A Statistical Taylor Theorem and Extrapolation of Truncated Densities

Constantinos Daskalakis, Vasilis Kontonis, Christos Tzamos, Emmanouil Zampetakis
Proceedings of Thirty Fourth Conference on Learning Theory, PMLR 134:1395-1398, 2021.

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 [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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v134-daskalakis21a, title = {A Statistical Taylor Theorem and Extrapolation of Truncated Densities}, author = {Daskalakis, Constantinos and Kontonis, Vasilis and Tzamos, Christos and Zampetakis, Emmanouil}, booktitle = {Proceedings of Thirty Fourth Conference on Learning Theory}, pages = {1395--1398}, year = {2021}, editor = {Belkin, Mikhail and Kpotufe, Samory}, volume = {134}, series = {Proceedings of Machine Learning Research}, month = {15--19 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v134/daskalakis21a/daskalakis21a.pdf}, url = {https://proceedings.mlr.press/v134/daskalakis21a.html}, 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 [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.} }
Endnote
%0 Conference Paper %T A Statistical Taylor Theorem and Extrapolation of Truncated Densities %A Constantinos Daskalakis %A Vasilis Kontonis %A Christos Tzamos %A Emmanouil Zampetakis %B Proceedings of Thirty Fourth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2021 %E Mikhail Belkin %E Samory Kpotufe %F pmlr-v134-daskalakis21a %I PMLR %P 1395--1398 %U https://proceedings.mlr.press/v134/daskalakis21a.html %V 134 %X 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.
APA
Daskalakis, C., Kontonis, V., Tzamos, C. & Zampetakis, E.. (2021). A Statistical Taylor Theorem and Extrapolation of Truncated Densities. Proceedings of Thirty Fourth Conference on Learning Theory, in Proceedings of Machine Learning Research 134:1395-1398 Available from https://proceedings.mlr.press/v134/daskalakis21a.html.

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