Heuristics on bias and variance for kernel density estimators

Consider the simple case of a moving histogram (which is a very simple kernel). The idea is to recall that where is the slope close to point . Then we use the empirical cumulative density to approximate the slope, i.e. which can also be writen Consider now the density seen as a random variable where the's are i.i.d. where , with Thus, observe that , but that's not what we're looking for... From Taylor's expansion, thus where the bias comes from the approximation of the density by som..