A First-Order Autoregressive Process with Size-Biased Lindley Marginals: Applications and Forecasting

Abstract

In this paper, a size-biased Lindley (SBL) first-order autoregressive (AR(1)) process is proposed, the so-called SBL-AR(1). Some probabilistic and statistical properties of the proposed process are determined, including the distribution of its innovation process, the Laplace transformation function, multi-step-ahead conditional measures, autocorrelation, and spectral density function. In addition, the unknown parameters of the model are estimated via the conditional least squares and Gaussian estimation methods. The performance and behavior of the estimators are checked through some numerical results by a Monte Carlo simulation study. Additionally, two real-world datasets are utilized to examine the model’s applicability, and goodness-of-fit statistics are used to compare it to several pertinent non-Gaussian AR(1) models. The findings reveal that the proposed SBL-AR(1) model exhibits key theoretical properties, including a closed-form innovation distribution, multi-step conditional measures, and an exponentially decaying autocorrelation structure. Parameter estimation via conditional least squares and Gaussian methods demonstrates consistency and efficiency in simulations. Real-world applications to inflation expectations and water quality data reveal a superior fit over competing non-Gaussian AR(1) models, evidenced by lower values of the AIC and BIC statistics. Forecasting comparisons show that the classical conditional expectation method achieves accuracy comparable to some modern machine learning techniques, underscoring its practical utility for skewed and fat-tailed time series.