We investigate the problem of direction of arrival (DOA) estimation using sparse linear arrays, such as co-prime and nested arrays, in the case of missing data resulting from sensor failures. We introduce a signal model where sensor failures occur after taking certain number of snapshots. We formulate a structured covariance estimation problem by exploiting the special geometry of sparse linear arrays, which also provides enhanced degrees of freedom. Numerical examples show that, by utilizing the information in both complete measurements and incomplete measurements, our method achieves better estimation accuracy than the traditional method using only complete measurements.