We study hypothesis testing for penalized estimators in settings where the full marginal distribution of a multivariate response is difficult to specify, such as longitudinal data with correlated measurements or high-dimensional heteroscedastic regression. Assuming that the conditional mean model is correctly specified, we establish that the penalized estimating equations admit a $\sqrt{n}$-consistent solution, even when the working covariance structure is misspecified. Our inferential target is a low-dimensional subvector of parameters associated with the mean model. We show that the resulting test statistic converges to a $χ^2$ distribution, and that its asymptotic power depends on the nuisance covariance function. To mitigate this dependence, we propose estimating the covariance function via cross-fitting, which provides a calibrated and robust procedure for inference.

翻译：我们研究了在多元响应变量的全边际分布难以指定（例如具有相关测量的纵向数据或高维异方差回归）的情况下，罚估计量的假设检验问题。假设条件均值模型被正确指定，我们证明当工作协方差结构被错误指定时，罚估计方程仍存在一个$\sqrt{n}$一致的解。我们的推断目标是均值模型中参数的低维子向量。我们证明所得到的检验统计量收敛于$\chi^2$分布，且其渐近功效依赖于冗余协方差函数。为减轻这种依赖性，我们提出通过交叉拟合来估计协方差函数，从而为推断提供一种校准后的稳健程序。