# Subsampling Algorithms for Irregularly Spaced Autoregressive Models
## 3 Full-Data Estimation for the IS-AR(1) Model  
Given a time series of length \(m\) from the IS-AR(1) model in (1), we present the full-data maximum likelihood estimates (MLEs) of the model parameters. Since \(m\) is assumed to be large, we can obtain the MLEs of the unknown parameters \(\boldsymbol{\theta}=(\phi,\sigma)^{\prime}\) by maximizing the _conditional_ log-likelihood function (which ignores the marginal distribution of the initial data point \(y_{0}\) at time \(t_{0}\)). Up to a normalizing constant, this has the form  
\[ \begin{split}\mathrm{M}_{m}(\boldsymbol{\theta})&= \frac{1}{m}\sum_{j=1}^{m}\ell(\boldsymbol{\theta};y_{j},y_{j-1},d_{j})\\ &=\frac{1}{m}\sum_{j=1}^{m}\left(-\log\sigma-\frac{1}{2}\log \frac{1-\phi^{2d_{j}}}{1-\phi^{2}}-\frac{(y_{j}-\phi^{d_{j}}y_{j-1})^{2}(1- \phi^{2})}{2\sigma^{2}(1-\phi^{2d_{j}})}\right).\end{split}  \tag{2}\]  
The MLE \(\hat{\boldsymbol{\theta}}=\arg\max_{\theta}\mathrm{M}_{m}(\boldsymbol{\theta})\) is the maximizer of (2) and must be solved numerically since an analytical solution is infeasible. Finding the MLE is challenging due to (a) the domain restriction (i.e., \(-1<\phi<1\)) and (b) the nonconcavity of the objective function. These challenges often cause convergence issues for the the classical Newton-Raphson algorithm. Since \(\phi\) is a correlation coefficient constrained to lie between \(-1\) and \(1\), an unconstrained gradient-based optimization method may not work well. To remedy this, we use [PERSON]'s \(z\)-transformation to map \(\phi\) from \((-1,1)\) onto \(\mathbb{R}\) so that an unconstrained optimizing algorithm can be used on the transformed parameter. Although the choice of the transformation is not unique and other transformations may be used, [PERSON]'s \(z\)-transformation provides a well-understood estimator of the correlation coefficient, especially when the error term \(\varepsilon_{j}\) follows a normal distribution ([PERSON] et al., 2018). Specifically, let  
\[z=\mathrm{atanh}(\phi)=\frac{1}{2}\log\biggl{(}\frac{1+\phi}{1-\phi}\biggr{)},\]  
so that  
\[\phi=\tanh(z)=\frac{\exp(2z)-1}{\exp(2z)+1}. \tag{3}\]  
We have  
\[\frac{\partial\phi}{\partial z}=\mathrm{sech}^{2}(z)=1-\phi^{2},\]and  
\[\frac{\partial^{2}\phi}{\partial z^{2}}=-2\text{tanh}(z)\text{sech}^{2}(z)=-2\phi( 1-\phi^{2}),\]  
where \(\text{tanh}(\cdot)\) and \(\text{sech}(\cdot)\) are, respectively, the hyperbolic tangent and secant functions. Below, we show the use of the coordinate ascent algorithm to maximize (2) in terms of \(z\); we then insert the estimate \(z\) into (3) to obtain the estimate of \(\phi\). We denote the gradient vector and Hessian matrix of the per-observation log-likelihood \(\ell(\mathbf{\theta};y_{j},y_{j-1},d_{j})\) in (2) as follows:  
\[\dot{\mathbf{\ell}}(\mathbf{\theta};y_{j},y_{j-1},d_{j})=\frac{\partial\ell(\mathbf{\theta };y_{j},y_{j-1},d_{j})}{\partial\mathbf{\theta}}= \begin{bmatrix}\dot{\ell}_{1}( \mathbf{\theta};y_{j},y_{j-1},d_{j})\\ \dot{\ell}_{2}(\mathbf{\theta};y_{j},y_{j-1},d_{j})\end{bmatrix} , \tag{4}\]  
and  
\[\dot{\mathbf{\ell}}(\mathbf{\theta};y_{j},y_{j-1},d_{j})=\frac{\partial^{2}\ell(\mathbf{ \theta};y_{j},y_{j-1},d_{j})}{\partial\mathbf{\theta}\partial\mathbf{\theta}^{\prime }}= \begin{bmatrix}\dot{\ell}_{11}(\mathbf{\theta};y_{j},y_{j-1},d_{j})&\dot{\ell}_ {12}(\mathbf{\theta};y_{j},y_{j-1},d_{j})\\ \dot{\ell}_{12}(\mathbf{\theta};y_{j},y_{j-1},d_{j})&\dot{\ell}_{22}(\mathbf{\theta};y _{j},y_{j-1},d_{j})\end{bmatrix} . \tag{5}\]  
The first and second derivatives of \(\ell(\mathbf{\theta};y_{j},y_{j-1},d_{j})\) with respect to \(z\) are, respectively, denoted by  
\[\dot{\ell}_{z1}(\mathbf{\theta})=(1-\phi^{2})\dot{\ell}_{1}(\mathbf{\theta})\text{ and }\dot{\ell}_{z11}(\mathbf{\theta})=(1-\phi^{2})^{2}\dot{\ell}_{11}(\mathbf{\theta})-2(1- \phi^{2})\phi\dot{\ell}_{1}(\mathbf{\theta}).\]  
We describe the coordinate ascent algorithm. Suppose \(\phi^{(k)}=\text{tanh}(z^{(k)})\) is obtained in the \(k\)-th step of the algorithm.