t) = µ(t/n) +ξ(t), t= 1, . . . , n withµ(s) =Pl

k=0

Ppk

j=1a_j,k(s−η_k)^b₊^j,k denoting a piecewise polynomial andξ(·) satisfying the assumptions of theorem 1.3.5.¹ and one of the conditions stated in definition 1.1.5.

Definition 5.3.3. We say that µ(s) = Pl k=0

Ppk

j=1a_j,k(s−η_k)^b₊^j,k is the minimal repre-sentation of µ if a_j,k 6= 0 for all j, k. Clearly, for each piecewise polynomial there is a unique minimal representation.

Definition 5.3.4. Let µ(s) = Pl k=0

Ppk

j=1a_j,k(s−η_k)^b₊^j,k be the minimal representation of µ. We then refer to l as the number of knots of µ.

1Consider remark 1.3.6 for a brief discussion about these assumptions.

Definition 5.3.5. Let l ∈N and let pk ∈ N\ {0} and bj,k ∈N for all (j = 1, . . . , pk, k = 0, . . . , l). Given a triple (l,(p_k),(b_j,k)), define the parametric family M= M_{l,(pk),(bj,k)} of piecewise polynomials by

M=

s: [0,1]→R, s(t) =

l

X

k=0 pk

X

j=1

a_j,k(t−κ_k)^b₊^j,k,(κ_k)^l_k=0 ∈ {0} ×S^l, a_j,k ∈R

. Likewise, define

M(∆)=

s: [0,1]→R, s(t) =

l

X

k=0 pk

X

j=1

a_j,k(t−κ_k)^b₊^j,k,(κ_k)^l_k=0 ∈ {0} ×S^l∆, a_j,k ∈R

with

S^l∆=

(κ₁, . . . κ_l)∈S^l:|κ_i−κ_j| ≥∆ . We refer to l as the number of knots of Mand to pM =l+Pl

k=0p_k as the total number of parameters of M. For the sake of simplicity, we omit the subscript l,(p_k),(b_j,k) if it is clearly determined from the context.

Givenk∈S^l, we define the design matrix corresponding toM by Mn,M(k) = M_{n,l,(pk),(bj,k)}(k),

see 4.2.5 for details.

Definition 5.3.6. Let M={M₁, . . . ,M_q} be a set of parametric families of piecewise polynomials. We say M is compatible with the piecewise polynomial µ, if there exists M_k∈ M such that

(i) µ∈ M_k,

(ii) The number of knots in M_k is equal to the number of knots of µ, (iii) pM_k < pM_j for all k 6=j such that µ∈ M_j.

In this case, we refer to M_k as the compatible model. Clearly, if M_k exists, then it is uniquely determined. If there exists no compatible model, we say that Misincompatible with µ.

Remark 5.3.7. Some examples may help to clarify the above definition. To that end, define

M₁ ={s(t) = a+btfor a, b∈R}

M₂ ={s(t) = a+bt+c(t−κ)₊ for a, b, c∈R and κ∈(0,1)}

M₃ =n

s(t) = a+P2

j=1b_jt^j+P2

j=1c_j(t−κ)^j₊ for a, b_j, c_j ∈R and κ∈(0,1)o M₄ =

s(t) =a+P4

t=1b_jt^j, a, b_j ∈R

Assume that µ(s) = (s−0.5)+ and M = {M1,M2,M3}. Then, M is compatible with µ and the compatible model is given by M₂.

Ifµ(s) =s²+ (s−0.5)₊,Mis still compatible with µwith M₃ being the compatible model.

Now, letµ(s) =s and M={M₂,M₄}. Then, Mis not compatible with µ. In fact, we have µ ∈ M2 and µ ∈ M4 with pM2 < pM4. But at the same time M2 violates condition (ii) and so it is not a compatible model. On the other hand, if µ(s) = s² then µ6∈ M₂. Thus, M₄ becomes the compatible model and Mis compatible with µ.

Definition 5.3.8. Letx(t) =µ _n^t

+ξ(t) as in 5.3.2. Define (r(n))_n∈

N ⊂R^∗+

as in theorems 5.2.3 and 5.2.4, i.e.

r²(n) = n²

D²(n) ∼n^αL^∗(n) with α ∈(0,2) and L^∗ slowly varying. Furthermore, define

(h(n))_n∈N⊂R^∗+ such thath(n)→ ∞.

