The quantile-crossing spectrum is the spectrum of quantile-crossing processes created from a time series by the indicator function that shows whether or not the time series lies above or below a given quantile at a given time. This bivariate function of frequency and quantile level provides a richer view of serial dependence than that offered by the ordinary spectrum. We propose a new method for estimating the quantile-crossing spectrum as a bivariate function of frequency and quantile level. The proposed method, called spline autoregression (SAR), jointly fits an AR model to the quantile-crossing series across multiple quantiles; the AR coefficients are represented as spline functions of the quantile level and penalized for their roughness. Numerical experiments show that when the underlying spectrum is smooth in quantile level the proposed method is able to produce more accurate estimates in comparison with the alternative that ignores the smoothness.

翻译：分位数穿越谱是由时间序列通过指示函数生成的穿越过程谱，该指示函数显示时间序列在给定时刻是否高于或低于特定分位数。这一关于频率与分位数水平的二元函数比普通谱提供了更丰富的序列依赖性视角。我们提出了一种估计分位数穿越谱的新方法，将其视为频率与分位数水平的二元函数。该方法称为样条自回归（SAR），通过联合拟合多个分位数水平下的穿越序列自回归模型实现；其自回归系数表示为分位数水平的样条函数，并对其粗糙度进行惩罚化处理。数值实验表明，当潜在谱在分位数维度上具有平滑性时，相比忽略平滑性的替代方法，本方法能获得更精确的估计结果。