## Advanced Studies in Pure Mathematics

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

### A limit theorem in singular regression problem

#### 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