Lastly, if ξ(·) has either long or short memory, define (λ(n))_n∈

N⊂R^∗+ such that λ(n)

r²(n) →0 and h(n) log log(n)

λ(n) →0.

and if ξ(·) is antipersistent, define (λ(n))_n∈

N ⊂R^∗+ such that λ(n)

n →0 and h(n) λ(n) →0.

For example, choose λ(n) = log(n) and h(n) = q

λ(n) log log(n). Definition 5.3.9. Letx(t) =µ _n^t

+ξ(t),t= 1, . . . , nas 5.3.2 and letM={M₁, . . . ,M_q} be a compatible set of parametric families of piecewise polynomials. For i ∈ {1, . . . , q}, define ˜σ_M²

i,h by

˜ σ²_M

i,h = inf (1

n

n

X

t=1

(x_t−f(t/n))² :f ∈ M_i h⁻¹(n) )

.

The determination of ˜σ²_Mi,h can be regarded as a restricted least squares estimation:

given an observed series x(1), . . . , x(n), we obtain the optimal fit ˆµ∈ M_i(h⁻¹(n)). The

space of admissible functions is restricted by a lower boundh⁻¹(n) for the minimal distance between knots. In fact, we have

˜

σ_M² _i = inf

1 n

x_n−pr_Mn,M

i(k)x_n

2

:k∈S^lh⁻¹(n)

with S^lh⁻¹(n) defined as above. Note that by the same arguments as in 5.1.1, there exists k˜ⁱ_n∈S^lh(n) such that

xn−pr_M

n,Mi(^˜kin)xn

2

= inf

1 n

xn−pr_Mn,M

i(k)xn

2

:k∈S^lh⁻¹(n)

.

Furthermore, note that h⁻¹(n)→ 0, since h(n)→ ∞. Now, the information criteria can be defined as follows:

Definition 5.3.10. If ξ(·) has either short or long memory, define IC(M_i) = log ˜σ²_Mi

+pMi

λ(n) r²(n). If ξ(·) is antipersistent, define

IC(M_i) = log ˜σ_M²

i

+p_Miλ(n) n .

Given a set a models {M₁, . . . ,M_q}, we estimate the true model by minimising IC(M_i) with respect toi. This estimation procedure is consistent in the following sense:

Theorem 5.3.11. Letx(t) = µ _n^t

+ξ(t),t = 1, . . . , nas 5.3.2 and letM={M1, . . . ,Mq} be a compatible set of families of piecewise polynomials. Denote by M_k the compatible model. Then, for all i6=k,

P IC(M_i)> IC(M_k)

→1 as n→ ∞.

Proof. Case 1: Leti6=k such that µ6∈ M_i. Then, in the case of long and short memory, IC(M_i)> IC(M_k)⇔log ˜σ²_Mi

+pM_i

λ(n)

r²(n) >log ˜σ_M²

k

+pM_k

λ(n) r²(n)

⇔σ˜²_Mi

˜ σ²_M

k

>exp

(pM_k−pM_i) λ(n) r²(n)

.

Likewise, in the case of antipersistence, IC(M_i)> IC(M_k)⇔ σ˜_M²

i

˜ σ_M²

k

>exp

(p_Mk −p_Mi)λ(n) n

.

It follows from lemma 5.3.13 and 5.1.4 that ˜σ²_Mk =σ²+op(1). On the other hand,

˜ σ²_M

i =n⁻¹

µ_n+e_n−pr_M

n,Mi(^˜kin)x_n

2

=n⁻¹ke_nk²+n⁻¹

µ_n−pr_M

n,Mi(^k˜in)x_n

2

+ 2n⁻¹D

e_n, µ_n−pr_M

n,Mi(^k˜in)x_nE . As in the proof of theorem 5.1.9, we have _n² D

e_n, µ_n−pr_M

n,Mi(^k˜in)x_nE

=o_p(1). However, since µ6∈ M_i, there is ∆>0 such that

lim inf

n

1 n

pr_M

n,Mi(^k˜in)x_n−µ_n

2

>∆ a.e.

