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From Reviews: 0MR4320950 62G05 60G22 62M10Caron, Emmanuel (F-AVIG-LM) ; Dedecker, Jérôme (F-UPARIS-LAM) ;Michel, Bertrand [Michel, Bertrand1 ] (F-NANTC-LM)
Gaussian linear model selection in a dependent context. (English summary)Electron. J. Stat. 15 (2021), no. 2, 4823–4867.

This article is concerned with a nonparametric fixed-design regression model with aGaussian dependent error. The model can be expressed as i∗Yi = f+ εi , i ∈ {1, . . . , n} ,nwhere f ∗ is an unknown function having support on [0, 1] and εi is a stationary Gaussian
process. The aim is to estimate f ∗ using n observations {Yi }ni=1 .
The article focuses on fitting the regressogram of order m to f ∗ , where m is chosen
via a model selection procedure. More specifically, the optimal order mb is determinedby
2b(1)mb ∈ arg min
Y − fm + pen(m) ,m∈{1,... ,n}nPn
where kXkn = n1 i=1 Xi2 , fbm is the regressogram of order m, and pen(m) is somepenalty term depending on m.
It is demonstrated that the form of pen(m) varies with the dependent structure of εi .
Details are given below.
• If εi is a short-memory process so that its spectral radius (the largest eigenvalueof its long-run variance) is bounded, then the penalty is in the formm
pen(m) = K
nfor some K > 0.
• If εi is a long-memory process so that its kth-order autocovariance satisfies|E(εi εi−k )| ≤ κk −γ for some κ > 0 and γ ∈ (0, 1), then the penalty becomes m γpen(m) = Knfor some K > 0.
• If εi is anti-persistent so that Var(Sn ) ≤ κn2−γ for the partial sum process
Sn = ε1 + · · · + εn , some κ > 0 and γ ∈ (1, 2), then the penalty is given by 
m γ log mpen(m) = K+nnfor some K > 0.
In practice, however, making the order selection via (1) fully operational requires one
to prespecify the dependent structure of εi and choose the constant K in a data-driven
manner. Now, if Var(Sn ) ∼ κn2−γ for some γ ∈ (0, 2), then the leading term of pen(m)is of order (m/n)γ . The exponent γ also relates to the Hurst index H of the partial sum
process Sn so that γ = 2 − 2H. In conclusion, it is recommended that the order selection
is implemented in the following two-step procedure:
1. Run the dimension jump algorithm [see J.-P. Baudry, C. Maugis-Rabusseau andB. Michel, Stat. Comput. 22 (2012), no. 2, 455–470; MR2865029] using the penaltycb 1 is the Whittleshape (m/n)2−2H1 to select the pre-model of order m
b 1 , where H
estimate of the Hurst index using the Y process.
c
2. Rerun the dimension jump algorithm using the penalty shape (m/n)2−2H2 tob 2 is the Whittle estimate of the Hurst
select the final model of order m,b where Hindex using the residual of the pre-model mb 1.
Masayuki Hirukawa
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