## Statistical Science

### Self-consistency: a fundamental concept in statistics

#### Abstract

The term "self-consistency" was introduced in 1989 by Hastie and Stuetzle to describe the property that each point on a smooth curve or surface is the mean of all points that project orthogonally onto it. We generalize this concept to self-consistent random vectors: a random vector Y is self-consistent for X if $\mathscr{E}[X|Y] = Y$ almost surely. This allows us to construct a unified theoretical basis for principal components, principal curves and surfaces, principal points, principal variables, principal modes of variation and other statistical methods. We provide some general results on self-consistent random variables, give examples, show relationships between the various methods, discuss a related notion of self-consistent estimators and suggest directions for future research.

#### Article information

Source
Statist. Sci., Volume 11, Number 3 (1996), 229-243.

Dates
First available in Project Euclid: 17 September 2002

https://projecteuclid.org/euclid.ss/1032280215

Digital Object Identifier
doi:10.1214/ss/1032280215

Mathematical Reviews number (MathSciNet)
MR1436648

Zentralblatt MATH identifier
0955.62540

#### Citation

Tarpey, Thaddeus; Flury, Bernard. Self-consistency: a fundamental concept in statistics. Statist. Sci. 11 (1996), no. 3, 229--243. doi:10.1214/ss/1032280215. https://projecteuclid.org/euclid.ss/1032280215

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