Consequently, by 5.1.4, ˜σ²_Mi =σ²+ ∆ +o_p(1) and thus, by definition of (λ(n))_n∈

N, P(IC(M_i)−IC(M_k)>0) =P

σ²+ ∆

σ² +o_p(1) >1 +o(1)

→1.

Case 2: Let i 6= k such that µ ∈ Mi. Since Mk is a compatible model, we have k =pM_i−pM_k >0. Thus

IC(M_i)> IC(M_k)⇔kλ(n)

r²(n) >−log σ˜_M² _i

˜ σ_M²

k

in the case of long and short memory, and

IC(M_i)> IC(M_k)⇔kλ(n)

n >−log σ˜_M² _i

˜ σ_M²

k

in the case of antipersistence. In the case of short and long memory, lemma 5.3.12 implies r²(n)

λ(n) log σ˜_M²

i

˜ σ_M²

k

=O_p

λ⁻¹(n)h(n) log log (n) . But by definition, h(n) log log(n)

λ(n) →0 and therefore

P(IC(M_i)> IC(M_k))→1.

In the case of antipersistence, we obtain n

λ(n)log σ˜²_M

i

˜ σ²_M

k

=O_p

λ⁻¹(n)h(n) . Again, by definition, _λ(n)^h(n) →0 which gives the desired result.

Lemma 5.3.12. Under the same assumptions and notations as in theorem 5.3.11, let M_i ∈ M such that µ∈ M_i.

Ifξ(·) has long or short memory, then log

σ˜_M²

i

˜ σ²_M

k

=Op

r⁻²(n)h(n) log log (n) . Ifξ(·) is antipersistent, then

log σ˜_M²

i

˜ σ_M²

k

=O_p

n⁻¹h(n) . Proof. Let x ∈ −¹₂,¹₂

. By Taylor’s theorem, there exists ˜x with |˜x| ≤ |x| such that log(1 +x) = x_(1+˜¹_x), hence the upper bound

|log (1 +x)| ≤ |2x|

for all x∈ −¹₂,¹₂ .

Assume that ξ(·) has long or short memory. Defining the set B=

˜ σ_M²

i−σ˜_M²

k

˜ σ_M²

k

< 1 2

, we get the inequality

P

r²(n)

h(n) log log (n)log σ˜_M²

i

˜ σ²_M

k

≥∆

≤

P(B^c) +P

r²(n) h(n) log log (n)2

˜

σ²_Mi−σ˜_M²

k

˜ σ_M²

k

>∆

. By lemma 5.3.13 and 5.1.4, P(Ξ^c) =o(1) and

r²(n) h(n) log log (n)

σ˜²_Mi−σ˜_M² _k

˜ σ_M²

k

=O_p(1),

which proves the assertion. The same argument applies for the antipersistent case.

Lemma 5.3.13. Under the same assumptions and notations as in theorem 5.3.11, let M_i ∈ M such that µ∈ M_i.

Ifξ(·) has long or short memory, then ˜σM_i−n⁻¹ke_nk²

=O_p

r⁻²(n)h(n) log log (n) . Ifξ(·) is antipersistent, then

˜σ_Mi−n⁻¹ke_nk²

=O_p

n⁻¹h(n) .

Proof. Letµ(s) =Pl k=0

Pp_k

j=1a_j,k(s−η_k)^b₊^j,k be the minimal representation ofµand denote by l_i the number of knots of M_i. Setting b∞ = max (bM_i,max_j,k(b_j,k)) and η_l+1 = 1, we immediately obtain

µ_n∈

l

M

k=0

sp F^η_b∞^k,η_,n^k+1 .

Likewise, for all k= (κ₁, . . . , κ_li)∈S^l_h(n)ⁱ , κ₀ = 0 and κ_li+1 = 1, sp (Mn,Mi(k))⊂

li

M

k=0

sp F^κ_b∞^k,κ_,n^k+1 . Denote

W_n(k) =

l

M

k=0

sp F^η_b^k,ηk+1

∞,n

+

li

M

k=0

sp F^κ_b^k,κk+1

∞,n

. (5.34)

Since µ ∈ M_i, it follows from the definition of the restricted least squares estimate that there exists N ∈Nsuch that ke_nk² ≥n˜σ_M² _i for all n≥N. Hence, for such n,

e_n

2 ≥nσ˜²_Mi =

e_n+µ_n−pr_M

n,Mi(^k˜in) [µ_n+e_n]

2 ≥

e_n+µ_n−pr_W

n(^k˜in) [µ_n+e_n]

2 = e_n

2− pr_W

n(^k˜in)e_n

2

and thus

0≥nσ˜²_Mi− e_n

2 ≥ − pr_W

n(^˜kin)e_n

2. In particular, we obtain for all n≥N

e_n

2−nσ˜²_M∗

≤

pr_W

n(^k˜in)e_n

2 ≤supn

pr_Wn(k)e_n

2 :k∈S^lh(n)ⁱ

o .

