Higher-order spectra (or polyspectra), defined as the Fourier Transform of a stationary process' autocumulants, are useful in the analysis of nonlinear and non Gaussian processes. Polyspectral means are weighted averages over Fourier frequencies of the polyspectra, and estimators can be constructed from analogous weighted averages of the higher-order periodogram (a statistic computed from the data sample's discrete Fourier Transform). We derive the asymptotic distribution of a class of polyspectral mean estimators, obtaining an exact expression for the limit distribution that depends on both the given weighting function as well as on higher-order spectra. Secondly, we use bispectral means to define a new test of the linear process hypothesis. Simulations document the finite sample properties of the asymptotic results. Two applications illustrate our results' utility: we test the linear process hypothesis for a Sunspot time series, and for the Gross Domestic Product we conduct a clustering exercise based on bispectral means with different weight functions.

翻译：高阶谱（或多谱）定义为平稳过程自累积量的傅里叶变换，在分析和处理非线性与非高斯过程中具有重要价值。多谱均值是多谱在傅里叶频率上的加权平均，其估计量可通过高阶周期图（一种基于数据样本离散傅里叶变换计算的统计量）的类似加权平均构造得到。本文推导了一类多谱均值估计量的渐近分布，获得了极限分布的精确表达式，该表达式既依赖于给定的加权函数，也依赖于高阶谱。其次，我们利用双谱均值定义了一种新的线性过程假设检验方法。仿真实验验证了渐近结果的有限样本性质。通过两个应用实例展示了本研究成果的实用性：对太阳黑子时间序列进行线性过程假设检验，并针对国内生产总值数据，基于采用不同加权函数的双谱均值进行了聚类分析。