Type:	Package
Title:	Accelerated Estimation of Robust Location and Scale
Version:	0.2.1
Description:	Estimates robust location and scale parameters using platform-specific Single Instruction, Multiple Data (SIMD) vectorization and Intel Threading Building Blocks (TBB) for parallel processing. Implements a novel variance-weighted ensemble estimator that adaptively combines all available statistics. Included methods feature logistic M-estimators, the estimators of Rousseeuw and Croux (1993), the Gini mean difference, the scaled Median Absolute Deviation (MAD), the scaled Interquartile Range (IQR), and unbiased standard deviations. Achieves substantial speedups over existing implementations through an 'Rcpp' backend and a unified dispatcher that automatically selects the optimal estimator based on sample size.
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), RcppParallel (≥ 5.0.0)
LinkingTo:	Rcpp, RcppParallel, BH
Suggests:	collapse, GiniDistance, Hmisc, revss, robustbase, testthat (≥ 3.0.0)
SystemRequirements:	GNU make, TBB
Config/testthat/edition:	3
NeedsCompilation:	yes
RoxygenNote:	7.3.3
Packaged:	2026-03-16 17:22:04 UTC; dd
Author:	Dennis Alexis Valin Dittrich [aut, cre, cph]
Maintainer:	Dennis Alexis Valin Dittrich <davd@economicscience.net>
Repository:	CRAN
Date/Publication:	2026-03-16 19:30:10 UTC

Faster Robustness: SIMD-Accelerated Estimation of Location and Scale

Description

A comprehensive suite of high-performance robust scale and location estimators. The package provides:

Details

Scale estimators:

• qn, sn — Rousseeuw & Croux (1993) with SIMD and TBB parallelism

• robScale — logistic M-estimator with Newton-Raphson and SIMD-accelerated tanh

• gmd — Gini mean difference (98% ARE)

• mad_scaled, iqr_scaled — scaled MAD and IQR with O(n) selection

• sd_c4 — unbiased standard deviation with c4 correction

• adm — average distance to the median (implosion fallback)

Location estimator:

• robLoc — logistic M-estimator of location

Unified API:

• scale_robust — adaptive dispatcher with variance-weighted ensemble for small samples, automatic GMD switching for larger samples

Value

None, this is a package documentation object.

Author(s)

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

References

Rousseeuw, P. J., and Croux, C. (1993). Alternatives to the Median Absolute Deviation. Journal of the American Statistical Association, 88, 1273–1283.

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:

• https://github.com/davdittrich/robscale

• doi:10.5281/zenodo.18828607

• Report bugs at https://github.com/davdittrich/robscale/issues

Average Distance to the Median

Description

Computes 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

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 sample median.

Statistical Properties. The default constant C = \sqrt{\pi/2} ensures that the ADM is a consistent estimator of the standard deviation \sigma under the Gaussian model. At the normal distribution, the ADM achieves an asymptotic relative efficiency (ARE) of 0.88 compared to the sample standard deviation.

While the ADM is less efficient than the standard deviation for purely Gaussian data, it offers superior resistance to "implosion" (the estimate collapsing to zero). Its implosion breakdown point is (n-1)/n, meaning it only collapses if all but one observation are identical. Conversely, its explosion breakdown point is 1/n, similar to the sample mean. These properties make the ADM the ideal implosion fallback for the M-estimator of scale in robScale.

Computational Performance. This implementation employs a tiered selection strategy: optimal sorting networks for very small samples (n \le 16) and a C++17 introselect algorithm for larger datasets. This approach provides an O(n) worst-case time complexity, typically outperforming pure-R implementations by an order of magnitude. Performance is further enhanced by avoiding the full sort required by the standard median function.

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)

Get Consistency Constant

Description

Returns the consistency constant or finite-sample correction factor for a given scale estimator and sample size.

Usage

get_consistency_constant(method, n = NULL)

Arguments

Details

For "c4", "sn", and "qn", the returned value depends on n. For "gmd", "mad", and "iqr", the returned value is the asymptotic constant (independent of n).

Value

A single numeric value: the consistency constant or correction factor.

Examples

get_consistency_constant("mad")      # 1.4826
get_consistency_constant("c4", 5)    # c4(5)
get_consistency_constant("qn", 10)   # finite-sample Qn factor

Gini Mean Difference

Description

Computes the Gini mean difference, scaled by a consistency constant for asymptotic normality under the Gaussian model.

