Mathematical Programming Approach to a Minimax Theorem of Statistical Discrimination Applicable to Pattern Recognition

Abstract

Let k _j’s be two famerlies of all possible p-variate distribution functions with specified mean vectors μ _i and non-degenerate variance-covariance matrices ∑_i, and π_i be prior probability or weight assigned to k _i for i=1, 2 (π₁ + π₂ = 1). We are supposed to discriminate whether an observation x is from a (true) distribution F ₁ ∈k ₁ or F ₂ ∈k ₂ A randomized decision rule is represented by a pair of functions Φ₁ (x) and ${\phi _2}(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} ) = 1 - {\phi _1}(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )\;(0\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } {\phi _1}(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } 1)$, based on which one decides, with probability ø_i x, that an observed value x. is a sample from some F _i in k _i (i=1,2). If the pair F = (F ₁, F ₂) is known, the error probability or classification error for the decision rule ø = (ø₁, ø₂) is clearly given by

$$e(\phi ,F) = {\pi _1}\int_{{R^p}} {\phi (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )d{F_1}(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} ) + {\pi _2}\int_{{R^p}} {{\phi _1}(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )d{F_2}(x)} } $$

((1.1))

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