We consider a new framework where a continuous, though bounded, random variable has unobserved bounds that vary over time. In the context of univariate time series, we look at the bounds as parameters of the distribution of the bounded random variable. We introduce an extended log-likelihood estimation and design algorithms to track the bound through online maximum likelihood estimation. Since the resulting optimization problem is not convex, we make use of recent theoretical results on Normalized Gradient Descent (NGD) for quasiconvex optimization, to eventually derive an Online Normalized Gradient Descent algorithm. We illustrate and discuss the workings of our approach based on both simulation studies and a real-world wind power forecasting problem.

翻译：我们考虑一种新框架，其中连续但有界的随机变量存在随时间变化的未观测边界。在单变量时间序列背景下，我们将这些边界视为有界随机变量分布的参数。我们提出扩展对数似然估计方法，并设计通过在线极大似然估计追踪边界的算法。由于最终优化问题非凸，我们利用近期关于拟凸优化的归一化梯度下降（NGD）理论成果，推导出在线归一化梯度下降算法。我们通过仿真研究及实际风电功率预测问题，说明并讨论了所提方法的运行机制。