Type: Package
Title: Fast Robust Estimation in Very Small Samples
Version: 0.1.1
Description: High-performance C++ implementation (via 'Rcpp') of the robust location and scale M-estimators described in Rousseeuw & Verboven (2002) <doi:10.1016/S0167-9473(02)00078-6> for very small samples. Provides numerically identical results to the 'revss' package with significantly improved performance through sorting networks and compiled iteration loops.
License: MIT + file LICENSE
URL: https://github.com/davdittrich/robscale, https://doi.org/10.5281/zenodo.18828607
BugReports: https://github.com/davdittrich/robscale/issues
Encoding: UTF-8
Imports: Rcpp (≥ 1.0.0)
LinkingTo: Rcpp
Suggests: tinytest, revss, microbenchmark
NeedsCompilation: yes
RoxygenNote: 7.3.3
Packaged: 2026-03-02 17:37:00 UTC; dittrd01
Author: Dennis Alexis Valin Dittrich ORCID iD [aut, cre, cph]
Maintainer: Dennis Alexis Valin Dittrich <davd@economicscience.net>
Repository: CRAN
Date/Publication: 2026-03-06 13:30:02 UTC

robscale: Fast Robust Estimation in Very Small Samples

Description

High-performance C++ implementation (via 'Rcpp') of the robust location and scale M-estimators described in Rousseeuw & Verboven (2002) doi:10.1016/S0167-9473(02)00078-6 for very small samples. Provides numerically identical results to the 'revss' package with significantly improved performance through sorting networks and compiled iteration loops.

Author(s)

Maintainer: Dennis Alexis Valin Dittrich davd@economicscience.net (ORCID) [copyright holder]

References

Rousseeuw, P. J. and Verboven, S. (2002) Robust estimation in very small samples. Computational Statistics & Data Analysis, 40(4), 741–758. doi:10.1016/S0167-9473(02)00078-6

Nair, K. R. (1947) A Note on the Mean Deviation from the Median. Biometrika, 34(3/4), 360–362. doi:10.2307/2332448

See Also

Useful links:

Average Distance to the Median

Description

Compute the mean absolute deviation from the median, scaled by a consistency constant for asymptotic normality under the Gaussian model.

Usage

adm(x, center, constant = 1.2533141373155, na.rm = FALSE)

Arguments

x

A numeric vector.

center

Optional numeric scalar giving the central value from which to measure the average absolute distance. Defaults to the median of x.

constant

Consistency constant for asymptotic normality at the Gaussian. Defaults to \sqrt{\pi/2} \approx 1.2533 (Nair, 1947). Set to 1 for the raw (unscaled) mean absolute deviation.

na.rm

Logical. If TRUE, NA values are stripped from x before computation. If FALSE (the default), the presence of any NA raises an error.

Details

The average distance to the median (ADM) is defined as

\mathrm{ADM}(x) \;=\; C \cdot \frac{1}{n}\sum_{i=1}^{n} |x_i - \mathrm{med}(x)|

where C is the consistency constant and \mathrm{med}(x) is the sample median. When center is supplied it replaces the median.

The default constant C = \sqrt{\pi/2} makes the ADM a consistent estimator of the standard deviation under the normal distribution. In large samples the ADM converges to \sigma at the Gaussian; however, this asymptotic property may not hold at the very small sample sizes (n = 3–8) for which this package is primarily intended.

The ADM is not robust against outliers in the explosion sense: its explosion breakdown point is 1/n. However, it is highly resistant to implosion, with an implosion breakdown point of (n-1)/n. It therefore serves as the fallback scale estimator in robScale when the MAD collapses to zero.

Value

A single numeric value: the scaled mean absolute deviation from the center. Returns NA if x has length zero after removal of NAs.