We now would like to apply lemma 5.3.14 and lemma 5.3.15 on the summands in (5.34) to conclude that

sup n

n⁻¹

pr_Wn(k)en

2 :k∈S^lh(n)ⁱ

o

=Op

r⁻²(n)h(n) log log (n)

. (5.35) However, as the sum in (5.34) is not orthogonal, this is not directly possible. In fact, de-pending on the value ofk, the collinearity betweenLl

k=0sp F^η_b∞^k,η_,n^k+1

andLli

k=0sp F^κ_b∞^k,κ_,n^k+1 can become arbitrarily close to 1. Note, however, that we may assume w.l.o.g. thatη_t6=κ_s for all t= 1, . . . , l and s= 1, . . . , l_i.²

To prove (5.35), we need to condition on the position of thefree knots (κ_s)^l_s=1ⁱ relative to thefixed knots (η_t)^l_t=0. More specifically, letv ∈N^l+1 such that|v|₁ =Pl+1

t=1v_t=l_i. We

2Indeed, if k∈S^l_h(n)ⁱ such thatη_s=η_t for some s andt, then the subspaces sp F^η_b^k,ηk+1

∞,n

^l

k=0 and sp F^κ_b^k,κk+1

∞,n

li

k=0 are linearly dependent and there exists k⁰ ∈S^l_h(n)ⁱ such that η_s⁰ 6= ηt for all s, t and Wn(k)⊂Wn(k⁰). In particular,

pr_Wn(k)en

2≤

pr_Wn(k0)en

2.

say that (κ1, . . . , κli)∈S^lh(n)ⁱ is of typev, ifPli

s=11(ηk−1,η_k)(κs) =vkfor allk = 1, . . . , l+ 1, i.e. if the number of free knots κ_s between ηk−1 and η_k is equal tov_k. Denote

S^l_hⁱ⁻¹_(n)(v) =n

k∈S^l_h(n)ⁱ :k is of type vo . Since S^l_hⁱ⁻¹_(n) =S

v∈N^l+1,|v|1=liS^l_hⁱ⁻¹_(n)(v), we have supn

pr_Wn(k)e_n

2 :k∈S^l_hⁱ⁻¹_(n) o

= max

v∈N^l+1,|v|1=li

supn

pr_Wn(k)e_n

2 :k∈S^l_hⁱ⁻¹_(n)(v)o . Hence, it suffices to show that for all v

supn

pr_Wn(k)e_n

2 :k∈S^lhⁱ⁻¹(n)(v)o

=O_p

r⁻²(n)h(n) log log (n) .

So, let v ∈ N^l+1 such that |v|1 = li and k = (κ1, . . . , κli) ∈ S^lh(n)ⁱ (v). Set u1 = 0 and u_t+1 =v_t+u_t for t >1 and defineW_n,t(k), (t = 1, . . . , l+ 1) by

W_n,t(k) =











sp F^η_n,b^t−1_∞^,ηt

v_t = 0, sp F^η_n,b^t−1_∞^,κut+1

⊕sp F^κ_n,b^ut_∞^+1,ηt

v_t = 1, sp F^η_n,b^t−1,κut+1

∞

⊕Lvt−1

k=1 sp F^κ_n,b^{ut+k,κut+k+1}

∞

⊕sp F^κ_n,b^ut+vt,ηt

∞

v_t >1.

Clearly, Wn,s(k) andWn,t(k) are orthogonal for s6=t and Wn(k) =

l+1

M

t=1

Wn,t(k). Therefore,

sup

k∈S^li_h(n)(v)

pr_Wn(k)e_n

2 ≤

l+1

X

t=1

sup

k∈S^li_h(n)(v)

pr_Wn,t(k)e_n

2

.

