Asymptotic cut-off point in linear discriminant rule to adjust the misclassification probability for large dimensions

作者：Yamada Takayuki^*; Himeno Tetsuto; Sakurai Tetsuro

来源：Hiroshima Mathematical Journal, 2017, 47(3): 319-348.

DOI：10.32917/hmj/1509674450

• Linear discriminant rule
• Cut-off point
• (n,p) asymptotic

This paper is concerned with the problem of classifying an observation vector into one of two populations Pi(1) : N-p (mu(1) , Sigma ) and Pi(2) : N-p (mu(2) , Sigma) . Anderson (1973, Ann. Statist.) provided an asymptotic expansion of the distribution for a Studentized linear discriminant function, and proposed a cut-off point in the linear discriminant rule to control one of the two misclassification probabilities. However, as dimension p becomes larger, the precision worsens, which is checked by simulation. Therefore, in this paper we derive an asymptotic expansion of the distribution of a linear discriminant function up to the order p(-1) as N-1, N-2, and p tend to infinity together under the condition that p / (N-1 + N-2 - (2)) converges to a constant in (0,1), and N-1 / N-2 converges to a constant in (0,infinity), where N-i means the size of sample drown from Pi(i) (i = 1, 2). Using the expansion, we provide a cut-off point. A small-scale simulation revealed that our proposed cut-off point has good accuracy.

• 出版日期2017-11