zhang: M-estimator theory pivotal bootstrap confidence region

Description

Compute the radius of a \(100(1-\alpha)\)% confidence region for the central orientation based on M-estimation theory.

Usage

zhang(x, estimator, alp = NULL, m = 300)
# S3 method for SO3
zhang(x, estimator, alp = NULL, m = 300)
# S3 method for Q4
zhang(x, estimator, alp = NULL, m = 300)

Value

Radius of the confidence region centered at the specified estimator.

Arguments

x: \(n\times p\) matrix where each row corresponds to a random rotation in matrix (\(p=9\)) or quaternion (\(p=4\)) form.
estimator: character string either "mean" or "median."
alp: alpha level desired, e.g. 0.05 or 0.10.
m: number of replicates to use to estimate the critical value.

Details

Compute the radius of a \(100(1-\alpha)\)% confidence region for the central orientation based on the projected mean estimator using the method due to Zhang & Nordman (2009) (unpublished MS thesis). By construction each axis will have the same radius so the radius reported is for all three axis. A normal theory version of this procedure uses the theoretical chi-square limiting distribution and is given by the chang option. This method is called "direct" because it used M-estimation theory for SO(3) directly instead of relying on transforming a result from directional statistics as prentice and fisheretal do.

Examples

Run this code

Rs <- ruars(20, rcayley, kappa = 100)

# The zhang method can be accesed from the "region" function or the "zhang" function
# They will be different because it is a bootstrap.
region(Rs, method = "direct", type = "bootstrap", alp = 0.1, estimator = "mean")
zhang(Rs, estimator = "mean", alp = 0.1)