Higher order asymptotic theory for discriminant analysis in exponential families of distributions

This paper deals with the problem of classifying a multivariate observation X into one of two populations Π₁: p(x; w⁽¹⁾) ∈ S and Π₂: p(x; w⁽²⁾) ∈ S, where S is an exponential family of distributions and w⁽¹⁾ and w⁽²⁾ are unknown parameters. Let I; be a class of appropriate estimators (ŵ⁽¹⁾, ŵ⁽²⁾) of (w⁽¹⁾, w⁽²⁾ based on training samples. Then we develop the higher order asymptotic theory for a class of classification statistics D = [Ŵ | Ŵ = log(p(X; ŵ⁽¹⁾)/p(X; ŵ⁽²⁾)), (ŵ⁽¹⁾, ŵ⁽²⁾) ∈ I;]. The associated probabilities of misclassification of both kinds M(ŵ) are evaluated up to second order of the reciprocal of the sample sizes. A classification statistic Ŵ is said to be second order asymptotically best in D if it minimizes M(Ŵ) up to second order. A sufficient condition for Ŵ to be second order asymptotically best in D is given. Our results are very general and give us a unified view in discriminant analysis. As special results, the Anderson W, the Cochran and Bliss classification statistic, and the quadratic classification statistic are shown to be second order asymptotically best in D in each suitable classification problem. Also, discriminant analysis in a curved exponential family is discussed.