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 Pₙ is the empirical distribution of the Xᵢ, 1 i n, and Qₙ is the MDIA of Pₙ relative to C, then Qₙ 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 Qₙ converges weakly to Q, the MDIA of the true P, with probability 1 and that Eₙ (D) = DdQₙ is an asymptotically normal and asymptotically unbiased estimate of E (D) = D dQ.