Maximum likelihood estimation for the multinomial distribution

is consistent while Huzurbazar (1949) showed, under Cramer's regularity conditions (Cramer, 1946), that with probability tending to unity a consistent root of the m.l. equation provides a local maximum of the likelihood. It was not known whether at such a root the likelihood attains an absolute maximum. It is shown in this paper that in the case of the multinomial distribution, a m.l. estimate is, with probability 1, a root of the likelihood equation, and provides a maximum of the likelihood when the parameter is restricted to the roots. We thus have a complete theory of the method of m.l. for the multinomial distribution. No assumption is made about the existence of the m.l. estimate, but the existence and uniqueness are deduced as a consequence of the regularity conditions. To make the arguments free from unnecessary complications only the one parameter case is considered. The additional complication in the proof for the multi parameter case is in establishing the existence of the roots of the likelihood equation. Once this is done, the rest of the argument is the same. In the development of this paper, the following scheme is adopted. First, the consistency of the m.l. estimates of the hypothetical frequencies of the multinomial distribution is established without any regularity conditions whatsoever. Second, the consistency of the m.l. estimate of a parameter occurring in the specification of the hypo thetical frequencies is established under a very natural restriction on the parameter,