Usage

gmd(x, constant = 0.886226925452758, na.rm = FALSE, ci = FALSE, level = 0.95)

Arguments

Details

The Gini mean difference is defined as

\mathrm{GMD}(x) = C \cdot \frac{2}{n(n-1)} \sum_{i=1}^{n} (2i - n - 1)\, x_{(i)}

where x_{(1)} \le \ldots \le x_{(n)} are the order statistics and C is the consistency constant. The computation requires a full sort (O(n log n)).

Statistical Properties. The default constant C = \sqrt{\pi}/2 ensures that the GMD is a consistent estimator of \sigma under the Gaussian model. The GMD achieves an asymptotic relative efficiency (ARE) of 0.98 compared to the sample standard deviation, making it the most efficient robust alternative in this package. Its breakdown point is approximately 29.3%.

Value

If ci = FALSE (default), a single numeric value: the scaled Gini mean difference. Returns 0 if n < 2; returns NA if x has length zero after removal of NAs. If ci = TRUE, an object of class "robscale_ci".

References

Gini, C. (1912) Variabilita e mutabilita. Bologna.

See Also

scale_robust for the unified dispatcher; qn and sn for high-breakdown scale estimators.

Examples

gmd(c(1:9))

x <- c(1, 2, 3, 5, 7, 8)
gmd(x)                      # with consistency constant
gmd(x, constant = 1)        # raw Gini mean difference

# Asymptotic confidence interval
gmd(x, ci = TRUE)

Scaled Interquartile Range

Description

Computes the interquartile range, scaled by a consistency constant for asymptotic normality under the Gaussian model.

Usage

iqr_scaled(
  x,
  constant = 0.741301109252801,
  na.rm = FALSE,
  ci = FALSE,
  level = 0.95
)

Arguments

Details

The scaled IQR is defined as

\mathrm{IQR}_s(x) = C \cdot (Q_{0.75} - Q_{0.25})

where Q_p denotes the Type 7 quantile (R default) and C is the consistency constant.

Statistical Properties. The default constant ensures consistency for \sigma under the Gaussian model. The IQR achieves an asymptotic relative efficiency (ARE) of 0.37 compared to the sample standard deviation. Its breakdown point is 25%.

Computational Performance. Unlike IQR, which requires a full sort, this implementation uses O(n) selection via the pdqselect algorithm for each quantile, providing a substantial speedup for large datasets.

Value

If ci = FALSE (default), a single numeric value: the scaled interquartile range. Returns 0 if n < 2; returns NA if x has length zero after removal of NAs. If ci = TRUE, an object of class "robscale_ci".

See Also

IQR for the base R (unscaled) interquartile range; scale_robust for the unified dispatcher; mad_scaled for the scaled median absolute deviation.

Examples

iqr_scaled(c(1:9))

x <- c(1, 2, 3, 5, 7, 8)
iqr_scaled(x)                 # with consistency constant
iqr_scaled(x, constant = 1)   # raw IQR

# Asymptotic confidence interval
iqr_scaled(x, ci = TRUE)

Scaled Median Absolute Deviation

Description

Computes the median absolute deviation from the median (or a user-supplied center), scaled by a consistency constant for asymptotic normality under the Gaussian model.

Usage

mad_scaled(
  x,
  center,
  constant = 1.4826022185056,
  na.rm = FALSE,
  ci = FALSE,
  level = 0.95
)

Arguments

Details

The scaled MAD is defined as

\mathrm{MAD}_s(x) = C \cdot \mathrm{med}_i\, |x_i - \mathrm{med}(x)|

Statistical Properties. The MAD achieves a 50% breakdown point, meaning it tolerates up to half the data being contaminated. Its asymptotic relative efficiency (ARE) is 36.8% compared to the sample standard deviation.

Computational Performance. Unlike mad, this implementation uses O(n) selection with adaptive algorithm dispatch: Floyd-Rivest for moderate n, pdqselect for large n (threshold derived from per-core L2 cache size at startup), and sorting networks for n \le 16.

Value

If ci = FALSE (default), a single numeric value: the scaled median absolute deviation. Returns 0 if n = 1; returns NA if x has length zero after removal of NAs. If ci = TRUE, an object of class "robscale_ci".

