# Subsampling Algorithms for Irregularly Spaced Autoregressive Models
## 5 Numerical Results
### Estimation Efficiency  
We use the empirical mean squared error (MSE) to measure the estimation efficiency of the subsample estimator with respect to the true parameter. We implement the subsampling methods discussed in Section 4 and repeat the simulation 1000 times to calculate the empirical MSE, which is defined as  
\[\text{MSE}=\frac{1}{1000}\sum_{s=1}^{1000}\left\|\mathbf{\theta}^{(s)}-\mathbf{\theta} \right\|^{2}=\frac{1}{1000}\sum_{s=1}^{1000}((\tilde{\mathbf{\theta}}^{(s)}-\phi)^ {2}+(\vartheta^{(s)}-\sigma)^{2}]=\text{MSE}_{\phi}+\text{MSE}_{\sigma}, \tag{15}\]  
where \(\mathbf{\tilde{\theta}}^{(s)}=(\tilde{\mathbf{\vartheta}}^{(s)},\tilde{\sigma}^{(s)})\) is the subsample estimate in the \(s\)-th repetition. Due to the nonconvex nature of the model, some methods failed to converge in a few repetitions. We drop the results in these repetitions when calculating the MSE. This does not affect the accuracy of the MSE because the nonconvergence rate is very low. The results in terms of the empirical MSE are presented in Figure 1. It is seen that all three proposed subsampling methods outperform the uniform sampling in all the cases. Overall, the **opt** algorithm displays the best performance, especially when \(\sigma\) or \(\phi\) is large. The deterministic methods, **iboss** and **thin**, have higher estimation efficiency when \(\sigma\) is small. The varying performances in empirical MSE among the proposed subsampling methods are partly due to the different optimality criteria adopted. The A-optimality criterion used by **opt** seeks to minimize the trace of the asymptotic variance matrix, whereas the D-optimality criterion used by both **iboss** and **thin** focuses on minimizing the determinant of the asymptotic variance matrix. As a result, **opt** places greater emphasis on parameter components with larger variances, while **iboss** and **thin** distribute their focus more evenly across all parameters. When \(\sigma\) is small (e.g., \(\sigma=1\)), the variance in estimating \(\sigma\) is lowacross all subsampling methods, allowing **iboss** and **thin** to express their advantage in reducing the variance of estimating \(\phi\), thereby outperforming **opt**. However, when \(\sigma=20\), the variance in estimating \(\sigma\) becomes the dominant contributor to the empirical MSE. Since **iboss** and **thin** do not prioritize the estimation of \(\sigma\), their performance weakens in this case. A similar reasoning applies to the differing performances across various values of \(\phi\). To better understand this, we further decompose the empirical MSEs for \(\sigma\) and \(\phi\) to validate our explanation. In order to further examine the performance of the subsampling methods on estimating different types of parameters, we plot \(\text{MSE}_{\phi}\) for estimating \(\phi\) in Figure 2 and \(\text{MSE}_{\sigma}\) for estimating \(\sigma\) in Figure 3. Irrespective of the values of \(\phi\) and \(\sigma\), **iboss** and **thin** outperform **opt** in estimating \(\phi\), while they fall short in estimating \(\sigma\) compared to **opt**. We observe that the proposed subsampling methods provide greater benefit for smaller \(r\). As \(r\) increases, the differences between the methods gradually diminish. This is to be expected, since all estimators converge to the full-data estimator when \(r\) gets closer to \(m\). Therefore, with larger datasets, the advantage of using optimal subsampling estimators becomes more pronounced. In the IS-AR(1) model, the predicted value at any given time point is directly influenced by the correlation coefficient \(\phi\). Therefore, more accurate estimates of \(\phi\) lead to better prediction accuracy. As shown in Figure 2, the optimal subsampling methods provide exceptional performance in estimating \(\phi\), especially when using the **iboss** and **thin** approaches. Notably, with a subsample size of \(r=5000\), both **iboss** and **thin** outperform the uniform sampling method, which uses a significantly larger subsample size of \(r=30,000\) for estimating \(\phi\).