Testing Stability in Functional Event Observations with an Application to IPO Performance

Many sequentially observed functional data objects are available only at the times of certain events. For example, the trajectory of stock prices of companies after their initial public offering (IPO) can be observed when the offering occurs, and the resulting data may be affected by changing circumstances. It is of interest to investigate whether the mean behavior of such functions is stable over time, and if not, to estimate the times at which apparent changes occur. Since the frequency of events may fluctuates over time, we propose a change point analysis that has two steps. In the first step, we segment the series into segments in which the frequency of events is approximately homogeneous using a new binary segmentation procedure for event frequencies. After adjusting the observed curves in each segment based on the frequency of events, we proceed in the second step by developing a method to test for and estimate change points in the mean of the observed functional data objects. We establish the consistency and asymptotic distribution of the change point detector and estimator in both steps, and study their performance using Monte Carlo simulations. An application to IPO performance data illustrates the proposed methods.诸多按序观测得到的函数型数据对象（functional data objects）仅能在特定事件发生的时刻被获取。例如，企业首次公开募股（Initial Public Offering，IPO）后的股价轨迹仅能在募股事件发生时被观测，而所得数据可能随周遭环境的变动受到影响。学界关注的核心问题是：此类函数的平均行为是否随时间保持稳定；若不稳定，则需估计表观变化发生的具体时刻。由于事件发生频率可能随时间波动，本文提出一种分两步进行的变点分析方法。第一步，我们针对事件频率设计全新的二元分割算法，将原序列划分为若干事件频率近似同质的子区间。基于各子区间内的事件频率对观测曲线完成校正后，第二步我们进一步提出一种方法，用于检验并估计函数型数据对象均值序列中的变点。我们论证了两步法中变点检测量与估计量的相合性及渐近分布，并通过蒙特卡洛（Monte Carlo）模拟研究了方法的实际性能表现。将本文所提方法应用于IPO绩效数据，可演示其实际应用效果。