A Classification Statistic from Anderson's Criterion in Discrimination and its Asymptotic Distribution in the Heteroscedastic Normal Model

SYNOPTIC ABSTRACT

T.W. Anderson (1958) introduced the likelihood ratio criterion which can be used in discrimination into one of two multivariate normal populations when the parameters of populations are estimated. This paper introduces a classification statistic based on Anderson's likelihood ratio criterion. A deductive classification statistic is used in discrimination into one of several normal populations with unequal covariance matrices. We discuss its asymptotic distribution and its asymptotic behaviour for large samples. The asymptotic distribution is shown by the moment generating function, and some moments are given. An asymptotical mean bias and an asymptotic correct classification probability give the asymptotic behaviour of the classification statistic.

Key Words and Phrases:

• Quadratic classification statistic
• heteroscedastic normal distribution
• likelihood ratio criterion
• asymptotic distribution
• moment generating function
• Kullback-Leibler information