On inferring the probability of misclassification by the linear discriminant function

Summary Let the two alternative populationsP ₁ andP ₂ from which the individual with measurements χ may have come beN(μ(1), Σ) andN(μ(2), Σ). Then the classification rule with minimum risk is to assign the individual toP ₁ orP ₂ according as (μ(2)-μ(1))′Σ^-1 x≶(1/2)(μ(2)-μ(1))′Σ^-1(μ(1)+μ(2))+c wherec is a constant depending on the prior probabilities ofP ₁ andP ₂ and the costs of the two kinds of misclassification. The probability of misclassifying an individual fromP ₂ by this rule is π₂₁=Φ(-δ/2+cδ^-1), where Φ(.) is the distribution function of anN(0, 1) and . (Since we are free to choose which population we shall callP ₂, it is not necessary to consider separately the probability of misclassifying an individual fromP ₁.) LetP ₂₁ denote the probability of misclassification of an individual fromP ₂ by the rule derived from the one mentioned by fixing μ(1), μ(2) and Σ at estimates andV and letP ₂₁ ^* be the probability of misclassification of an individual fromP ₂ when the classification rule is the one with minimum risk among those based on . The fiducial distributions of π₂₁,P ₂₁ andP ₂₁ ^* are determined. Point estimates and confidence intervals for π₂₁,P ₂₁ andP ₂₁ ^* are derived. Only easily available tables are needed to make fiducial inferences. An incidental result of some interest elsewhere as well is the distribution of a linear combination of a chi and an independent normal variable.