Asymptotic Error Rate of Linear, Quadratic and Logistic Rules in Multi-Group Discriminant Analysis

A Monte Carlo study was performed to assess the asymptotic error rate of linear, quadratic and logistic rules in 2, 3 and 5-group discriminant analyses. The simulation design that was considered took into account the overlap of the populations (e=0.05, e=0.1, e=0.15), their common distribution (Normal, Chi-square with 12, 8, and 4 df) and their heteroscedasticity degree, , measured by the value of the power function, 1- of the homoscedasticity test related to  (1-=0.05, 1-=0.4, 1-=0.6, 1-=0.8). For each combination of these factors, the asymptotic error rate of the 3 rules was computed using large samples of size 20,000. The efficiency parameter of the rules was their relative error, re with regard to the optimal error rate. The results showed the overall best performance of the quadratic rule for Normal heteroscedastic cases. For Normal homoscedastic populations, the three rules have the same efficiency. The linear rule seemed to be more robust to an increased number of groups than the two other rules. The logistic rule was less affected by the distribution of the populations. Moreover high overlap favored linear and quadratic rules. The logistic rule seemed less influenced by the overlap of the populations. For small size samples, the three rules become less efficient.

Source de publication: International journal of applied mathematics and statistics 2010, 18(10): 69-81

Contacts du ou des auteurs: R. Glèlè Kakaï (glele.romain@gmail.com), D. R. Pelz