Asymptotic distributions of linear combinations of logs of multinomial parameter estimates

1 Formulas Suppose that Y has a multinomial distribution with parameters (p1, . . . , pm). (That is to say, suppose that Pr(Y = i) = pi for an integer i such that 1 ≤ i ≤ m.) Given a sample Y1, . . . , Yn of multinomial random variables with vector parameter (p1, . . . , pm), the maximum–likelihood (and method of moments) estimator of pi is p̂i = #{j : Yj = i}/n = ni/n (1) where ni = #{j : Yj = i} is the number of sample indices j such that Yj = i. The dispersion matrix of the p̂i is defined by Cov[p̂i, pj ] = { n−1pi(1− pi), if i = j, −n−1pipj , if i 6= j. (2) We aim to estimate the ln pi with a view to estimating linear combinations of these logs. For each i, we have ∂ ∂pi ln pi = 1 pi , (3)