Limit behavior of G with one rate change point

In this section, we extend Theorem 3.4 allowing for one rate change point, while testing the null hypothesis of variance homogeneity. Assuming a process with at most one rate change point, the processGcan be shown to converge against a limit process Le (Theorem 3.6), which

3. The MFT for testing variance homogeneity

is, likeL, a zero-mean 2h-dependent Gaussian process with unit variance (Corollary 3.7). It differs fromL only in the covariance in the 3h-neighborhood of a change pointc (see Section 5.2 and Fig. 5.4 C,D).

Theorem 3.6. One rate change point: Convergence of G

LetΞ⁽ⁿ⁾ ∈M (Def. 3.3) with at most one rate change and no variance change, as follows. Let Ξ₁(µ₁, σ₁², ν₁²),Ξ₂(µ₂, σ₂², ν₂²)∈R withµ₁ 6=µ₂, σ₁²=σ₂². For c∈(0, T]and n= 1,2, . . . let Ξ⁽ⁿ⁾ := Ξ₁|_[0,nc]+ Ξ₂|_(nc,nT_], (3.9) meaning that Ξ⁽ⁿ⁾ fulfills H₀. Assume a consistent estimator ˆc of c with

|ˆc−c|=o_P(1/√

n) (3.10)

where o_P(·) is the small o-notation with respect to convergence in probability. Let G⁽ⁿ⁾ be the filtered derivative process associated withΞ⁽ⁿ⁾ using the empirical means µˆ^ˆ^c₁, µˆ^ˆ^c₂ estimated in the intervals [0,ˆc) and [ˆc, T]. Then with Le from (3.11), as n→ ∞, we have

G⁽ⁿ⁾ −−→^d L,e

where −→^d denotes weak convergence in the Skorokhod topology. The marginals Le_h,t of the limit process Le equalL outside the h-neighborhood of c and are given by

Leh,t=











L√_h,t, |t−c|> h,

(µriν2)²/(µ2h²)(Wt+h−Wc)+

√

(µriν1)²/(µ1h²)(Wc−Wt)−√

µ1ν12/h²(Wt−Wt−h)

s⁽¹⁾_t , t∈[c−h, c],

√

µ2ν22/h²(W_t+h−Wt)−

√

(µ_leν2)²/(µ2h²)(Wt−Wc)−

√

(µ_leν1)²/(µ1h²)(Wc−Wt−h)

s⁽¹⁾_t , t∈(c, c+h],

(3.11) for a standard Brownian motion (Wt)t≥0. The functions µri := µri,h,t, µle :=µle,h,t are the limits of the empirical means µˆ_ri,µˆ_le and are given by µ_ri,h,t:=µ₁ for t≤c−h, µ_ri,h,t:=µ₂ for t > cand

µ_ri,h,t:= h

(c−t)/µ1+ (t+h−c)/µ2

, (3.12)

fort ∈(c−h, c] and analogously for µ_le. The true order of scaling

(s⁽ⁿ⁾_t )²

t∈τh

is defined by _nh/µ^2ν¹²

1 for t < c−h, by _nh/µ^2ν²²

2 for t > c+h and for |t−c| ≤ h by the following linear interpolation

(s⁽ⁿ⁾_t )² := (s⁽ⁿ⁾_h,t)² :=







1 n

µ1ν12

h +^(c−t)_h2µ1(µriν1)²+^(t+h−c)_h2µ2 (µriν2)²

, if c−h≤t≤c,

1 n

_{(c−(t−h))}

h²µ1 (µ_leν₁)²+^(t−c)_h2µ2(µ_leν₂)²+ ^µ²_h^ν²²

, if c < t≤c+h.

(3.13) Sketch of proof: Again, we sketch the proof and refer for the detailed proof version to Section 4.2.

The key ingredients are the Anscombe-Donsker-Theorem and continuous mapping. In addition to the proof of Theorem 3.4, a change point in the rate requires separate considerations for

3. The MFT for testing variance homogeneity

different intervals in the neighborhood of a change point. Like in the proof of Theorem 3.4 we first assume known process parameters and use the modified filtered derivative process Γ

Γ⁽ⁿ⁾_t = Γ⁽ⁿ⁾_ri,t −Γ⁽ⁿ⁾_le,t, (3.14) which is comparable to the process Γ⁽ⁿ⁾_t defined in equation (3.6). The detailed definitions are given in Section 4.2. Moreover, we decompose the limit processLe∼Leri−Le_le, where∼ denotes equality in distribution and refer again for detailed definitions to Section 4.2. The first step of the proof will be to show convergence of the processes

Γe⁽ⁿ⁾_ri :=

N_n(t+h)−Nnt−1 nh/µri

·Γ⁽ⁿ⁾_ri and Γele against

Leri,Lele

using the Donsker-Ascombe-Theorem and continuous mapping.

