This paper proposes a new observation-driven model for continuous positive-valued time series, called the new XLindley autoregressive moving average (NXL–ARMA) model. The model is built by combining an ARMA-type recursion for the conditional mean with the new XLindley distribution as the conditional law of the observations. This framework is suitable for variables such as trading volume, realized volatility, durations, waiting times, precipitation amounts, energy consumption, and other positive measurements, where the data often show persistence, skewness, and variability that changes with the level of the series.
The proposed model preserves positivity by construction and remains relatively simple to handle from both theoretical and computational points of view. We derive the conditional moments in closed form and show that the conditional variance is proportional to the square of the conditional mean. This quadratic variance–mean relationship gives a natural motivation for using likelihood and quasi-likelihood estimation methods, including Gaussian, exponential, and geometric quasi-maximum likelihood approaches. We also establish conditions for strict stationarity and ergodicity, and prove the strong consistency of the proposed estimators under suitable regularity assumptions. Monte Carlo simulations are used to study the finite-sample behavior of the estimators, while empirical applications with benchmark comparisons illustrate the practical usefulness of the NXL–ARMA model for positive-valued financial time series.
Citation: Amin Guerouah, Bassant Elkalzah, Halim Zeghdoudi, Majdah Mohammed Badr. A new XLindley ARMA model for positive time series[J]. AIMS Mathematics, 2026, 11(9): 28762-28800. doi: 10.3934/math.20261145
This paper proposes a new observation-driven model for continuous positive-valued time series, called the new XLindley autoregressive moving average (NXL–ARMA) model. The model is built by combining an ARMA-type recursion for the conditional mean with the new XLindley distribution as the conditional law of the observations. This framework is suitable for variables such as trading volume, realized volatility, durations, waiting times, precipitation amounts, energy consumption, and other positive measurements, where the data often show persistence, skewness, and variability that changes with the level of the series.
The proposed model preserves positivity by construction and remains relatively simple to handle from both theoretical and computational points of view. We derive the conditional moments in closed form and show that the conditional variance is proportional to the square of the conditional mean. This quadratic variance–mean relationship gives a natural motivation for using likelihood and quasi-likelihood estimation methods, including Gaussian, exponential, and geometric quasi-maximum likelihood approaches. We also establish conditions for strict stationarity and ergodicity, and prove the strong consistency of the proposed estimators under suitable regularity assumptions. Monte Carlo simulations are used to study the finite-sample behavior of the estimators, while empirical applications with benchmark comparisons illustrate the practical usefulness of the NXL–ARMA model for positive-valued financial time series.
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Amin Guerouah, Bassant Elkalzah, Halim Zeghdoudi, Majdah Mohammed Badr. A new XLindley ARMA model for positive time series[J]. AIMS Mathematics, 2026, 11(9): 28762-28800. doi: 10.3934/math.20261145
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