References

Nair, K. R. (1947) A Note on the Mean Deviation from the Median. Biometrika, 34(3/4), 360–362. doi:10.2307/2332448

Rousseeuw, P. J. and Verboven, S. (2002) Robust estimation in very small samples. Computational Statistics & Data Analysis, 40(4), 741–758. doi:10.1016/S0167-9473(02)00078-6

See Also

mad for the median absolute deviation from the median; robScale for the M-estimator of scale that uses the ADM as an implosion fallback.

Examples

adm(c(1:9))

x <- c(1, 2, 3, 5, 7, 8)
adm(x)                      # with consistency constant
adm(x, constant = 1)        # raw mean absolute deviation

# Supply a known center
adm(x, center = 4.0)

Robust M-Estimate of Location

Description

Compute the robust M-estimate of location for very small samples using the logistic \psi function of Rousseeuw & Verboven (2002).

Usage

robLoc(
  x,
  scale = NULL,
  na.rm = FALSE,
  maxit = 80L,
  tol = sqrt(.Machine$double.eps)
)

Arguments

x

A numeric vector.

scale

Optional numeric scalar giving a known scale. When supplied, the MAD is replaced by this value and the minimum sample size for iteration is lowered from 4 to 3 (see ‘Details’).

na.rm

Logical. If TRUE, NA values are stripped from x before computation. If FALSE (the default), the presence of any NA raises an error.

maxit

Maximum number of Newton–Raphson iterations. Defaults to 80.

tol

Convergence tolerance. Iteration stops when the absolute Newton step falls below tol. Defaults to sqrt(.Machine$double.eps).

Details

The location estimator T_n is the solution to the M-estimating equation

\frac{1}{n}\sum_{i=1}^{n}\psi_{\mathrm{log}} \!\left(\frac{x_i - T_n}{S_n}\right) = 0

where S_n is a fixed auxiliary scale estimate and \psi_{\mathrm{log}} is the logistic psi function (Rousseeuw & Verboven 2002, Eq. 23):

\psi_{\mathrm{log}}(x) = \frac{e^x - 1}{e^x + 1} = \tanh(x/2)

This function is bounded in (-1, 1), smooth (C^\infty), and strictly monotone. Boundedness provides robustness against outliers; smoothness avoids the corner artifacts of Huber's \psi at small n.

Iteration scheme. The estimating equation is solved by Newton–Raphson iteration. The derivative of the logistic psi satisfies \psi'(x) = 1 - \psi^2(x), so the Newton step requires no additional transcendental function evaluations beyond those already computed for the numerator. Starting value: T^{(0)} = \mathrm{med}(x). Auxiliary scale: S = \mathrm{MAD}(x) unless scale is supplied.

Decoupled estimation. Location and scale are estimated separately: robLoc holds the auxiliary scale fixed at \mathrm{MAD}(x) throughout iteration, following the decoupled approach of Rousseeuw & Verboven (2002, Sec. 4.1). This avoids the positive-feedback instability of simultaneous location–scale iteration (Huber's Proposal 2) in small samples.

Fallback. When the sample is too small for reliable iteration the function returns the median directly:

Value

A single numeric value: the robust M-estimate of location. Returns NA if x has length zero (after removal of NAs when na.rm = TRUE).

References

Rousseeuw, P. J. and Verboven, S. (2002) Robust estimation in very small samples. Computational Statistics & Data Analysis, 40(4), 741–758. doi:10.1016/S0167-9473(02)00078-6

See Also

median for the starting value; mad for the auxiliary scale; robScale for the companion scale estimator.

Examples

robLoc(c(1:9))

x <- c(1, 2, 3, 5, 7, 8)
robLoc(x)

# Known scale lowers the minimum sample size to 3
robLoc(c(1, 2, 3), scale = 1.5)

# Outlier resistance
x_clean <- c(2.0, 3.1, 2.7, 2.9, 3.3)
x_dirty <- replace(x_clean, 5, 100)
c(robLoc(x_clean), robLoc(x_dirty))   # barely moves
c(mean(x_clean), mean(x_dirty))       # destroyed

Robust M-Estimate of Scale

Description

Compute the robust M-estimate of scale for very small samples using the \rho function of Rousseeuw & Verboven (2002).

