We propose a kernel-based partial permutation test for checking the equality of functional relationship between response and covariates among different groups. The main idea, which is intuitive and easy to implement, is to keep the projections of the response vector $\boldsymbol{Y}$ on leading principle components of a kernel matrix fixed and permute $\boldsymbol{Y}$'s projections on the remaining principle components. The proposed test allows for different choices of kernels, corresponding to different classes of functions under the null hypothesis. First, using linear or polynomial kernels, our partial permutation tests are exactly valid in finite samples for linear or polynomial regression models with Gaussian noise; similar results straightforwardly extend to kernels with finite feature spaces. Second, by allowing the kernel feature space to diverge with the sample size, the test can be large-sample valid for a wider class of functions. Third, for general kernels with possibly infinite-dimensional feature space, the partial permutation test is exactly valid when the covariates are exactly balanced across all groups, or asymptotically valid when the underlying function follows certain regularized Gaussian processes. We further suggest test statistics using likelihood ratio between two (nested) GPR models, and propose computationally efficient algorithms utilizing the EM algorithm and Newton's method, where the latter also involves Fisher scoring and quadratic programming and is particularly useful when EM suffers from slow convergence. Extensions to correlated and non-Gaussian noises have also been investigated theoretically or numerically. Furthermore, the test can be extended to use multiple kernels together and can thus enjoy properties from each kernel. Both simulation study and application illustrate the properties of the proposed test.
翻译:我们提出一个基于内核的部分偏移测试, 用于检查不同组别之间响应和共变之间功能关系的平等性。 主要的想法是( 直观且易于执行), 将响应矢量 $\boldsymbol{Y}$ 的预测保留在内核矩阵中的主要主构件上, 固定和 permute $\boldsymol{Y}} 剩余主构件的预测中。 提议的测试允许对内核的不同选择, 与无效假设下不同类别的函数相对应。 首先, 使用线性或多式内核内核, 我们部分的逻辑调整测试测试完全有效, 以直线性或多式回归模型为样本; 类似的结果直接延伸至内核内核矩阵的内核部分。 第二, 允许内核特性空间与内核大小不同, 拟议的有用性功能类别。 第三, 普通内核内核, 可能具有无限的内核空间, 部分的内核内核内核内核, 也包含部分的内核内核特性的内核反应, 测试测试, 当我们使用正常的内核的内核的内核, 的内核, 的内核, 的内核, 的内核, 的内核, 也显示的内核, 的内核的内核的内核的内核的内核, 的内核的内核的内核的内核的内核的内核, 的内核, 的内核, 的内核, 的内核的内核, 的内核的内核的内核, 的内核的内核的内核的内核的内核的内核的内核的内核, 的内核的内核, 的内核, 的内核, 的内核, 的内核, 的内核, 的内核, 的内核, 的内核, 的内核, 的内核, 的内核, 的内核, 的内核, 的内核, 的内核, 的内核, 的内核, 的内核, 的内核, 的内核, 的内核, 的内核,