The state of a thrusting/ballistic object moving in a 3-D space can be determined uniquely by a 4-D parameter vector (launch azimuth and elevation angles, drag coefficient, and specific thrust, with the latter two assumed constant) given the location of the launch point (LP). Using the first line-of-sight (LoS) measurement and the known launch altitude, one can reconstruct the LP, which, if taken as the true LP, can be used to estimate the above parameter vector. However, the resulting estimate ignores the fact that the LP based on the first LoS measurement is noisy. To make the estimate account for the LP location uncertainty, we augment the parameter vector with the ground coordinates of the LP, yielding a 6-D vector to be estimated. The estimation errors of the resulting maximum likelihood estimates are shown to meet the Cramér-Rao lower bound, i.e., they are efficient.