The present paper clarifies that the variance of the maximum likelihood estimator (MLE) of a parameter does not reach the Cramer-Rao lower bound (CRLB) when fitting a straight-line to observed two-dimensional data. In addition, the variance of the MLE can be shown to be equal to the CRLB only if observed noise reduces to a one-dimensional Gaussian variable. For most practical applications, it can be assumed that noise is added only to the range direction. In this case, the MLE is clearly an asymptotically effective estimator. However, even if we assume such a noise model, ML line-fitting to the data from many points of view has a high computational cost. The present paper proposes an alternative fitting method in order to provide a cost-effective unbiased estimator. The reliability of this new method is analyzed statistically and by computer simulation.