Errors in discrimination with monotone missing data from multivariate normal populations

The behavior of the linear discriminant function is studied, when it is used for the classification of an observation X   into one of two independent multivariate normal populations Np(μ(ν),Σ)Npμ(ν),Σ, with distinct mean vectors μ(ν)μ(ν), ν=1,2ν=1,2 and a common covariance matrix ΣΣ. The effect of the estimation of the parameters, on the basis of random 2-step monotone training samples, is studied, in three stages of increasing complexity. Asymptotic expressions for the distribution functions of the probabilities of misclassification are derived. Moreover, numerical and simulation results are presented in order to study the effect to the distribution of the probabilities of misclassification using different estimation procedures and missingness rate in the data. Two extensions, related to the case of kk-step monotone missing training samples and the case of completely unknown heteroscedastic normal populations are also discussed.