Recently, several spectra have emerged, designed to encapsulate the distributional characteristics of non-Gaussian stationary processes. This article introduces parametric families of generalized spectra based on the characteristic function, alongside inference procedures enabling $\sqrt{n}$-consistent estimation of the unknown parameters in a broad class of parametric models. These spectra capture non-linear dependencies without requiring that the underlying stochastic processes satisfy any moment assumptions. Crucially, this approach facilitates frequency domain analysis for heavy-tailed time series, including possibly non-causal Cauchy autoregressive models and discrete-stable integer-valued autoregressive models. To the best of our knowledge, the latter models have not been studied theoretically in the literature. By estimating parameters across both causal and non-causal parameter spaces, our method automatically identifies the causal or non-causal structure of Cauchy autoregressive models. Furthermore, our estimator does not depend on smoothing parameters since it is based on the integrated periodogram associated with the generalized spectrum. As applications, we develop goodness-of-fit tests, moving average unit-root tests, and tests for non-invertibility. We study the finite-sample performance of the proposed estimators and tests via Monte Carlo simulations, and apply the methodology to estimation and forecasting of a measles count dataset. We evaluate finite-sample performance using Monte Carlo simulations and illustrate the practical value of the procedure with an application to measles case-count estimation and forecasting.

翻译：摘要：近年来，出现了几种旨在刻画非高斯平稳过程分布特征的新谱。本文基于特征函数引入参数化广义谱族，并给出相应的推断程序，使得在广泛参数模型类别中能够对未知参数进行$\sqrt{n}$一致估计。这些谱无需底层随机过程满足任何矩条件即可捕捉非线性依赖关系。关键在于，该方法实现了重尾时间序列（包括可能非因果的柯西自回归模型与离散稳定整数值自回归模型）的频域分析。据我们所知，后两种模型在文献中尚未得到理论研究。通过在因果与非因果参数空间上估计参数，我们的方法能自动识别柯西自回归模型的因果或非因果结构。此外，由于估计量基于广义谱关联的积分周期图，其不依赖于平滑参数。在应用方面，我们开发了拟合优度检验、移动平均单位根检验及不可逆性检验。通过蒙特卡洛模拟研究了所提估计量与检验的有限样本表现，并将该方法应用于麻疹病例计数数据集的估计与预测。我们使用蒙特卡洛模拟评估有限样本表现，并通过麻疹病例估计与预测实例说明该方法的实用价值。