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Minimax estimation for time series models

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The minimax principle is very important for all the fields of statistical science. The minimax approach is to choose an estimator which protects against the largest risk possible. In this paper we show that the Whittle estimator becomes a minimax estimator for the prediction error loss. It is shown that the Whittle estimator is a Bayes estimator for Jeffreys’ prior. Because the minimax approach is very immature in time series analysis, the result shows another advantage of the Whittle estimator.

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We are grateful to two anonymous referees for their careful reading original manuscript and providing very valuable and constructive comments for authors.

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Correspondence to Yan Liu.

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Taniguchi was supported by JSPS Grant-in-Aid for Scientific Research (S) 18H05290. Liu was supported by JSPS Grant-in-Aid Scientific Research (C) 20K11719.



\(\underline{\mathrm{Derivation\,of\,}(2)}\). Note that

$$\begin{aligned} \frac{\partial }{\partial \xi _{jk}}\log \det \{f_\varXi (\lambda )\}&=-\left[ \frac{\partial }{\partial \xi _{jk}} \log \det (I_p-\varXi { \mathrm {e} }^{{ \mathrm {i} }\lambda }) +\frac{\partial }{\partial \xi _{jk}} \log \det (I_p-\varXi { \mathrm {e} }^{-{ \mathrm {i} }\lambda })\right] \nonumber \\&= \mathrm {tr}\left[ (I_p-\varXi { \mathrm {e} }^{{ \mathrm {i} }\lambda })^{-1}(\delta _{jk}{ \mathrm {e} }^{{ \mathrm {i} }\lambda })\right] + \mathrm {tr}\left[ (I_p-\varXi { \mathrm {e} }^{-{ \mathrm {i} }\lambda })^{-1} (\delta _{jk}{ \mathrm {e} }^{-{ \mathrm {i} }\lambda })\right] \nonumber \\&= \{I_p-\varXi { \mathrm {e} }^{{ \mathrm {i} }\lambda }\}^{-1}_{(k,j)}{ \mathrm {e} }^{{ \mathrm {i} }\lambda } + \{I_p-\varXi { \mathrm {e} }^{-{ \mathrm {i} }\lambda }\}^{-1}_{(k,j)}{ \mathrm {e} }^{-{ \mathrm {i} }\lambda }. \end{aligned}$$

Then, it follows from (10) that

$$\begin{aligned}&\frac{1}{4\pi }\int _{-\pi }^\pi \frac{\partial }{\partial \xi _{jk}} \log \det \{f_\varXi (\lambda )\} \frac{\partial }{\partial \xi _{lm}} \log \det \{f_\varXi (\lambda )\}\mathop {}\!\mathrm {d}\lambda \nonumber \\ =&\frac{1}{4\pi }\int _{-\pi }^\pi \Bigg [\{I_p-\varXi { \mathrm {e} }^{{ \mathrm {i} }\lambda }\}^{-1}_{(k,j)}\{I_p-\varXi { \mathrm {e} }^{-{ \mathrm {i} }\lambda }\}^{-1}_{(m,l)} \nonumber \\&\qquad \qquad + \{I_p-\varXi { \mathrm {e} }^{-{ \mathrm {i} }\lambda }\}^{-1}_{(k,j)}\{I_p-\varXi { \mathrm {e} }^{{ \mathrm {i} }\lambda }\}^{-1}_{(m,l)} \Bigg ]\mathop {}\!\mathrm {d}\lambda . \end{aligned}$$

Transforming \({ \mathrm {e} }^{{ \mathrm {i} }\lambda }=z\), (11) is equal to

$$\begin{aligned}&\frac{1}{4\pi }\left[ \int _{|z|=1} \left( \{I_p-\varXi z\}^{-1}_{(k,j)} \{I_p-\varXi z^{-1}\}^{-1}_{(m,l)}\right. \right. \nonumber \\&\quad \left. \left. +\{I_p-\varXi z^{-1}\}^{-1}_{(k,j)} \{I_p-\varXi z\}^{-1}_{(m,l)}\right) \frac{1}{{ \mathrm {i} }z}\right] \mathop {}\!\mathrm {d}z. \end{aligned}$$

Let us consider the second integrand in (12), which is equal to

$$\begin{aligned} \frac{1}{4\pi { \mathrm {i} }} \int _{|z|=1} \{z I_p-\varXi \}^{-1}_{(k,j)} \{I_p-\varXi z\}^{-1}_{(m,l)} \mathop {}\!\mathrm {d}z. \end{aligned}$$

Let G be

$$\begin{aligned} G_{\{(j,k):(l,m)\}}(z) = \{z I_p-\varXi \}^{-1}_{(k,j)} \{I_p-\varXi z\}^{-1}_{(m,l)}. \end{aligned}$$

Under Assumption 1, \(\{I_p-\varXi z\}^{-1}_{(m,l)}\) does not have poles in the unit circle. On the other hand,

$$\begin{aligned} \{z I_p-\varXi \}^{-1}_{(k,j)} = \frac{ \mathrm{adj}\{z I_p-\varXi \}_{(k, j)}}{\det \{z I_p-\varXi \}}, \end{aligned}$$

where \(\mathrm{adj}(A)\) denotes the adjugate matrix of A. It is straightforward to see that

$$\begin{aligned} \det \{z I_p-\varXi \} = (z - \theta _1) \cdots (z - \theta _p), \end{aligned}$$

where \(\theta _1, \dots , \theta _p\) are eigenvalues of \(\varXi \). By use of the residue theorem (e.g. [9, p. 167]), we obtain

$$\begin{aligned} \frac{1}{4\pi { \mathrm {i} }} \int _{|z|=1} \{z I_p-\varXi \}^{-1}_{(k,j)} \{I_p-\varXi z\}^{-1}_{(m,l)} \mathop {}\!\mathrm {d}z = \frac{1}{2} \sum _{r=1}^p \mathrm{Res}(G_{\{(j,k):(l,m)\}}, \theta _r), \end{aligned}$$

where \(\mathrm{Res}(F, \tau )\) is the residue of F at \(\tau \). Especially, if \(\theta _1, \dots , \theta _p\) are distinct eigenvalues, then

$$\begin{aligned} \mathrm{Res}(G_{\{(j,k):(l,m)\}}, \theta _r) = \frac{\mathrm{adj}\{\theta _r I_p-\varXi \}_{(k, j)}\{I_p-\varXi \theta _r\}^{-1}_{(m,l)}}{\prod _{i \not = r}(\theta _r - \theta _i)}. \end{aligned}$$

With a similar computation for the first integrand in (12), we obtain (2).

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Liu, Y., Taniguchi, M. Minimax estimation for time series models. METRON 79, 353–359 (2021).

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