## The Annals of Statistics

### $M$-Estimates of Rigid Body Motion on the Sphere and in Euclidean Space

#### Abstract

This paper calculates the influence functions and asymptotic distributions of $M$-estimators of the rotation $A$ in a spherical regression model on the unit sphere in $p$ dimensions with isotropic errors. The problem arises in the reconstruction of the motion of a rigid body on the surface of the sphere. The comparable model for $p$-dimensional Euclidean space data is that $(\nu_1, \ldots, \nu_n)$ are independent with $\nu_i$ symmetrically distributed around $\gamma A \cdot u_i + b, u_i$ known, where the real constant $\gamma > 0, p \times p$ rotation matrix $A$ and $p$-vector $b$ are the parameters to be estimated. This paper also calculates the influence functions and asymptotic distributions of $M$-estimators for $\gamma, A$ and $b$. Besides rigid body motion, this problem arises in image registration from landmark data. Particular attention is paid to how the geometry of the rigid body or landmarks affects the statistical properties of the estimators.

#### Article information

Source
Ann. Statist., Volume 23, Number 5 (1995), 1823-1847.

Dates
First available in Project Euclid: 11 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176324325

Digital Object Identifier
doi:10.1214/aos/1176324325

Mathematical Reviews number (MathSciNet)
MR1370309

Zentralblatt MATH identifier
0843.62064

JSTOR
links.jstor.org

#### Citation

Chang, Ted; Ko, Daijin. $M$-Estimates of Rigid Body Motion on the Sphere and in Euclidean Space. Ann. Statist. 23 (1995), no. 5, 1823--1847. doi:10.1214/aos/1176324325. https://projecteuclid.org/euclid.aos/1176324325