Inference for High-dimensional Maximin Effects in Heterogeneous Regression Models Using a Sampling Approach

Heterogeneity is an important feature of modern data sets and a central task is to extract information from large-scale and heterogeneous data. In this paper, we consider multiple high-dimensional linear models and adopt the definition of maximin effect (Meinshausen, B{u}hlmann, AoS, 43(4), 1801--1830) to summarize the information contained in this heterogeneous model. We define the maximin effect for a targeted population whose covariate distribution is possibly different from that of the observed data. We further introduce a ridge-type maximin effect to simultaneously account for reward optimality and statistical stability. To identify the high-dimensional maximin effect, we estimate the regression covariance matrix by a debiased estimator and use it to construct the aggregation weights for the maximin effect. A main challenge for statistical inference is that the estimated weights might have a mixture distribution and the resulted maximin effect estimator is not necessarily asymptotic normal. To address this, we devise a novel sampling approach to construct the confidence interval for any linear contrast of high-dimensional maximin effects. The coverage and precision properties of the proposed confidence interval are studied. The proposed method is demonstrated over simulations and a genetic data set on yeast colony growth under different environments.