A Statistical Taylor Theorem and Extrapolation of Truncated Densities

TL;DR

This paper introduces a statistical version of Taylor's theorem for non-parametric density estimation from truncated samples, enabling efficient approximation of distributions over entire intervals using limited truncated data, applicable in multiple dimensions.

Contribution

It presents the first non-parametric density estimation method from truncated samples under the hard truncation model, extending to multiple dimensions and improving upon previous single-dimensional approaches.

Findings

First method for density estimation from truncated samples in multiple dimensions

Efficient approximation of distributions over entire support from partial data

Improves with the smoothness of the underlying density function

Abstract

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 \cite{woodroofe1985estimating, stute1993almost}. 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 \emph{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.