# Subsampling Algorithms for Irregularly Spaced Autoregressive Models
## 1 Introduction  
Time series analysis is useful in many application domains for understanding patterns over time and for forecasting into the future. Most time series methods have been developed for regularly spaced time series, where the gaps between two consecutive time points are fixed and equal. Regular time series include hourly, daily, monthly, or annually recorded time series. Recently, there has been increasing interest in developing methods for analyzing and forecasting irregularly spaced time series, where the gaps between subsequent observations are not the same. Such irregular time series are observed in diverse fields such as astronomy [1], climatology [2], finance [3], etc. For instance, intra-day transaction-level data in finance consist of prices of a financial asset recorded at each trade within a trading day, resulting in irregular time intervals (in seconds, say) between consecutive trades. An example in real estate consists of the time series of sale prices of houses, where the discrete time gaps (in days or weeks) between subsequent sale dates are typically nonconstant. Consider an irregularly spaced time series \(y_{j},\ j=0,\ldots,m\) consisting of observations at discrete times \(t_{j}\), where \(t_{0}<t_{1}<\cdots<t_{m}\). The gaps between consecutive time points, denoted by \(d_{j}=t_{j}-t_{j-1},\ j=1,\ldots,m\), are not constant. To model irregularly spaced time series with discrete gaps, [PERSON] et al. (2019) proposed the stationary gap time autoregressive (Gap AR(1)) model:  
\[y_{0}=\sigma\sqrt{\frac{1}{1-\phi^{2}}}\epsilon_{0},\quad y_{j}=\phi^{d}y_{j-1}+ \sigma\sqrt{\frac{1-\phi^{2 dj}}{1-\phi^{2}}}\epsilon_{j}\text{ for }j=1,\ldots,m, \tag{1}\]  
where \(\epsilon_{j}\) is the error term with the standard normal distribution, i.e., \(\epsilon_{j}\sim\mathcal{N}(0,1)\), \(\phi\in(-1,1)\) is the autoregressive parameter, and \(\sigma>0\) is a scale parameter. They used (1) to model and forecast house prices and then constructed a house price index. Other models have also been constructed in the literature for irregularly spaced time series. [PERSON] et al. (2019) described a nonstationary irregularly spaced autoregressive (NIS-AR(1)) model and illustrated its use on astronomy data. [PERSON] et al. (2019) described Bayesian modeling for an irregular stochastic volatility autoregressive conditional duration (IR-SV-ACD) model and used this for estimating and forecasting inter-transaction gaps and the volatility of financial log-returns. In this paper, we refer to the Gap AR(1) model in (1) as the irregularly spaced AR(1) or the IS-AR(1) model. Estimation in this model can be handled by the method of maximum likelihood. While a model for irregularly spaced time series enables handling a wider range of temporal scenarios, the statistical analysis involves considerable computational complexity. Specifically, the task of capturing temporal structures by estimating the model parameters becomes more challenging due to the irregular gaps. One potential solution to address these computational challenges is to perform the computations of interest on _subsamples_ that are selected from the full data. Despite extensive recent research on subsampling techniques, subsampling methods tailored for irregularly spaced time series data are sparse. In this article, we fill this gap by proposing novel subsampling methods specifically designed for irregularly spaced time series data analyzed by the IS-AR(1) model. In comparison to other subsampling methods, our proposed algorithms are founded upon the optimality criteria under classical statistical frameworks. It should be noted that the term \"subsample\" in time series analysis generally refers to a subset of data that are in the form of multiple series, blocks, or sequences, and the main objective of their analysis is to provide estimates of the variance of the full-data estimator; see [PERSON] (2019), [PERSON] (2019), [PERSON] (2019).