Adjustment by Minimum Discriminant Information

Minimum discriminant information adjustment has primarily been used in the analysis of multinomial data; however, no such restriction is necessary. Let P be a distribution on Rᵃ, and let C be a convex set of distributions on Rᵃ. Let Xᵢ, 1 ≤ i ≤ n, be independent and identically distributed observations with common distribution P. The minimum discriminant information adjustment (MDIA) of P relative to C is the element Q of C that is closest to P in the sense of Kullback-Leibler discriminant information. If P̄ₙ is the empirical distribution of the Xᵢ, 1 ≤ i ≤ n, and Q̄ₙ is the MDIA of P̄ₙ relative to C, then Q̄ₙ is the maximum likelihood estimate in C. Let C consist of distributions A on Rᵃ such that ∫ T dA = t, where T is a measurable transformation from Rᵃ to Rᵇ and t ∈ Rᵇ. It is shown that under mild regularity conditions Q̄ₙ converges weakly to Q, the MDIA of the true P, with probability 1 and that Ēₙ(D) = ∫ DdQ̄ₙ is an asymptotically normal and asymptotically unbiased estimate of E(D) = ∫ D dQ.