This paper explores the nonparametric estimation of the volatility component in a heteroscedastic scalar-on-function regression model, where the underlying discrete-time process is ergodic and subject to a missing-at-random mechanism. We first propose a simplified estimator for the regression and volatility operators, constructed solely from the observed data. The asymptotic properties of these estimators, including the almost sure uniform consistency rate and asymptotic distribution, are rigorously analyzed. Subsequently, the simplified estimators are employed to impute the missing data in the original process, enhancing the estimation of the regression and volatility components. The asymptotic behavior of these imputed estimators is also thoroughly investigated. A numerical comparison of the simplified and imputed estimators is presented using simulated data. Finally, the methodology is applied to real-world data to model the volatility of daily natural gas returns, utilizing intraday EU/USD exchange rate return curves sampled at a 1-hour frequency.

翻译：本文探讨了异方差标量对函数回归模型中波动率分量的非参数估计问题，其中基础离散时间过程具有遍历性，并受到随机缺失机制的影响。我们首先提出了一种简化的回归与波动率算子估计量，该估计量仅基于观测数据构建。我们严格分析了这些估计量的渐近性质，包括几乎必然一致收敛速率与渐近分布。随后，利用简化估计量对原始过程中的缺失数据进行插补，从而提升回归与波动率分量的估计精度。本文亦深入研究了插补后估计量的渐近性质。通过模拟数据对简化估计量与插补估计量进行了数值比较。最后，将该方法应用于实际数据，以日内1小时频率采样的欧元/美元汇率收益率曲线为函数型协变量，对日度天然气收益率的波动率进行建模。