See Also

mad for the base R implementation; scale_robust for the unified dispatcher; robScale for the M-estimate of scale that uses the MAD as a starting value.

Examples

mad_scaled(c(1:9))

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

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

# Asymptotic confidence interval
mad_scaled(x, ci = TRUE)

Robust Estimator of Scale Qn

Description

Computes the robust estimator of scale Q_n proposed by Rousseeuw and Croux (1993).

Usage

qn(
  x,
  constant = 2.2191,
  finite.corr = TRUE,
  na.rm = FALSE,
  ci = FALSE,
  level = 0.95
)

Arguments

Details

The Q_n estimator is defined as the k-th order statistic of the \binom{n}{2} absolute pairwise differences |x_i - x_j| for i < j. Specifically, Q_n = C \cdot d_{(k)} where d is the set of absolute differences and k = \binom{h}{2} with h = \lfloor n/2 \rfloor + 1.

Statistical Properties. Q_n is a highly robust estimator with a 50% breakdown point. Unlike the Median Absolute Deviation (MAD), Q_n does not require a prior location estimate, making it more robust in asymmetric distributions. At the Gaussian distribution, it achieves an asymptotic relative efficiency (ARE) of 0.82, significantly higher than the 0.37 achieved by the MAD.

Computational Performance. While the naive calculation of Q_n requires O(n^2) space and time, this implementation employs a specialized O(n \log n) algorithm. The package utilizes a tiered execution strategy:

• Optimal sorting networks for very small samples (n \le 16). These networks eliminate branch misprediction in the target regimes of extremely small samples.

• A specialized Johnson–Mizoguchi selection algorithm for medium-sized datasets.

• Cache-aware parallelization via Intel TBB (Threading Building Blocks) for large-scale data, with thresholds derived from the detected per-core L2 cache size.

Value

If ci = FALSE (default), the Q_n estimator of scale (a scalar). If ci = TRUE, an object of class "robscale_ci".

References

Rousseeuw, P. J., and Croux, C. (1993). Alternatives to the Median Absolute Deviation. Journal of the American Statistical Association, 88(424), 1273–1283. doi:10.1080/01621459.1993.10476408

Akinshin, A. (2022). Finite-sample Rousseeuw-Croux scale estimators. arXiv preprint arXiv:2209.12268.

See Also

sn for the S_n scale estimator; robScale for the M-estimator of scale; adm for the average distance to median.

Examples

qn(c(1:9))
x <- c(1, 2, 3, 5, 7, 8)
qn(x)

# Asymptotic confidence interval
qn(x, ci = TRUE)

Robust M-Estimate of Location

Description

Computes 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

Details

The M-estimate of location T_n is defined as the solution to the estimating equation

\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 (defaulting to the MAD) and \psi_{\mathrm{log}} is the logistic psi function:

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

Statistical Properties. The logistic psi function is bounded, smooth (C^\infty), and strictly monotone. These properties ensure that the resulting M-estimator is both robust to outliers and numerically stable. At the Gaussian distribution, the logistic M-estimator of location achieves high efficiency, with an asymptotic relative efficiency (ARE) of 0.98 compared to the sample mean.

Small-Sample Strategy. Following Rousseeuw & Verboven (2002), location and scale are estimated separately. In robLoc, the auxiliary scale S_n remains fixed throughout the Newton–Raphson iteration. This "decoupled" approach avoids the instabilities often encountered in small samples when using simultaneous location–scale iteration (e.g., Huber's Proposal 2).

Numerical Computation. The estimating equation is solved via Newton–Raphson iteration starting from the sample median. Because the derivative of the logistic psi satisfies \psi'(x) = \frac{1}{2}(1 - \psi^2(x)), the Newton step is computationally efficient, requiring no additional transcendental calls beyond the tanh evaluations used for the psi function itself.

Performance and SIMD. The underlying C++ core utilizes platform-specific SIMD (Single Instruction, Multiple Data) backends (SLEEF on Linux, Apple Accelerate on macOS) to vectorize the tanh evaluations. This architectural choice delivers substantial performance gains, particularly for large-scale or high-throughput workflows.

Fallback Mechanism. For extremely small samples where iteration may be unreliable, the function returns the median directly:

• If scale is unknown: n < 4 (since the MAD of 3 points is insufficiently robust).

• If scale is supplied: n < 3.

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; qn and sn for high-efficiency scale estimators.