With Lemma 4.5, we can then conclude

Γ⁽ⁿ⁾_ri ,Γ⁽ⁿ⁾_le _d

−

−→

Leri,Lele

, and using continuous mapping again yields fort∈τh

Γ⁽ⁿ⁾= Γ⁽ⁿ⁾_ri −Γ⁽ⁿ⁾_le −−→^d Le_ri−Le_le∼L.e

In step two of the proof, we first replace the true means µ1, µ2 in the numerator by their estimators to define the process ˆΓ⁽ⁿ⁾ and show

Γˆ⁽ⁿ⁾−Γ⁽ⁿ⁾−−→^P (0)_t. (3.15) Then, we use Lemma 4.6 to substitute the scalings⁽¹⁾_t used in ˆΓ by the estimator ˆs⁽ⁿ⁾_t to prove the assertion.

We show thatLe is a Gaussian process with zero mean and unit variance.

Corollary 3.7. Marginal distribution of Le

Let Le be defined as in (3.11). For all t∈τh it holds Leh,t ∼N(0,1).

Proof: As the increments of a standard Brownian motion (Wt)t≥0are independent and Gaussian distributed,Le is Gaussian distributed. Moreover, the increments have zero expectation and thus the zero mean follows by the linearity of expectation. To show the unit variance, we analyse three cases. We use the independence of increments and the property that for 0≤s≤t: Var(Wt−Ws) =t−s. For |t−c|> h the assertion follows directly. Now, let t∈[c−h, c]. We obtain

Var(Le_h,t) =

µ²_riν₂²

µ2h²(t+h−c) + ^µ_µ²^ri^ν²¹

1h²(c−t) +^µ_h¹^ν2¹²h

µ1ν12

h + ^(µ_h^ri2^νµ¹1⁾²(c−t) +^(µ_h^ri2^νµ²2⁾²(t+h−c) = 1.

The caset∈(c, c+h] is shown analogously.

Note that analogous results to Theorem 3.6 hold if there are several rate change points with pairwise distances each larger than 2h as shown in Figure 3.3 for two rate change points.

3. The MFT for testing variance homogeneity

In theh-neighborhood of each rate change point the marginals of the corresponding limit process are similar to the marginals ofLe aroundc and in between the marginals are identical to the marginals ofL. In case of rate change points with distance smaller than or equal to 2h, the structure of the limit process becomes more complicated but is still comparable to the structure ofL. Furthermore, note that the proof of Theorem 3.6 is based on a Functionale Central Limit Theorem and a consistent estimator ofst. Therefore the result can be shown not only for renewal processes but also for a subclass of renewal processes with varying variance (RPVV) as introduced in Messer et al. (2014).

c₁ c₁+h

c₁−h c₂−h c₂c₂+h

L L L

marginals of limit process similar to

Figure 3.3: Limit process in the case of two rate cps with distance larger than 2h. The marginals of the limit process resemble in the h-neighborhoods of the two rate change pointsc₁, c₂ the marginals of the processL˜ around its rate change point, and outside they are identical to the marginals of L.

As the marginals of L andLe differ only in the h-neighborhood of cand both processes are 2h-dependent, their covariance structures differ in the 3h-neighborhood of the rate change pointc, which is illustrated in Figure 3.4. Our simulations in Section 5.2 suggest that the differences between L and Le are typically small with respect to the 95%-quantile of their absolute maxima. We therefore suggest to use the parameter independent limit processLalso in the situation of potential rate change points for the derivation of the rejection threshold in the statistical test. The simulations in Section 5 show that the MFT using L instead of Le keeps the asymptotic significance level for most combinations ofµand σ even for the case of multiple unknown rate change points.

c c+h

c−h c+3h c−3h

impact of

c−h L~

c+h

Figure 3.4: Impact of a rate change point on the covariance of L.e The processes L and Le are 2h-dependent as visualized by orange arrows in the graph. As the marginals of L and Le differ in the h-neighborhood of the rate change point c (colored orange), the covariance of Le differs in a 3h-neighborhood of the rate change point from the covariance of L.

Chapter 4

Proofs of Theorems 3.4 and 3.6

In this section, we prove the key theoretical results of the first part of this thesis in detail. The proof of Theorem 3.4 in Section 4.1 uses the Anscombe-Donsker-Theorem, continuous mapping and the consistency of the estimator ˆs². In Section 4.2, we prove Theorem 3.6 basically using the same ideas as for the proof of Theorem 3.4.

