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Consider the simple case of a moving histogram (which is a very simple kernel). The idea is to recall thatwhereis the slope close to point .Then we use the empirical cumulative density to approximate the slope, i.e.which can also be writenConsider now the density seen as a random variablewhere the's are i.i.d. where , withThus, observe that , but that's not what we're looking for... From Taylor's expansion,thuswhere the bias comes from the approximation of the density by some string. About the variance,thus, since ,i.e.We can observe thatis decreasing as , while the variance is increasing as . This is the standard bias-variance tradeoff in statistics.