On minimax identification of nonparametric autoregressive models (Q1964758)

The authors study the nonparametric autoregressive process \[ y_t= f(y_{t-1},\dots, y_{t-d}+ e_t) \] with \(f:\mathbb{R}\to \mathbb{R}\) bounded measurable and \(e_t\), \(t= 0, 1, 2,\dots\), independent Gaussian with mean zero. They investigate the statistical problem of estimating the unknown function \(f\) from data \(y_t\), \(t= -d+ 1,\dots, N\), where \(f\) is assumed to lie in the Barron class of functions \[ f(x)= \int_{\mathbb{R}^d} \exp (iw^Tx) \widehat{F} (dx) \] where \(\widehat{F}\) has bounded total variation. For any function \(f\) in this class, \textit{A.R. Barron} [IEEE Trans. Inf. Theory 39, No. 3, 930-945 (1993; Zbl 0818.68126)] could prove that for any probability distribution \(\mu\) on \(\mathbb{R}^d\) there exist Fourier frequencies \(\omega_1,\dots, \omega_M\) and amplitudes \(\lambda_1,\dots, \lambda_M\) with \(\sum_{k=1}^M |\lambda_k|\leq\|\widehat F\|_{TV}\) such that \[ \int \Biggl|f(x)- \sum_{k=1}^M \lambda_k \exp(i\omega_k^T x) \Biggr|d\mu(x)\leq L^2/M. \] The estimation algorithm proposed in this paper chooses first an \(\varepsilon\)-net of Fourier frequencies \(\omega_1,\omega_M\) from a suitably large ball in \(\mathbb{R}^d\). Then a stochastic approximation algorithm is applied to obtain estimates of the coefficients \(\lambda_1,\dots,\lambda_M\). The authors show that the resulting estimator has \(L_2\)-risk \(O(\log N/N)^{1/4}\) and that this rate is optimal.