This paper aims to investigate nonparametric estimation of the volatility component in a heteroscedastic scalar-on-function regression model when the underlying discrete-time process is ergodic and affected by a missing at random mechanism. First, we introduce a simplified estimator of the regression and volatility operators based on observed data only. We study their asymptotic properties, such as almost sure uniform consistency rate and asymptotic distribution. Then, the simplified estimators are used to impute the missing data in the original process in order to improve the estimation of the regression and volatility components. The asymptotic properties of the imputed estimators are also investigated. A numerical comparison between the estimators is discussed through simulated data. Finally, a real-data analysis is conducted to model the volatility of daily Brent crude oil returns using intraday, 1-minute frequency, natural gas returns.

翻译：本文旨在研究异方差标量-函数回归模型中波动率成分的非参数估计问题，其中底层离散时间过程是遍历的且受随机缺失机制影响。首先，我们基于仅观测到的数据引入回归算子与波动率算子的简化估计量，并研究其渐近性质，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，以改进回归成分与波动率成分的估计，并进一步探究插补估计量的渐近性质。通过模拟数据对各类估计量进行数值比较。最后，基于实际数据应用，利用日内1分钟频率的天然气收益率数据对布伦特原油日收益率波动率进行建模分析。