Analytic expressions for the first order bias and second order covariance of a maximum-likelihood estimate (MLE) are applied to the problem of localizing an acoustic source in range and depth in a shallow water waveguide with a vertical hydrophone array. These expressions are then used to determine necessary conditions on sample size, or equivalently signal-to-noise ratio (SNR), for the localization MLE to become asymptotically unbiased and attain minimum variance as expressed by the Cramer-Rao lower bound (CRLB). These analytic expressions can be applied in a similar fashion to any ocean-acoustic inverse problem involving random data. Both deterministic and completely randomized signals embedded in independent and additive waveguide noise are investigated. As the energy ratio of received signal to additive noise (SANR) descends to the lower operational range of a typical passive localization system, source range and depth estimates exhibit significant biases and have variances that can exceed the CRLB by orders of magnitude. The spatial structure of the bias suggests that acoustic range and depth estimates tend to converge around particular range and depth cells for moderate SANR values.