Finite-Sample Distortion in Kernel Specification Tests: A Perturbation Analysis of Empirical Directional Components

This paper provides a new theoretical lens for understanding the finite-sample performance of kernel-based specification tests, such as the Kernel Conditional Moment (KCM) test. Rather than introducing a fundamentally new test, we isolate and rigorously analyze the finite-sample distortion arising from the discrepancy between the empirical and population eigenspaces of the kernel operator. Using perturbation theory for compact operators, we demonstrate that the estimation error in directional components is governed by local eigengaps: components associated with small eigenvalues are highly unstable and contribute primarily noise rather than signal under fixed alternatives. Although this error vanishes asymptotically under the null, it can substantially degrade power in finite samples. This insight explains why the effective power of omnibus kernel tests is often concentrated in a low-dimensional subspace. We illustrate how truncating unstable high-frequency components--a natural consequence of our analysis--can improve finite-sample performance, but emphasize that the core contribution is the diagnostic understanding of \textit{why} and \textit{when} such instability occurs. The analysis is largely non-asymptotic and applies broadly to reproducing kernel Hilbert space-based inference.