A Remark on the Roots of the Maximum Likelihood Equation

The statistical literature combines two types of investigations concerning the consistency of maximum likelihood (M.L.) estimates. A few of these, such as the most excellent paper of A. Wald 1, do prove directly the consistency of M.L. estimates. However, most investigators seem to have concentrated their efforts on proving the existence and consistency of suitably selected roots of the successive likelihood equations. Some authors, see 2, for example, add the supplementary remark that such consistent roots will eventually be unique in suitably small neighborhoods of the true value and will achieve a local maximum. It is the purpose of the present note to point out by means of examples that this second mode of attack is not adequate. In the examples given below, the "usual regularity conditions" of Cramer 3 or Wald 4 are satisfied, but the M.L. estimates are not consistent. It should also be pointed out that the direct proofs of existence of roots, simple in the case of a unidimensional parameter, become unwieldy in more than one dimension. On the other hand, if one has proved the consistency of the M.L. estimates, the existence of roots follows trivially from the fact that when a differentiable function reaches its maximum in an open set, the derivatives vanish at that point.