scaleM: M-scale estimator

In msalibian/RobStatTM: Robust Statistics: Theory and Methods

M-scale estimator

Description

This function computes an M-scale, which is a robust scale (spread) estimator. M-estimators of scale are a robust alternative to the sample standard deviation. Given a vector of residuals r, the M-scale estimator s solves the non-linear equation mean(rho(r/s, cc))=delta, where delta determines the breakdown point of the estimator, and cc is a tuning parameter calculated to obtain consistency under a Gaussian model. The breakdown point of the estimator is min(delta, 1-delta), so the optimal choice for delta is 0.5. To obtain a consistent estimator the constant cc is chosen such that E(rho(Z, cc)) = delta, where Z is a standard normal random variable.

Usage

scaleM(
  u,
  delta = 0.5,
  family = "bisquare",
  max.it = 100,
  tol = 1e-06,
  tolerancezero = .Machine$double.eps,
  tuning.chi = lmrobdet.control(family = family, bb = delta)$tuning.chi
)

Arguments

u	vector of residuals
delta	the right hand side of the M-scale equation
family	string specifying the name of the family of loss function to be used (current valid options are "bisquare", "opt" and "mopt").
max.it	maximum number of iterations allowed
tol	relative tolerance for convergence
tolerancezero	smallest (in absolute value) non-zero value accepted as a scale. Defaults to .Machine$double.eps
tuning.chi	the tuning object as returned by lmrobdet.control, bisquare, mopt, or opt. It defaults to the value that results in a consistent scale estimator for the specified family of loss functions and breakdown point as set by delta.

Details

The iterative algorithm starts from the scaled median of the absolute values of the input vector, and then cycles through the equation s_{k+1}^2 = s_k^2 * mean(rho(r/s_k, cc)) / delta.

Value

The scale estimate value at the last iteration or at convergence.

Author(s)

Matias Salibian-Barrera, [e-mail]

Examples

set.seed(123)
r <- rnorm(150, sd=1.5)
scaleM(r)
sd(r)
# 10% of outliers, sd of good points is 1.5
set.seed(123)
r2 <- c(rnorm(135, sd=1.5), rnorm(15, mean=-5, sd=.5))
scaleM(r2, family='opt')
sd(r2)

msalibian/RobStatTM documentation built on Jan. 5, 2025, 2:19 a.m.