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

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

Usage

robScale(
  x,
  loc = NULL,
  fallback = c("adm", "na"),
  implbound = 1e-04,
  na.rm = FALSE,
  maxit = 80L,
  tol = sqrt(.Machine$double.eps),
  ci = FALSE,
  level = 0.95
)

Arguments

Details

The M-estimate of scale S_n is defined as the solution to the estimating equation

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

where the location T_n is fixed at the sample median, \beta = 0.5 is the expected value of \rho under the Gaussian model, and \rho is a smooth rho function (Rousseeuw & Verboven, 2002, Sec. 4.2):

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

The tuning constant c = 0.373941121 is chosen to satisfy E_\Phi[\rho(u)] = 0.5.

Statistical Properties. This estimator is designed for high robustness and efficiency. It achieves a 50% breakdown point, meaning the estimate remains reliable even if half the sample is contaminated by outliers. At the Gaussian distribution, the logistic M-estimator of scale achieves an asymptotic relative efficiency (ARE) of 0.55 compared to the sample standard deviation.

Numerical Computation. The estimating 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)}

The algorithm starts at the Median Absolute Deviation (MAD). Because location is held fixed at the sample median, the estimator follows a "decoupled" approach that avoids the positive-feedback instabilities often seen in simultaneous location–scale estimation (Proposal 2) at very small sample sizes.

Performance and SIMD. The C++ implementation leverages platform-specific SIMD (Single Instruction, Multiple Data) backends (SLEEF on Linux, Apple Accelerate on macOS) to \psi_{\mathrm{log}} evaluations (via tanh). This specialized architecture typically yields an 11–39x speedup over pure-R code for samples of size n \le 20.

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 Mechanism and Implosion. Robust scale estimators like the MAD can "implode" (collapse to zero) if more than 50 or the sample size is too small for reliable iteration (n < 4, or n < 3 if location is known):

• If fallback = "adm" (default), the function returns the scaled Average Distance to the Median (adm). The ADM is highly resistant to implosion (breakdown point (n-1)/n).

• If fallback = "na", the function returns NA.

Value

If ci = FALSE (default), 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) or if the MAD collapses and fallback = "na". If ci = TRUE, an object of class "robscale_ci".

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; qn and sn for high-efficiency scale estimators.

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()

# Asymptotic confidence interval
robScale(x, ci = TRUE)

Robust Ensemble Scale Estimation

Description

Unified dispatcher for robust scale estimation. Automatically selects between a variance-weighted ensemble of 7 estimators (for small samples) and the Gini Mean Difference (for large samples), or returns a specific estimator by name.

Usage

scale_robust(
  x,
  method = c("ensemble", "gmd", "sd", "mad", "iqr", "sn", "qn", "robScale"),
  auto_switch = TRUE,
  threshold = 20L,
  n_boot = 200L,
  na.rm = FALSE,
  ci = FALSE,
  level = 0.95,
  boot_method = c("auto", "bca", "percentile", "parametric")
)

Arguments

Details

Ensemble method. When method = "ensemble", the function computes a variance-weighted combination of 7 scale estimators:

1. sd_c4 — unbiased standard deviation

2. gmd — Gini mean difference

3. mad_scaled — median absolute deviation

4. iqr_scaled — scaled interquartile range

5. sn — Sn estimator of Rousseeuw & Croux

6. qn — Qn estimator of Rousseeuw & Croux

7. robScale — logistic M-estimator of scale

The weights are determined by bootstrap resampling: each estimator's inverse variance across n_boot resamples determines its contribution. Estimators with lower sampling variance receive higher weight.

Automatic switching. When auto_switch = TRUE and n >= threshold, the function returns gmd(x) directly. The GMD achieves 98% asymptotic relative efficiency at the Gaussian while being computationally cheaper than the ensemble, making it the optimal choice for moderate-to-large samples.

Individual methods. When a specific method is requested, scale_robust dispatches to the corresponding function with default parameters.

Value

If ci = FALSE (default), a single numeric value: the scale estimate (NA when n < 2). If ci = TRUE with a single method, a "robscale_ci" object. If ci = TRUE with method = "ensemble", a "robscale_ensemble_ci" object containing the ensemble estimate, bootstrap CI, and a table of per-estimator results.

See Also

Individual estimators: sd_c4, gmd, mad_scaled, iqr_scaled, sn, qn, robScale; get_consistency_constant for the underlying constants.

