Misclassification Probabilities for Quadratic Discrimination

Two p-variate normal populations Nₚ (μ ₁ ,Σ ₁ ), Nₚ (μ ₂ ,Σ ₂ ) with parameters known and μ ₁ ≠ μ ₂, Σ ₁ ≠ Σ ₂ are considered. A method is given to calculate, to a desired accuracy, the probabilities of misclassification (PM’s) for quadratic discrimination by using Davies [1980]. Monte Carlo studies, assuming equal priors and equal costs, are conducted. These studies demonstrate the dependence of the overall PM on a measure m of separation between the populations, used by Marks [1970], Marks and Dunn [1974], and a measure ρ of the divergence between Σ ₁ and Σ ₂ and discussed by Rao [1977]. Tables for dimensions $p = 10$, 25, and 40 displaying the dependence are constructed from these studies. An iterative algorithm for the calculation of m is given. Application of the above results in the large sample situation is discussed.