But it follows from a case-by-case application of lemma 5.3.14, lemma 5.3.15 and remark 5.3.16 that

sup

k∈S^li_h(n)(v)

pr_Wn,t(k)e_n

2

=O_p

r⁻²(n)h(n) log log (n)

(t= 1, . . . , l+ 1)

if ξ(·) has short or long memory, and sup

k∈S^li_h(n)(v)

pr_Wn,t(k)e_n

2

=O_p

n⁻¹h(n)

(t= 1, . . . , l+ 1)

if ξ(·) is antipersistent which proves our claim.

Lemma 5.3.14. Under the same assumptions and notations as in theorem 5.3.11, we have

sup

a,b∈S² b−a>h⁻¹(n)

1 n

pr_V^p,n

a,be_n

2

=O_p r⁻²(n)h(n) log log(n)

. (5.36)

Proof. We may w.l.o.g. assume thatV_a,b^p,n has full column rank. Indeed, ifa < bsuch that rank V_a,b^p,n

< p, then there exists a⁰ < b⁰ such that rank V^p,n_a0,b⁰

= p and sp V^p,n_a,b

⊂ sp V_a^p,n0,b⁰

, thus pr_V^p,n

a,be_n ≤

pr_V^p,n

a0,b0e_n

. Hence D

e_n,pr_V^p,n

a,be_nE

=D

e_n,V^p,n_a,b V^p,n_a,b⁰V_a,b^p,n−1

V^p,n_a,b⁰e_nE

= D

V^p,n_a,b⁰e_n, V^p,n_a,b⁰V^p,n_a,b⁻¹

V^p,n_a,b⁰e_nE

≤

V^p,n_a,b⁰e_n

2·λ_maxh

V^p,n_a,b⁰V_a,b^p,n−1i . Note that the maximum eigenvalue λ_maxh

V_a,b^p,n0V^p,n_a,b⁻¹i

is bounded: since

V^p,n_a,b⁰V_a,b^p,n

jk =

Pbbnc

t=banc+1(t/n− banc/n)^k−1+j−1 q

Pbbnc

t=banc+1(t/n− banc/n)^2j−2q Pbbnc

t=banc+1(t/nbanc/n)^2k−2

=

=

Pbbnc−banc

t=1 (t/n)^k−1+j−1 q

Pbbnc−banc

t=1 (t/n)^2j−2 q

Pbbnc−banc

t=1 (t/n)^2k−2

=

=

Pbbnc−banc t=1

t bbnc−banc

^k−1+j−1 r

Pbbnc−banc t=1

t bbnc−banc

^2j−2r

Pbbnc−banc t=1

t bbnc−banc

^2k−2

ifbbnc−banc→∞

−−−−−−−−−→

R1

0 s^k+j−2ds qR1

0 s^2k−2ds qR1

0 s^2j−2ds , there exists ε >0 such that inf_{n∈N,a,b∈S}²,bbnc−banc≥pλ_min

V^p,n_a,b⁰V^p,n_a,b

> ε.

For proving (5.36) it thus suffices to verify that sup

a,b∈S², b−a>h⁻¹(n)

1 n D

v^a,b_j,n,e_nE

2

=O_p r⁻²(n)h(n) log log(n)

(5.37)

for j = 1, . . . , p. Here, v^a,b_j,n denotes the j-th column of V_a,b^p,n, cf. definition 4.2.5.

To begin with, we need to find convenient upper bound of D

v^a,b_j,n,e_nE

. To that end, define Y_t =Pt

k=1ξ_k. Then,

hv^a,b_j,n,eni =

Pbbnc−banc

t=1 (t/n)^j−1ξt+banc

q

Pbbnc−banc

t=1 (t/n)^2j−2

=

Pbbnc−banc

t=1 t^j−1

Yt+banc−Ybanc− Yt−1+banc−Ybanc

q

Pbbnc−banc t=1 t^2j−2

≤

Pbbnc−banc−1 t=1

h

t^j−1−(t+ 1)^j−1i

Y_t+banc−Y_banc q

Pbbnc−banc t=1 t^2j−2 +

(bbnc − banc)^j−1

Y_bbnc−banc−Y_banc q

Pbbnc−banc t=1 t^2j−2

≤

2 sup^bbnc−banc_t=1

Yt+banc−Ybanc

(bbnc − banc)^j−1 q

Pbbnc−banc t=1 t^2j−2

=

2 sup^bbnc−banc_t=1

Yt+banc−Ybanc

(bbnc − banc)^−1/2 q

(bbnc − banc)⁻¹Pbbnc−banc

t=1 (t/(bbnc − banc))^2j−2 .

