A Novel First-Order Autoregressive Moving Average Model to Analyze Discrete-Time Series Irregularly Observed

TL;DR

This paper introduces a new first-order autoregressive moving average model tailored for analyzing irregularly observed discrete-time series, demonstrating its statistical properties, estimation methods, and practical applications in medical and astronomical data.

Contribution

It presents a novel ARMA model for irregular time series, including stationarity analysis, state-space representation, maximum likelihood estimation, and real-world applications.

Findings

Model is strictly stationary and ergodic under Gaussianity.

Estimation bias and error decrease with longer series.

Method performs well even with small sample sizes.

Abstract

A novel first-order autoregressive moving average model for analyzing discrete-time series observed at irregularly spaced times is introduced. Under Gaussianity, it is established that the model is strictly stationary and ergodic. In the general case, it is shown that the model is weakly stationary. The lowest dimension of the state-space representation is given along with the one-step linear predictors and their mean squared errors. The maximum likelihood estimation procedure is discussed, and their finite-sample behavior is assessed through Monte Carlo experiments. These experiments show that bias, root mean squared error, and coefficient of variation are smaller when the length of the series increases. Further, the method provides good estimations for the standard errors, even with relatively small sample sizes. Also, the irregularly spaced times seem to increase the estimation variability. The application of the proposed model is made through two real-life examples. The first is concerned with medical data, whereas the second describes an astronomical data set analysis.