On variance estimation under shifts in the mean
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On variance estimation under shifts in the mean Ieva Axt1 · Roland Fried1 Received: 18 April 2019 / Accepted: 18 March 2020 © The Author(s) 2020
Abstract In many situations, it is crucial to estimate the variance properly. Ordinary variance estimators perform poorly in the presence of shifts in the mean. We investigate an approach based on non-overlapping blocks, which yields good results in changepoint scenarios. We show the strong consistency and the asymptotic normality of such blocks-estimators of the variance under independence. Weak consistency is shown for short-range dependent strictly stationary data. We provide recommendations on the appropriate choice of the block size and compare this blocks-approach with difference-based estimators. If level shifts occur frequently and are rather large, the best results can be obtained by adaptive trimming of the blocks. Keywords Blockwise estimation · Change-point · Trimmed mean
1 Introduction We consider a sequence of random variables Y1 , … , YN generated by the model
Yt = Xt +
K ∑
hk It≥tk .
(1)
k=1
Most ( ) of the time (we) assume that X1 , … , XN are i.i.d. random variables with E Xt = 𝜇 and Var Xt = 𝜎 2 , but this will be relaxed occasionally to allow for a short-range dependent strictly stationary sequence. The observed data y1 , … , yN are affected by an unknown number K of level shifts of possibly different heights h1 , … , hK at different time points t1 , … , tK . Our goal is the estimation of the variance 𝜎 2 . Without loss of generality, we will set 𝜇 = 0 in the following.
* Ieva Axt [email protected]‑dortmund.de Roland Fried [email protected]‑dortmund.de 1
TU Dortmund University, Dortmund, Germany
13
Vol.:(0123456789)
I. Axt, R. Fried
In Sect. 2 we analyse estimators of 𝜎 2 from the sequence of observations (Yt )t≥1 by combining estimates obtained from splitting the data into several blocks. Without the need of explicit distributional assumptions the mean of the blockwise estimates turns out to be consistent if the size and the number of blocks increases, and the number of jumps increases slower than the number of blocks. If many jumps in the mean are expected to occur, an adaptively trimmed mean of the blockwise estimates can be used, see Sect. 3. In Sect. 4 a simulation study is conducted to assess the performance of the proposed approaches. In Sect. 5 the estimation procedures are applied to real data sets, while Sect. 6 summarizes the results of this paper.
2 Estimation of the variance by averaging When dealing with independent identically distributed data the sample variance is the common choice for estimation of 𝜎 2 . However, if we are aware of a possible presence of level shifts at unknown locations, it is reasonable to divide the sample Y1 , … , YN into m non-overlapping blocks of size n = ⌊N∕m⌋ and to calculate the average of the m sample variances derived from the different blocks. A similar approach has been used in Dai et al. (2015) in the context of repeated measurements data and in Rooch et al. (2019) for estimation
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