Penalized spline smoothing of time series and its asymptotic properties are studied. A data-driven algorithm for selecting the smoothing parameter is developed. The proposal is applied to define a semiparametric extension of the well-known Spline-GARCH, called a P-Spline-GARCH, based on the log-data transformation of the squared returns. It is shown that now the errors process is exponentially strong mixing with finite moments of all orders. Asymptotic normality of the P-spline smoother in this context is proved. Practical relevance of the proposal is illustrated by data examples and simulation. The proposal is further applied to value at risk and expected shortfall.

翻译：正在研究时间序列及其无症状特性的惩罚性滑动样条及其无症状特性。开发了用于选择平滑参数的数据驱动算法。该提议用于根据平方返回的日志数据转换确定著名的Spline-GARCH(称为P-Spline-GARCH)的半参数扩展。显示现在错误过程与所有定单的有限时刻混合成指数性强。在此情况下,P-spline光滑器的无症状常化得到了证明。该提议的实际相关性通过数据实例和模拟加以说明。该提案还被进一步应用于风险价值和预期短缺。