We consider testing zero pricing errors in high-dimensional linear factor pricing models. Existing methods are mainly based on either an $L_2$ statistic, which is effective under dense alternatives, or an $L_\infty$ statistic, which is powerful under very sparse alternatives. To bridge these two regimes, we develop a class of $L_q$-based tests for finite $q$, including the practically useful $L_4$ and $L_6$ cases. We show that larger $q$ leads to greater sensitivity to sparse alternatives. We further establish the asymptotic independence between the $L_\infty$ statistic and the $L_q$ statistic for any finite $q$, which motivates a Cauchy combination test that adapts to a broad range of sparsity levels. Simulation studies and a real-data analysis show that the proposed methods are more robust to the unknown sparsity of the alternative and can outperform existing procedures in finite samples.

翻译：本文研究高维线性因子定价模型中的零定价误差检验问题。现有方法主要基于两种统计量：$L_2$统计量（适用于密集替代假设）和$L_\infty$统计量（适用于极端稀疏替代假设）。为弥合这两种情形，我们针对有限$q$值（包括实际应用中有价值的$L_4$和$L_6$情形）发展了一类基于$L_q$的检验方法。研究表明，较大的$q$值可增强对稀疏替代假设的敏感性。我们进一步证明，对于任意有限$q$值，$L_\infty$统计量与$L_q$统计量具有渐近独立性，这促使我们提出一种能适应广泛稀疏水平的柯西组合检验。模拟研究及实证分析表明，所提方法对替代假设未知的稀疏性具有更强的稳健性，且在有限样本中可优于现有方法。