Since bbnc − banc ≥p, the denominator of the last fraction is bounded away from zero.

Furthermore, _bbnc−banc^(b−a)n ≤ bbnc−banc+1

bbnc−banc ≤ ^p+1_p . Hence, there exists a constant K such that sup

a,b∈S², b−a>h⁻¹(n)

√1 n

D

v_j,n^a,b,e_nE

≤ sup

a,b∈S², b−a>h⁻¹(n)

K sup^bbnc−banc_t=1

Y_t+banc−Y_banc np

(b−a) ≤

sup

a,b∈S²

K

ph(n) sup^bbnc−banc_t=1

Yt+banc−Ybanc

n ≤

sup

a,b∈S²

K 2p

h(n) sup^bbnc_t=1 |Yt|

n ≤K 2p

h(n) supⁿ_t=1|Yt|

n .

The assertion thus follows if we can show that r(n)

ph(n) log log(n)

ph(n) supⁿ_t=1|Y_t|

n = supⁿ_t=1|Yt| plog log(n)D²(n)

is tight. To do so, note that by theorem 1.3.5, in conjunction with remark 1.3.7, P

∞

\

k=1

[

t≥k

( 1

pD²(t) log log (t)|Y_t|>1 )!

= 0, i.e. for all ε >0 exists N ∈Nsuch that

P [

t>N

( 1

pD²(t) log log (t)|Y_t|>1 )!

≤ε. (5.38)

Assuming n > N, we obtain for any ∆>0 P supⁿ_t=1|Y_t|

plog log(n)D²(n) >∆

!

≤P

n

[

t=1

|Y_t|

pD²(n) log log (n) >∆

!

≤P

N

[

t=1

|Y_t|

pD²(n) log log (n) >∆

! +P

n

[

t=N+1

|Y_t|

pD²(n) log log (n) >∆

! .

Since _D2(n) log log(n)¹ →0, the former summand is o(1). It follows from (1.2) that supⁿ_{t=1 D}^D2²(n)^(t)

is bounded. Thus, the latter summand is smaller thanεfor sufficiently large ∆ andn.

Lemma 5.3.15. Under the same assumptions and notations as in theorem 5.3.11, let a∈(0,1). Then, if ξ(·) has short or long memory

sup

b∈(a,1)

1 n

pr_V^p,n

a,be_n

2

=O_p r⁻²(n) log log(n)

. (5.39)

If ξ(·) is antipersistent, then sup

b∈(a,1)

1 n

pr_V^p,n

a,be_n

2

=O_p n⁻¹

, (5.40)

Proof. We argue similar as in the proof of lemma 5.3.14. As there, we may w.l.o.g. assume that V^p,n_a,b has full column rank and obtain

D

e_n,pr_V^p,n

a,be_nE

≤

V_a,b^p,n0e_n

2·λ_maxh

V^p,n_a,b⁰V^p,n_a,b⁻¹i with sup_n∈N,b∈(a,1),bbnc−banc≥pλmax

h

V^p,n_a,b⁰V^p,n_a,b⁻¹i

being bounded.

In case of short and long memory, it is thus again sufficient to show that sup

b∈(a,1)

1 n D

v_j,n^a,b,e_nE

2

=O_p r⁻²(n) log log(n) for all j = 1, . . . , p. But by the same argument as before, we obtain

sup

b∈(a,1)

√1 n

D

v_j,n^a,b,e_nE

≤ sup

b∈(a,1)

K· sup^bbnc−banc_t=1

Y_t+banc−Y_banc n√

b−a .

However, since a is fixed this time andξ(·) is strictly stationary, we have sup

b∈(a,1)

K sup^bbnc−banc_t=1

Yt+banc−Ybanc

n√

b−a

= supd b∈(a,1)

K sup^bbnc−banc_t=1 |Y_t| n√

b−a ≤K supⁿ_t=1|Y_t|

√nt . Again, we apply the law of the iterated logarithm in order to show that

r(n) plog log(n)

supⁿ_t=1|Y_t|

√tn

is tight: Let N ∈Nas in (5.38). Assuming n > N, we obtain for any ∆>0 P r(n)

plog log (n)

supⁿ_t=1|Y_t|

√tn > ∆ 2K

!

