失配自激过程的拟合优度检验
Goodness-of-Fit Test for Mismatched Self-Exciting Processes
摘要：
最近有许多研究工作在开发自激点过程的生成模型，部分原因是它们对现实世界的应用具有广泛的适用性。然而，我们很少能量化生成模型捕捉自然或地面真相的程度，因为它通常是未知的。挑战通常在于生成模型通常至多提供对地面实况的良好近似（例如，通过神经网络丰富的代表性力量），但它们不能准确地成为地面真理。因此，我们不能使用经典的拟合优度（GOF）测试框架来评估其性能。在本文中，我们开发了一个GOF测试生成模型的自激过程，使一个新的连接到这个问题与经典的统计理论的准最大似然估计（QMLE）。我们提出了一个非参数的自归一化统计的GOF测试：广义分数（GS）统计，并明确捕获模型误设定时，建立GS统计量的渐近分布。数值模拟和实际数据实验验证了我们的理论和GS测试的良好性能。
英文摘要：
Recently there have been many research efforts in developing generative models for self-exciting point processes, partly due to their broad applicability for real-world applications. However, rarely can we quantify how well the generative model captures the nature or ground-truth since it is usually unknown. The challenge typically lies in the fact that the generative models typically provide, at most, good approximations to the ground-truth (e.g., through the rich representative power of neural networks), but they cannot be precisely the ground-truth. We thus cannot use the classic goodness-of-fit (GOF) test framework to evaluate their performance. In this paper, we develop a GOF test for generative models of self-exciting processes by making a new connection to this problem with the classical statistical theory of Quasi-maximum-likelihood estimator (QMLE). We present a non-parametric self-normalizing statistic for the GOF test: the Generalized Score (GS) statistics, and explicitly capture the model misspecification when establishing the asymptotic distribution of the GS statistic. Numerical simulation and real-data experiments validate our theory and demonstrate the proposed GS test's good performance.
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