Z Z PtoTTFifV lzeln

with a and ft equal to 8 or m = (m(l), . . . , - l)).29 Since Amm is negative definite, and Aem is in general non-null, the matrix A«« is larger than V, in the sense that A^1 — V is non-null and positive semidefinite.

A third approach to the estimation of 0* not requiring knowledge of p begins with the identity

je C

where, it will be recalled, the conditional distribution q is defined by /(/, z) = P(i\z, 6*)p(z) = Q(i)q(z\i).

Observe first that if both of the distributions Q and q were a priori known, the value 0* could be determined directly as the unique solution to the set of equations

We note that the uniqueness of 0* as the solution to these equations is guaranteed by assumption 1.2 and that the solution does not require the sample data (/, z)„, it = 1, . . . , N.

In general the distribution q, like p, will not be a priori known.30

29. The constraint IieC m(i)Q(i) = 1 is used to eliminate m(M) prior to differentiation. The derivative is evaluated at 0* and m(i) — H(i)/Q(i).

30. In a recent paper Carroll and Relies (1976) assumed that the distributions q(z |j), je C, each fall within the multivariate normal family (see also Warner 1963). Given what is assumed to be a random sample of observations, they estimate the parameters for each such distribution and subsequently estimate the choice probabilities by

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