≤P

n

[

t=1

√n|Y_t|

pD²(n)t log log (n) > ∆ 2K

!

≤P

N

[

t=1

√n|Y_t|

pD²(n)t log log (n) > ∆ 2K

! +P

n

[

t=N+1

√n|Y_t|

pD²(n)t log log (n) > ∆ 2K

!

Since _D2(n) log log(n)ⁿ →0, the former summand is o(1). We can estimate the latter summand by

P

n

[

t=N+1

√n|Y_t|

pD²(n)t log log (n) > ∆ 2K

!

≤P

n

[

t=N+1

|Y_t|

pD²(t) log log (t) >

ptD²(n) log log (n) pnD²(t) log log (t)

∆ 2K

!

Suppose that ξ(·) has short memory. Then ^D2_n⁽ⁿ⁾ →P

k∈Zγ(k) = c, say. Hence, we may w.l.o.g. assume that ¹₂c < ^D2_t^(t) <2cfor all t > N. Therefore,

tD²(n) log log (n)

nD²(t) log log (t) ≥ tD²(n) nD²(t) ≥

1 2c 2c = 1

4 for all t > N and thus P

Sn

t=N+1

√n|Y_t|

√

D²(n)tlog log(n) > _2K^∆

≤ε for ∆ sufficiently large, by (5.38).

Suppose now that ξ(·) has long memory, i.e. D²(t) = t^2−αL(t) for some α ∈ (0,1) and a slowly varying function L. The function L^∗(t) = L(t) log log (t) is a product of two slowly varying function and therefore slowly varying itself. Hence

supn t=1

nD²(t) log log (t)

tD²(n) log log (n) =supⁿ

t=1

t^1−α n^1−α

L^∗(t) L^∗(n)

≤

supn t=1

t n

1−α

L^∗(t) L^∗(n)−

t n

1−α

+ 1≤q^1,1−α_n + 1 →1, by 1.1.4, in other words lim_ninfⁿ_t=1 ^tD2(n) log log(n)

nD²(t) log log(t) = 1. As above, we conclude from (5.38) that P

Sn

t=N+1

√n|Yt|

√

D²(n)tlog log(n) > _2K^∆

≤ε for ∆ sufficiently large.

Ifξ(·) is antipersistent, must show that sup

b∈(a,1)

1 n D

v_j,n^a,b,e_nE

2

=O_p n⁻¹

(5.41)

Once more, the law of the iterate logarithm paves the way:

P √

nsupⁿ_t=1|Y_t|

√tn > ∆ 2K

≤P ^N

[

t=1

|Y_t|

√t > ∆ 2K

+P

ⁿ [

t=N+1

|Y_t|

√t > ∆ 2K

for all n > N and all ∆>0. As ^D√2(t)

t →0, both summands are can be made arbitrarily small for appropriate choices of N and ∆.

Remark 5.3.16. Since ξ(·) is strictly stationary, we have under the assumptions of the-orem 5.3.11 for all b ∈(0,1)

sup

a∈(0,b)

1 n

pr_V^p,n

a,be_n

2

=O_p r⁻²(n) log log(n)

. (5.42)

5.3.17. There are several unsolved problems in relation with theorem 5.3.11. First of all, the result seems incomplete since in the case of antipersistent noise the normalising sequence r²(n) has to be replaced by n. Furthermore, in the case of long and short memory we have ^{log log(n)}_λ(n) →0 which is not in line with the results of Hannan and Quinn [HQ79] who also admit λ(n) = log log(n). In practical application, it is not clear how the normalisation sequencer²(n) is calculated since the corresponding parameterα has to be estimated from the data. Moreover, the sequences h and λ have to be determined.

Finally, note that by means of theorem 1.3.14 an analogous results can derived ifξ(t) is subordinated to a Gaussian process in the sense of 1.3.8.

6 Simulation studies

Im Dokument Piecewise polynomial regression with fractional residuals for the analysis of calcium imaging data (Seite 124-137)