Relative efficiency of non-parametric error rate estimators in multi-group linear discriminant analysis

Abstract

A Monte Carlo study was achieved to assess the relative efficiency of ten non-parametric error rate estimators in 2-, 3- and 5-group linear discriminant analysis. The simulation design took into account the number of variables (4, 6, 10, 18) together with the size sample n so that: n/p = 1.5, 2.5 and 5. Three values of the overlap, e of the populations were considered (e = 0.05, e = 0.1, e = 0.15) and their common distribution was Normal, Chi-square with 12, 8, and 4 df; the heteroscedasticity degree, Γ was measured by the value of the power function, 1- β of the homoscedasticity test related to (1-β = 0.05, 1-β = 0.4, 1-β = 0.6, 1-β = 0.8). For each combination of these factors, the actual error rate was empirically computed as well as the ten estimators. The efficiency parameter of the estimators was their relative error, bias and efficiency with regard to the actual error rate, empirically computed. The results showed the overall best performance e632 estimator. On the contrary, e0, epp, eppCV and eA recorded the lowest performance in terms of mean relative error and mean relative bias. The ranks of the estimators were not influenced by the number of groups but for high values of the later, the mean relative bias of the estimators tend to zero.

Keywords: Error rate; Estimation; Efficiency; Multi-group; Linear rule; Simulation