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Dette, Holger (D-BCHMM) ; Eckle, Theresa (D-BCHMM) ;Vetter, Mathias (D-KIEL-MS)
Multiscale change point detection for dependent data. (English summary)
Scand. J. Stat. 47 (2020), no. 4, 1243–1274.

To study the problem of detecting multiple abrupt changes in time-series data, thisarticle concentrates on the simultaneous multiscale change point estimator (SMUCE)of the signal part in the model i∗
Yi = ϑ+ εi , i = 1, . . . , n.
nThe signal ϑ∗ : [0, 1] → R is defined as an unknown piecewise linear function∗∗ϑ (t) =KX

θk∗ 1[τ ∗ ,τ ∗ ) (t),k k+1k=0∗∗
where K ∗ is the unknown number of change points, 0 = τ0∗ < τ1∗ < · · · < τK
∗ < τK ∗ +1 = 1∗∗
are the change point locations, and θ0∗ , . . . , θK
∗ are the function values of ϑ . The vector∗∗∗∗of change points is denoted as J(ϑ ) = (τ1 , . . . , τK ∗ ) with |J(ϑ )| being its dimension.
Consistency of the SMUCE of ϑ∗ for an i.i.d. Gaussian error εi has already been
established. This article aims at demonstrating consistency of the estimator when theerror process εi is a so-called physically dependent sequence [W. B. Wu, Proc. Natl.
Acad. Sci. USA 102 (2005), no. 40, 14150–14154; MR2172215]. The change point
location estimates bτk are assumed to take only the values {0, 1/n, . . . , (n − 1)/n}. The
set of such functions is denoted as Sn , and the estimator of ϑ∗ is chosen among the classPKϑ(·) = k=0 θk 1[τk ,τk+1 ) (·) ∈ Sn . For this purpose, let the multiscale statistic ber1penjVn (Y, ϑ) = max
supj − i + 1 Y i − θk − 2 log,0≤k≤K nτk ≤i≤j≤nτk+1
σbj −i+1j−i+1≥ncn

Pj
jwhere cn > 0 is a sequence converging to zero and Y i = `=i Y` /(j − i + 1)
is a local mean. In addition, σb2 is a long-run variance estimator of theerror process, and a difference-based one is considered. Let the entire sample {Yi }ni=1 be partitioned into mn = bn/kn c sub-samples {Y1 , . . . , Ykn },{Ykn +1 , . . . , Y2kn }, . . . , {Y(mn −1)kn +1 , . . . , Ymn kn } with equal length kn . For a local avPkn
erage of the sub-sample Ai = j=1
Yikn +j /kn , the difference-based variance estimatoris given bymXn −1kn|Ai − Ai−1 |2 .
σb2 =2(mn − 1) i=1
The estimation procedure of the piecewise linear function ϑ∗ takes two steps. First,for a threshold q chosen via the limiting null distribution of Vn given below, the numberof change points K ∗ is estimated as
b = K(Vb n , q) =Kinfϑ∈SnVn (Y,ϑ)≤q|J(ϑ)|.

Second, ϑ∗ is estimated as 2n Xi
Yi − ϑnϑ∈C(Vn ,q) i=1

b = arg minϑb and Vn (Y, ϑ) ≤ q}. The only difference in thefor the set C(Vn , q) = {ϑ ∈ Sn : |J(ϑ)| = K
estimation procedure from the Gaussian error case is to employ the long-run varianceestimator σ
b2 in Vn .
If kn  n1/3 and other regularity conditions hold, then Vn has the limiting nulldistributionr|B(t) − B(s)|ed√,Vn (Y, ϑ∗ ) → max ∗sup− 2 log
0≤k≤K τ ∗ ≤s<t≤τ ∗
t−s
t−s
k+1k
where {B(t)}t∈[0,1] is the standard Brownian motion. This limiting distribution coincides
with the one for the Gaussian error case. Subsequently, the following two results on
consistency of SMUCE are established:nob n , qn ) = K ∗ = 1lim Pr K(Vn→∞√for a sequence qn satisfying 1/qn + qn / n → 0; and)(lim Prn→∞

sup|τk∗ − τk | > cn=0ϑ∈C(Vn ,qn )for k = 1, . . . , K ∗ . The first and second statements demonstrate consistencies of model
selection and change point locations in ϑ∗ , respectively.
Masayuki Hirukawa
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