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If the Bayes decision boundary is linear and the underlying distributional assumptions are Normal, we expect LDA to perform better than QDA on the test set. But if the Bayes decision boundary is linear and the underlying distributional assumptions are not Normal, can QDA ever perform better than LDA (I know QDA is also not appropriate if the normality assumption is not met, just trying to compare this with LDA in such a setting)?

Similarly, when the Bayes decision boundary is nonlinear (and not necessarily quadratic), can LDA ever perform better than QDA?

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I'm pretty sure the answer is yes in both cases. You can create an example with, say, uniform distributions with linear boundaries, in which the shapes and extensions of the different classes are vastly different, which can be much better modelled by flexible covariance matrices as in QDA. On the other hand, obviously one can create situations with nonlinear boundaries that are pretty close to linear where it matters, and a low enough number of points in one of the classes so that estimating a full covariance matrix for it as required for QDA is bad.

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