Sparse Direction-of-Arrival (DOA) estimators depend on the regularization parameter λ which is often empirically tuned. In this work, conducted under the vectorized covariance matrix model, we are looking for theoretical equivalence between the Maximum Likelihood (ML) and sparse estimators. We show that under mild conditions, λ can be chosen thanks to the distribution of the minimum of the ML criterion. The corresponding λ choice is θ-invariant, only requiring an upper bound on the number of sources. Furthermore, it guarantees the global minimum of the sparse ℓ0-regularized criterion to be the ML solution.