Abstract

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 N_p(μ^(ν),Σ), with distinct mean vectors μ^(ν), ν=1,2 and a common covariance matrix Σ. 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 k-step monotone missing training samples and the case of completely unknown heteroscedastic normal populations are also discussed.

Keywords: Monotone missing data; Discriminant analysis; Errors of misclassification; Distribution of errors

Article Outline

1. Introduction

2. Preliminaries

3. Main results

3.1. The parameter μ⁽²⁾ unknown

3.1.1. Distribution of

3.1.2. Distribution of

3.2. The parameters μ⁽¹⁾ and μ⁽²⁾ unknown

3.3. All the parameters unknown

4. Numerical results

4.1. Effectiveness of case-wise deletion method

4.1.1. The parameter μ⁽²⁾ unknown

4.1.2. The parameters μ⁽¹⁾ and μ⁽²⁾ unknown

4.1.3. All the parameters unknown

4.2. Effectiveness of the missingness rate

4.2.1. The parameter μ⁽²⁾ unknown

4.2.2. The parameters μ⁽¹⁾ and μ⁽²⁾ unknown

4.2.3. All the parameters unknown

4.3. Effectiveness of simulation technique

4.3.1. The parameter μ⁽²⁾ unknown

4.3.2. The parameters μ⁽¹⁾ and μ⁽²⁾ unknown

4.3.3. All the parameters unknown

4.4. Summary of numerical results

5. Concluding remarks and extensions

Acknowledgements

References