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 [Woo85, Stu93]. 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 + ε] ⊂ [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. 1 1Accepted for presentation at the Conference on Learning Theory (COLT) 2021. 1 ar X iv :2 10 6. 15 90 8v 1 [ m at h. ST ] 3 0 Ju n 20 21