Examples

x <- c(1, 2, 3, 5, 7, 8)
scale_robust(x)                          # ensemble (n < 20)
scale_robust(x, method = "qn")           # specific method

set.seed(42)
y <- rnorm(50)
scale_robust(y)                          # auto-switches to GMD (n >= 20)
scale_robust(y, auto_switch = FALSE)     # forces ensemble

# Ensemble with bootstrap CIs
scale_robust(x, ci = TRUE)

# Single-method analytical CI
scale_robust(x, method = "sn", ci = TRUE)

Unbiased Standard Deviation

Description

Computes the sample standard deviation corrected by the c_4(n) factor to remove the small-sample bias of the square-root estimator.

Usage

sd_c4(x, na.rm = FALSE, ci = FALSE, level = 0.95)

Arguments

Details

The standard sd function computes s = \sqrt{s^2} where s^2 is the unbiased sample variance. However, E[s] \neq \sigma for finite n; the square root introduces a downward bias.

The c_4(n) correction factor removes this bias:

\hat\sigma = \frac{s}{c_4(n)} = \frac{s}{\sqrt{2/(n-1)} \cdot \Gamma(n/2) / \Gamma((n-1)/2)}

Statistical Properties. This is a classical (non-robust) estimator with 100% ARE by construction, but 0% breakdown point — a single outlier can make it arbitrarily large.

Numerical Stability. Uses Welford's online algorithm for numerically stable variance computation, avoiding catastrophic cancellation that affects naive two-pass formulas.

Value

If ci = FALSE (default), a single numeric value: the bias-corrected standard deviation. Returns NA if n < 2. If ci = TRUE, an object of class "robscale_ci" with an exact chi-squared confidence interval.

See Also

sd for the (biased) sample standard deviation; scale_robust for the unified dispatcher; robScale for the robust M-estimate of scale.

Examples

sd_c4(c(1:9))

# Compare with base sd() --- difference is small for large n
x <- rnorm(1000)
c(sd_c4 = sd_c4(x), sd = sd(x))

# Exact chi-squared confidence interval
sd_c4(c(1:9), ci = TRUE)

Robust Estimator of Scale Sn

Description

Computes the robust estimator of scale S_n proposed by Rousseeuw and Croux (1993).

Usage

sn(
  x,
  constant = 1.1926,
  finite.corr = TRUE,
  na.rm = FALSE,
  ci = FALSE,
  level = 0.95
)

Arguments

Details

The S_n estimator is defined as the median of medians:

S_n = C \cdot \text{med}_i \left\{ \text{med}_j |x_i - x_j| \right\}

Statistical Properties. S_n achieves a 50% breakdown point and provides superior statistical efficiency compared to the Median Absolute Deviation (MAD). At the Gaussian distribution, it has an asymptotic relative efficiency (ARE) of 0.58, which is significantly higher than the 0.37 of the MAD. Unlike M-estimators of scale, S_n is an explicit function of the data and does not require an iterative solution or a prior location estimate.

Computational Performance. This implementation provides a highly optimized O(n \log n) algorithm, avoiding the O(n^2) complexity of the naive definition. The execution strategy is tiered for maximum efficiency:

• Optimal sorting networks are used for the target regime of very small samples (n \le 16). These networks minimize control flow overhead and maximize pipeline utilization.

• Highly tuned kernels are employed for general datasets, leveraging C++17 features for cache-aware computation.

• Cache-aware parallelization via Intel TBB (Threading Building Blocks) for large-scale data, with thresholds derived from the detected per-core L2 cache size.

Value

If ci = FALSE (default), the S_n estimator of scale (a scalar). If ci = TRUE, an object of class "robscale_ci".

References

Rousseeuw, P. J., and Croux, C. (1993). Alternatives to the Median Absolute Deviation. Journal of the American Statistical Association, 88(424), 1273–1283. doi:10.1080/01621459.1993.10476408

Akinshin, A. (2022). Finite-sample Rousseeuw-Croux scale estimators. arXiv preprint arXiv:2209.12268.

See Also

qn for the Q_n scale estimator; robScale for the M-estimator of scale; adm for the average distance to median.

Examples

sn(c(1:9))
x <- c(1, 2, 3, 5, 7, 8)
sn(x)

# Asymptotic confidence interval
sn(x, ci = TRUE)