Testing High-Dimensional Means under Sparse Missing Pattern

This paper introduces a new test for the population mean vector in high-dimensional data with missing observations. Our method overcomes key limitations of existing approaches, including the need for finite fourth-order moments and restrictive assumptions on the probabilities of missing data indicating dense missing pattern. The proposed test statistic is standardized using the conditional variance given the observed pattern of missing values, eliminating the requirement that the probabilities of missing observations be bounded away from one and accommodating sparse missing structure.In contrast to existing methods, our approach requires only the existence of finite (2+ζ)-th moments for some ζ>0, making it applicable to heavy-tailed distributions where fourth order moment is not finite. The procedure also accommodates weak dependence across variables via geometric α-mixing and does not rely on the assumption of identically distributed observations. This significantly improves the versatility of the proposed test. Theoretical guarantees are provided through the asymptotic normality of the test statistic, and we establish consistency and correct asymptotic size under appropriate conditions.