# Subsampling Algorithms for Irregularly Spaced Autoregressive Models
## 4 Subsampling Methods for the IS-AR(1) Model
### Information-Based Optimal Subdata Selection  
The **iboss** method is a deterministic selection approach to obtain a subsample. Its basic motivation is to maximize the determinant of the Fisher information matrix conditioned on the covariates. Under the model specified in (2), the conditional Fisher information for the observed data at time \(t_{j}\) is  
\[ \begin{split} I_{j}(\mathbf{\theta}|y_{j-1},d_{j})&=- \mathbb{E}\left(\left[\begin{array}{cc}\left(\frac{\partial\ell_{j}}{ \partial\theta}\right)^{2}&\frac{\partial\ell_{j}}{\partial\theta}\frac{ \partial\ell_{j}}{\partial\theta}\\ \frac{\partial\ell_{j}}{\partial\theta}\frac{\partial\ell_{j}}{\partial\theta }&\left(\frac{\partial\ell_{j}}{\partial\theta}\right)^{2}\end{array} \right]\left|y_{j-1},d_{j}\right)\\ &=\left[2c_{j}(\mathbf{\theta})^{2}\!+\!\frac{d_{j}^{2}\phi^{2d_{j}-2}y_{j-1}^{2} (1-\phi^{2})}{\sigma^{2}(1-\phi^{2})}-\frac{2}{\sigma}c_{j}(\mathbf{\theta}) \right],\\ &\quad\quad\quad\quad\quad-\frac{2}{\sigma}c_{j}(\mathbf{\theta})\quad\frac{2}{ \sigma^{2}}\end{split}  \tag{10}\]  
where  
\[c_{j}(\mathbf{\theta})=\frac{d_{j}\phi^{2d_{j}-1}}{1\!-\!\phi^{2d_{j}}}-\frac{\phi }{1\!-\!\phi^{2}}.\]  
The information matrix \(I_{j}(\mathbf{\theta}|y_{j-1},d_{j})\) has full rank. We select data points with the \(r\) largest values of \(\det\{I_{j}(\mathbf{\theta}|y_{j-1},d_{j})\}\), \(j=1,\dots,m\). This can be undertaken efficiently by established methods like some partition-based selection algorithms [30]. We implement the procedure via the following steps:1. Obtain a pilot estimate \(\mathbf{\theta}^{(\text{ptl})}\) as in Section 4.1. 2. Calculate \(\det(I_{j}(\theta^{(\text{ptl})}|y_{t_{j-1}},d_{j}))\) for \(j=1,\ldots,m\). 3. Take the \(r\) data points with the largest \(\det(I_{j}(\theta^{(\text{ptl})}|y_{j-1},d_{j}))\) using the Quickselect algorithm. Denote their inclusion indicators as \(\delta_{j}^{(\text{iboss})}\)s.
4. Obtain \(\mathbf{\theta}^{(\text{iboss})}\) by maximizing the following target function: \[\text{M}_{r}^{(\text{iboss})}(\mathbf{\theta})=\frac{1}{r}\sum_{j=1}^{m}\delta_{j} ^{(\text{iboss})}\ell(\mathbf{\theta};y_{j},y_{j-1},d_{j}).\]
5. Obtain the final subsample estimator \(\mathbf{\tilde{\theta}}^{(\text{iboss})}\) by \[ \begin{split}\mathbf{\tilde{\theta}}^{(\text{iboss})}=& \Big{(}r_{0}\tilde{\text{M}}_{r_{0}}^{(\text{ptl})}(\mathbf{\tilde{ \theta}}^{(\text{iboss})})+r\tilde{\text{M}}_{r}^{(\text{iboss})}(\mathbf{\tilde{ \theta}}^{(\text{iboss})})\Big{)}^{-1}\\ &\times\Big{(}r_{0}\tilde{\text{M}}_{r_{0}}^{(\text{ptl})}(\mathbf{ \tilde{\theta}}^{(\text{iboss})})\times\mathbf{\theta}^{(\text{ptl})}+r\tilde{ \text{M}}_{r}^{(\text{iboss})}(\mathbf{\tilde{\theta}}^{(\text{iboss})})\times\bm {\tilde{\theta}}^{(\text{iboss})}\Big{)},\end{split} \] (11) where \(\tilde{\text{M}}_{r_{0}}^{(\text{ptl})}(\mathbf{\theta})\) is defined in (8) and \[\tilde{\text{M}}_{r}^{(\text{iboss})}(\mathbf{\tilde{\theta}}^{(\text{iboss})})= \frac{1}{r}\sum_{j=1}^{m}\delta_{j}\hat{\ell}(\mathbf{\tilde{\theta}}^{(\text{iboss })};y_{j},y_{j-1},d_{j}).\] We present the **iboss** method in Algorithm 3. * **Step 1--Pilot Estimates and Preparations:** 1. Take a simple random sample of size \(r_{0}\) by sampling without replacement and set indicators \(\{\delta_{j}^{0}\}_{j=1}^{m}\). 2. \(\mathbf{\theta}^{(\text{ptl})}\leftarrow\arg\max_{\mathbf{\theta}}\sum_{j=1}^{m}\delta _{j}^{0}\ell(\mathbf{\theta};y_{j},y_{j-1},d_{j})\)
* **Step 2--Subdata Selection:** 1. **for**\(j=1\) to \(m\) \(\Big{|}I_{j}\leftarrow\det(I_{j}(\mathbf{\theta}^{(\text{ptl})}|y_{j-1},d_{j}))\) **end** 2. \(I_{(r)}\gets r\)-th largest \(I_{j}\) for \(j=1,\ldots,m\) 3. **for**\(j=1\) to \(m\) **if**\(I_{j}\geq I_{(r)}\) **then**\(\delta_{j}^{(\text{iboss})}\gets 1\) **else**\(\delta_{j}^{(\text{iboss})}\gets 1\) **end**
* **Step 3--Estimation:** 1. \(\mathbf{\tilde{\theta}}\leftarrow\arg\max_{\mathbf{\theta}}\sum_{j=1}^{m}\delta_{j}^{ (\text{iboss})}\ell(\mathbf{\theta};y_{j},y_{j-1},d_{j})\)
2.