We specify the sets of indices

Iˆ_le:={N_n(t−h)+ 2, N_n(t−h)+ 3, . . . , N_nt} and Iˆ_ri :={N_nt+ 2, N_nt+ 3, . . . , N_n(t+h)}.

The Anscombe-Donsker invariance principle is of central importance for the proofs. We state it here following Gut (2009, Theorem 2.1, p.158).

Theorem 4.1. Anscombe-Donsker-Theorem

Let(ξ_k)k≥1 be i.i.d. random variables with zero mean and variance σ² <∞, ( ˜N(t))t≥0 be a nondecreasing, right-continuous family of positive, integer valued random variables and set

Z_n(t) := 1 σ√

N˜nt

k=1

ξ_k.

Suppose that

N˜(t) t

−−−→a.s.

t→∞ κ (0< κ <∞).

Then we have in (D[0,∞), d_sk)

Z_n

√κ

−−−→d t→∞ W, with W as standard Brownian motion.

Proof: Compare, e.g., Gut (2009).

4. Proofs of Theorems 3.4 and 3.6

4.1 Proof of Theorem 3.4

Recall the sketch of proof directly following the statement of Theorem 3.4. The processes Γ⁽ⁿ⁾_le and Γ⁽ⁿ⁾_ri in equation (3.6) are in detail defined as

Γ⁽ⁿ⁾_ri,t := 1 Step 1: weak process convergence for known parameters

RecallV_i= (ξ_i−µ)² for the life times (ξ_i)i≥1 of a point process Ξ with a known meanµ. We apply the Anscombe-Donsker-Theorem to the process defined by

Y_t⁽ⁿ⁾:= 1 Here,W denotes a standard Brownian motion.

Letϕ: (D[0,∞), d_SK)→(D[h, T −h]×D[h, T −h], d_SK⊗d_SK) be defined by

This function is continuous. Mapping √

µY⁽ⁿ⁾ via ϕ, the first component is given by



By Slutsky’s theorem this also implies

 see Lemma A.3.2 in Messer et al. (2014). Thus by Slutsky’s theorem,

(Γ⁽ⁿ⁾_ri,t)t∈τ_h d asn→ ∞ by continuous mapping which is the assertion for a known mean µ.

4. Proofs of Theorems 3.4 and 3.6

Note that the expectationE[V₁] vanishes, since it appears in both summands.

Step 2: replacement of parameters by their estimators

In a second step we use the estimated mean ˆµnT. We show that for

Thus, the same arguments as in step 1 can be applied to show the assertion G⁽ⁿ⁾ −→ L.

Additionally, by Slutsky’s theorem and Corollary 4.4 (given below), we can exchange the factor (nh/2µν²)^1/2 with the estimator 1/ˆs(nt, nh) with the convergence holding true.

To show (4.1), we rewrite ˆY_t⁽ⁿ⁾=Y_t⁽ⁿ⁾+R⁽ⁿ⁾_t , with R given by This decomposition holds true since

Nnt convergence in (4.1) follows by Slutsky’s theorem.

It suffices to show that sup_t∈[0,T_]R⁽ⁿ⁾_t vanishes in probability. Recall σ²=Var(ξ₁) and set

Then, by the Anscombe-Donsker-Theorem we find in (D[0, T], dSK) that

(Z_t⁽ⁿ⁾)_t∈[0,T_] → (Wt)_t∈[0,T_] weakly as n → ∞, such that sup_t∈[0,T_]Z_t⁽ⁿ⁾ → sup_t∈[0,T]Wt

weakly. Applying the reflection principle, it is known that the distribution of sup_t∈[0,T]W_t directly derives from the normal distribution (e.g., Billingsley, 1968). Now, we first focus on the second summand ofR⁽ⁿ⁾_t and show that its square root vanishes

sup

4. Proofs of Theorems 3.4 and 3.6

asn→ ∞. This holds true since in (D[0, T], d_SK) it holds almost surely that (N_nt)_t∼(nt/µ)_t asn→ ∞, compare, e.g., Lemma A.3.2 in Messer et al. (2014). Thus, the first factor within the supreme itself vanishes such that the entire expression tends to zero. By continuous mapping theorem, this holds true for the squared expression, such that the second summand inR⁽ⁿ⁾_t uniformly tends to zero in probability. For the first summand inR⁽ⁿ⁾_t , we decompose (ˆµ_nT−µˆnt) = (ˆµ_nT−µ)−(ˆµnt−µ) and apply the same argument as before to both summands.

As a resultR uniformly vanishes in probability.

Now, we state the consistency of the variance estimator ˆsin Corollary 4.4. Therefore, we need the consistency of the local estimators forµand ν as stated in the following lemmata.

Lemma 4.2. Let Ξbe an element ofR with mean µ. Let T >0,h∈(0, T /2] and µˆ_le and µˆ_ri be defined as the empirical means of the life times in the left and the right window, respectively.

