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Risk-efficient sequential estimation of multivariate random coefficient autoregressive process. (English) Zbl 1429.62399

A \(p\)th-order vector autoregressive model of dimension \(k\) is defined as: \(\mathbf{X}_t=\sum_{i=1}^{p}(B_i + \boldsymbol{\Gamma}_{ti}) \mathbf{X}_{i-1}+{\boldsymbol{\epsilon}}_t\), where \(\mathbf{{X}}_t\) is the observation vector of order \(k \times 1\) at time \(t\), \(B_i\)’s are the fixed autoregressive parameter matrices of order \(k\times k\), \(\boldsymbol{\Gamma}_{ti}\)’s are the random matrices of order \(k\times k\) associated with \(\mathbf{X}_{t-i}\), and \(\boldsymbol{\epsilon}_t\)’s are the random error vector of order \(k\times 1\). The preceding model can be rewritten as: \(\mathbf{X}_t=B \mathbf{Y}_{t-1}+\mathbf{u}_t\), where \(\mathbf{Y}_t=(\mathbf{X}_t',\mathbf{X}_{t-1}',\dots,\mathbf{X}_{t-p+1}')'\) is a \(pk\)-vector, \(B=(B_1, B_2,\dots,B_p)\) is a \(k\times kp\) matrix, and similarly \(\mathbf{u}_t\) is defined. Let \(\tilde{B}\) be a suitable estimate of \(B\), the predicted value of \(\mathbf{X}_t\), based on observations of the previous time points, is \(\tilde{\mathbf{X}}_t=\tilde{B} \mathbf{Y}_{t-1}\). Let \(c>0\) be the reciprocal of the cost of a single observation, and let \(\mathcal F_t\) be the \(\sigma\)-field generated by \(\{\mathbf{X}_s;s\leq t \}\). Then the loss function for \(n\) time points under consideration is: \[ L_n(\tilde{B})=\frac{c}{n} \sum_{t=1}^{n}\{\tilde{\mathbf{X}}_t-E(\mathbf{X}_t|\mathcal F_{t-1})\}'\{\tilde{\mathbf{X}} -E(\mathbf{X}_t|\mathcal F_{t-1})\} + n . \] The predictive risk is defined as \(R_n(\tilde{B})=EL_n(\tilde{B})\). Using \(\hat{B}_n\) as the least squares estimate of the parameter matrix \(B\), the authors consider the asymptotic (as \(n\rightarrow \infty)\) normal distribution of \(\sqrt{n}(\hat{B}_n - B)\) due to D. F. Nicholls and B. G. Quinn [Random coefficient autoregressive models: an introduction. Cham: Springer (1982; Zbl 0497.62081)]. This is used to approximate the risk \(R_n(\hat{B}_n)\), which in turn is minimized to describe the oracle procedure. The minimization of the predictive risk \(R_n(\hat{B}_n)\) gives the optimal sample size \(n_0 \approx \sqrt{c \psi}\) where \(\psi\) depends on some nuisance parameters assumed to be known in the oracle procedure.
In general, the nuisance parameters are unknown and hence the best fixed sample size \(n_0\) cannot be used. Therefore, sequential procedure is usually the best option. Often the stopping time of the sequential procedure is defined by sequentializing the fixed sample size. Thus, for a suitable initial sample size, and suitable estimate \(\hat{\psi}\) at each stage, the stopping time is defined as: \(N=\inf\{n\geq n_c: n \geq \sqrt{c \hat{\psi}} \}\). Recalling that \(c\) is the reciprocal of the cost of sampling per observation, the authors study various asymptotic (as \(c\rightarrow \infty\)) properties of \(N=N(c)\). For example, they prove (i) \(\frac{N}{\sqrt{c \psi}} \rightarrow 1\) a.s., (ii) \(\frac{EN}{\sqrt{c \psi}} \rightarrow 1\), and (iii) \(\frac{R}{R_0} \rightarrow 1\), as \(c \rightarrow \infty\), where \(R\) and \(R_0\) are the risks based on stopping times and optimal fixed sample size. They also obtain asymptotic (as \(c \rightarrow \infty\)) mean and variance of \(N\), and the asymptotic normal (properly normalized) distribution of \(N\).

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62H12 Estimation in multivariate analysis
62L12 Sequential estimation

Citations:

Zbl 0497.62081
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References:

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