A Statistical Taylor Theorem and Extrapolation of Truncated Densities

We show a statistical version of Taylor’s theorem and apply this result to non-parametric density estimation from truncated samples, which is a classical challenge in Statistics [Woodroofe 1985, Stute 1993]. The single-dimensional version of our theorem has the following implication: "For any distribution P on [0, 1] with a smooth log-density function, given samples from the conditional distribution of P on [a, a + \varepsilon] \subset [0, 1], we can efficiently identify an approximation to P over the whole interval [0, 1], with quality of approximation that improves with the smoothness of P". To the best of knowledge, our result is the first in the area of non-parametric density estimation from truncated samples, which works under the hard truncation model, where the samples outside some survival set S are never observed, and applies to multiple dimensions. In contrast, previous works assume single dimensional data where each sample has a different survival set $S$ so that samples from the whole support will ultimately be collected.