This study considers a goodness of fit test based on the quadratic distance (QD) in composite hypotheses. Lindsay et al. [Quadratic distances on probabilities: a unified approach. Ann Statist. 2008;36:983–1006] established a general theory of QD measures for the goodnees of fit test. Using the spectral decomposition of centred kernels, they verified that the QD test asymptotically follows a sum of weighed chi-square distributions. In this study special attention is paid to a smoothing kernel-based QD test and its bootstrap version. Their performances are compared via Monte Carlo simulations with those of the Bickel-Rosenblatt test and those of the Fisher's dispersion test for the normality and the testing for the Poisson distribution in IID samples and AR(1) models. The comparison results demonstrate the validity of our method.

本研究考虑了基于复合假设中的二次距离 (QD) 的拟合优度检验。林赛等人。[概率的二次距离：一种统一的方法。安统计员。2008;36:983-1006] 为拟合检验的优点建立了 QD 度量的一般理论。使用中心核的谱分解，他们验证了 QD 检验渐近地遵循加权卡方分布的总和。在这项研究中，特别关注基于平滑内核的 QD 测试及其引导程序版本。它们的性能通过 Monte Carlo 模拟与 Bickel-Rosenblatt 检验和 Fisher 色散正态性检验以及 IID 样本和 AR(1) 模型中的泊松分布检验进行比较。比较结果证明了我们方法的有效性。