Then it holds in (D[h, T −h], d_||·||)) almost surely as n→ ∞ that

(ˆµ_le)t∈τ_h−→(µ)t∈τ_h and (ˆµri)t∈τ_h−→(µ)t∈τ_h. Proof: This is a weaker version of Lemma A.15 in Messer et al. (2014).

Lemma 4.3. Let Ξ be an element ofR with ν² =Var((ξ1−µ)²). Let T >0, h ∈ (0, T /2]

and νˆ_le² and νˆ_ri² be defined as in (3.5) using the estimated global mean ˆ

µ⁽ⁱ⁾= ˆµnT = (1/NnT)PNnT

i=1 ξi ∀i≥1. Then it holds in(D[h, T −h], d||·||))almost surely as n→ ∞ that

(ˆν_le²)t∈τ_h−→(ν²)t∈τ_h and (ˆν_ri²)t∈τ_h−→(ν²)t∈τ_h. (4.2) Proof: For a known mean µ, i.e., V_i instead of ˆV_i convergences (4.2) are shown using the same techniques as for the consistencies of (ˆµ)t and (ˆσ²)tin Messer et al. (2014). A complete proof can be found in Albert (2014). To extend the result to the estimated global mean ˆµ_nT we derive

ν_ri² = 1

N_n(t+h)−N_nt−1

N_n(t+h)

i=Nnt+2

((ξi−µˆ_nT)²−σˆ_ri²)²

= 1

N_n(t+h)−Nnt−1

N_n(t+h)

i=Nnt+2

(ξ_i²−2ξiµˆnT + ˆµ²_nT −σˆ_ri²)²

Nn(t+h)

i=Nnt+2

(ξ_i⁴−4ξ_i³µˆ_nT + 6ξ_i²µˆ²_nT −2ξ_i²σˆ_ri² −4ξ_iµˆ³_nT + 4ξ_iµˆ_nTσˆ_ri² + ˆµ⁴_nT −2ˆµ²_nTσˆ_ri² + ˆσ⁴_ri)

N_n(t+h)−N_nt−1 .

(4.3) Using the summand withξ³_iµˆ_nT as an example, we explain that the difference between the summands with the estimators forµandσ in the latter display and with the known values vanishes a.s. asymptotically forn→ ∞. It holds

N_n(t+h)−Nnt−1

Nn(t+h)

i=Nnt+2

ξ³_iµˆnT − 1

N_n(t+h)−Nnt−1

Nn(t+h)

i=Nnt+2

ξ_i³µ

= 1

N_n(t+h)−N_nt−1

N_n(t+h)

i=Nnt+2

ξ³_i(ˆµ_nT −µ). (4.4)

4. Proofs of Theorems 3.4 and 3.6

The strong consistency (ˆµ_nT)t∈τ_h → (µ)t∈τ_h a.s. asn → ∞ follows directly by Lemma 4.2 (note that in our situation regarding the fixed point in timeT is sufficient) and thus it follows that (ˆµ_nT −µ)t∈τ_h vanishes a.s. Moreover, withξ_i∈L⁴ and the exact same techniques as in the consistency proof of ˆσ_ri (Lemma A.16 in Messer et al., 2014) it can be shown that almost surely asn→ ∞





N_n(t+h)−N_nt−1

N_n(t+h)

i=Nnt+2

ξ_i³





t∈τ_h

−→(c)t∈τ_h

for some constant c >0. This implies that the expression in (4.4) vanishes a.s. in (D[h, T −h], d_||·||)).

Similar arguments using also the strong consistency of ˆσ_ri (where the replacement of the estimator ˆµ_ri by the estimator ˆµ_nT neither changes the correctness of Lemma A.16 in Messer et al. (2014) nor the arguments in its proof) hold for all other summands in equation (4.3).

Consequently, the assertion follows by standard application of Slutsky’s theorem.

Lemma 4.3 directly implies the consistency of the variance estimator ˆs².

Corollary 4.4. Let Ξ∈R with ν² =Var((ξ1−µ)²). Let T >0, h∈(0, T /2] and ˆs²(t, h) be defined as in (3.4). Then it holds in (D[τ_h], d_||·||) almost surely as n→ ∞ that

nsˆ²(nt, nh)

t∈τ_h−→

2ν² h/µ

t∈τh

Proof: Recall equation (3.4)

s²_t := νˆ_ri² h/ˆµri

+ νˆ_le² h/ˆµ_le.

The assertion follows from the consistency of ˆµri,µˆle (Lemma 4.2) and ˆν_ri²,νˆ_le² (Lemma 4.3) by application of Slutsky’s theorem.

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