Advanced Studies in Pure Mathematics

A limit theorem in singular regression problem

Sumio Watanabe

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Abstract

In statistical problems, a set of parameterized probability distributions is often used to estimate the true probability distribution. If the Fisher information matrix at the true distribution is singular, then it has been left unknown what we can estimate about the true distribution from random samples. In this paper, we study a singular regression problem and prove a limit theorem which shows the relation between the accuracy of singular regression and two birational invariants, a real log canonical threshold and a singular fluctuation. The obtained theorem has an important application to statistics, because it enables us to estimate the generalization error from the training error without any knowledge of the true probability distribution.

Article information

Source
Probabilistic Approach to Geometry, M. Kotani, M. Hino and T. Kumagai, eds. (Tokyo: Mathematical Society of Japan, 2010), 473-492

Dates
Received: 15 January 2009
Revised: 7 July 2009
First available in Project Euclid: 24 November 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1543086333

Digital Object Identifier
doi:10.2969/aspm/05710473

Mathematical Reviews number (MathSciNet)
MR2648274

Zentralblatt MATH identifier
1210.62102

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Singular regression real log canonical threshold singular fluctuation resolution of singularities generalization error

Citation

Watanabe, Sumio. A limit theorem in singular regression problem. Probabilistic Approach to Geometry, 473--492, Mathematical Society of Japan, Tokyo, Japan, 2010. doi:10.2969/aspm/05710473. https://projecteuclid.org/euclid.aspm/1543086333


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