When choosing a classiÿcation rule, it is important to take into account the amount of sample data available. This paper examines the performances of classiÿers of di ering complexities in relation to the complexity of feature-label distributions in the case of small samples. We deÿne the distributional complexity of a feature-label distribution to be the minimal number of hyperplanes necessary to achieve the Bayes classiÿer if the Bayes classiÿer is achievable by a ÿnite number of hyperplanes, and inÿnity otherwise. Our approach is to choose a model and compare classiÿer e ciencies for various sample sizes and distributional complexities. Simulation results are obtained by generating data based on the model and the distributional complexities. A linear support vector machine (SVM) is considered, along with several nonlinear classiÿers. For the most part, we see that there is little improvement when one uses a complex classiÿer instead of a linear SVM. For higher levels of distributional complexity, the linear classiÿer degrades, but so do the more complex classiÿers owing to insu cient training data. Hence, if one were to obtain a good result with a more complex classiÿer, it is most likely that the distributional complexity is low and there is no gain over using a linear classiÿer. Hence, under the model, it is generally impossible to claim that use of the nonlinear classiÿer is beneÿcial. In essence, the sample sizes are too small to take advantage of the added complexity. An exception to this observation is the behavior of the three-nearest-neighbor (3NN) classiÿer in the case of two variables (but not three) when there is very little overlap between the label distributions and the sample size is not too small. With a sample size of 60, the 3NN classiÿer performs close to the Bayes classiÿer, even for high levels of distributional complexity. Consequently, if one uses the 3NN classiÿer with two variables and obtains a low error, then the distributional complexity might be large and, if such is the case, there is a signiÿcant gain over using a linear classiÿer.