Usage

robScale(
  x,
  loc = NULL,
  implbound = 0.0001,
  na.rm = FALSE,
  maxit = 80L,
  tol = sqrt(.Machine$double.eps)
)

Arguments

x

A numeric vector.

loc

Optional numeric scalar giving a known location. When supplied, the observations are centered at loc and the minimum sample size for iteration is lowered from 4 to 3 (see ‘Details’).

implbound

Implosion bound: the smallest value the MAD is allowed to take before it is considered to have “imploded” (collapsed to zero). Defaults to 1e-4.

na.rm

Logical. If TRUE, NA values are stripped from x before computation. If FALSE (the default), the presence of any NA raises an error.

maxit

Maximum number of multiplicative iterations. Defaults to 80.

tol

Convergence tolerance. Iteration stops when the multiplicative update factor satisfies |v - 1| \le \mathrm{tol}. Defaults to sqrt(.Machine$double.eps).

Details

The scale estimator S_n solves the M-estimating equation

\frac{1}{n}\sum_{i=1}^{n}\rho\!\left(\frac{x_i - T_n}{S_n} \right) = \beta

where T_n is fixed at the sample median, \beta = 0.5, and \rho is a smooth rho function defined as the square of the logistic psi (Rousseeuw & Verboven, 2002, Sec. 4.2):

\rho_{\mathrm{log}}(x) = \psi_{\mathrm{log}}^2\!\left( \frac{x}{c}\right)

with the tuning constant c = 0.37394112142347236 chosen so that

\int\rho(u)\,d\Phi(u) = 0.5

yielding a 50% breakdown point.

Iteration scheme. The equation is solved by multiplicative iteration (Rousseeuw & Verboven, 2002, Eq. 27):

S^{(k+1)} = S^{(k)} \cdot \sqrt{2 \cdot \frac{1}{n}\sum \psi_{\mathrm{log}}^2\!\left(\frac{x_i - T}{c \cdot S^{(k)}}\right)}

Starting value: S^{(0)} = \mathrm{MAD}(x). The logistic psi values are computed via the algebraic identity \psi_{\mathrm{log}}(x) = \tanh(x/2).

Decoupled estimation. Scale is estimated with location held fixed at \mathrm{med}(x), following the decoupled approach of Rousseeuw & Verboven (2002, Sec. 4.2). This avoids the positive-feedback instability of Huber's Proposal 2 in small samples.

Known location. When loc is supplied, the observations are centered as x_i - \mu and the initial scale is set to 1.4826 \cdot \mathrm{med}(|x_i - \mu|) rather than the MAD. This lowers the minimum sample size from 4 to 3 (Rousseeuw & Verboven, 2002, Sec. 5).

Fallback. When n is below the minimum for iteration:

Value

A single numeric value: the robust M-estimate of scale. Returns NA if x has length zero (after removal of NAs when na.rm = TRUE).

References

Rousseeuw, P. J. and Verboven, S. (2002) Robust estimation in very small samples. Computational Statistics & Data Analysis, 40(4), 741–758. doi:10.1016/S0167-9473(02)00078-6

See Also

adm for the implosion fallback; mad for the starting value and classical alternative; robLoc for the companion location estimator.

Examples

robScale(c(1:9))

x <- c(1, 2, 3, 5, 7, 8)
robScale(x)
robScale(x, loc = 5)           # known location

# Outlier resistance
x_clean <- c(2.0, 3.1, 2.7, 2.9, 3.3)
x_dirty <- replace(x_clean, 5, 100)
c(robScale(x_clean), robScale(x_dirty))   # barely moves
c(sd(x_clean), sd(x_dirty))               # destroyed

# MAD implosion: identical values cause MAD = 0
robScale(c(5, 5, 5, 5, 6))     # falls back to adm()