Pairwise likelihood offers a useful approximation to the full likelihood function for covariance estimation in high-dimensional context. It simplifies high-dimensional dependencies by combining marginal bivariate likelihood objects, thereby making estimation more manageable. In certain models, including the Gaussian model, both pairwise and full likelihoods are known to be maximized by the same parameter values, thus retaining optimal statistical efficiency, when the number of variables is fixed. Leveraging this insight, we introduce the estimation of sparse high-dimensional covariance matrices by maximizing a truncated version of the pairwise likelihood function, which focuses on pairwise terms corresponding to nonzero covariance elements. To achieve a meaningful truncation, we propose to minimize the discrepancy between pairwise and full likelihood scores plus an L1-penalty discouraging the inclusion of uninformative terms. Differently from other regularization approaches, our method selects whole pairwise likelihood objects rather than individual covariance parameters, thus retaining the inherent unbiasedness of the pairwise likelihood estimating equations. This selection procedure is shown to have the selection consistency property as the covariance dimension increases exponentially fast. As a result, the implied pairwise likelihood estimator is consistent and converges to the oracle maximum likelihood estimator that assumes knowledge of nonzero covariance entries.

翻译：在高维背景下，成对似然为协方差估计的全似然函数提供了一种有效的近似方法。该方法通过组合边缘二元似然对象来简化高维依赖关系，从而使估计更易处理。在某些模型（包括高斯模型）中，当变量数量固定时，已知成对似然与全似然会在相同参数值处达到最大化，从而保持了最优统计效率。基于这一认识，我们提出通过最大化成对似然函数的截断版本来估计稀疏高维协方差矩阵，该截断版本专注于非零协方差元素对应的成对项。为实现有效的截断，我们建议最小化成对似然与全似然得分之间的差异，并添加L1惩罚项以排除非信息性项。与其他正则化方法不同，我们的方法选择完整的成对似然对象而非单个协方差参数，从而保留了成对似然估计方程固有的无偏性。该选择过程被证明具有选择一致性性质，即使协方差维度呈指数级增长时亦然。因此，导出的成对似然估计量具有一致性，并收敛于假定已知非零协方差条目的理想最大似然估计量。