We examine the behaviour of a certain kind of transformation based kernel density estimator (TKDE). The transformation is a Taylor series approximation to a smoothed empirical cumulative distribution function computed from a pilot estimate. In this way a whole class of estimators is introduced, with a different number of terms m in the Taylor series. The case m = 1 corresponds to a standard varying bandwidth estimator, while the case m = ∞ corresponds to a TKDE with the smoothed empirical c.d.f. as transformation. We give an asymptotic expansion for any number of m. When m = 1,2, the rate of convergence is the same as for an ordinary kernel density estimator using a second order kernel, and when m ≥3 the rate of a fourth order kernel is obtained.