The problem is target motion analysis (TMA) in situations where the variance (standard deviation) of additive white Gaussian measurement noise is unknown and time-varying. In particular, the paper examines a somewhat surprising result from the theoretical analysis based on the Cramer-Rao bound, which suggests that the best-achievable (second-order) error in target state estimation is unaffected by the lack of knowledge of the measurement noise variance. In order to examine this result, the paper develops three recursive Bayesian filters for TMA, which jointly estimate the target state and the measurement variance. The basis of all filters is the Cubature Kalman filter for bearings-only tracking, combined with (i) the variational Bayesian approach, (ii) the Rao-Blackwellised particle filter, and (iii) the interactive multiple-model (IMM), to deal with the unknown time-varying measurement variance. The paper presents extensive numerical simulation results and comparisons, which confirm that the lack of knowledge of the measurement noise variance is by no means a handicap for TMA.