Shamos: Shamos estimator

In rQCC: Robust Quality Control Chart

Shamos estimator

Description

Calculates the conventional Shamos, unbiased Shamos and unbiased squared Shamos estimators. The conventional Shamos is calculated by shamos which is Fisher-consistent under the normal distribution. Note that it is not unbiased with a sample of finite size. The unbiased Shamos estimator under the normal distribution is calculated by shamos.unbiased with a finite-sample unbiasing factor. The unbiased squared Shamos estimator under the normal distribution is calculated by shamos2.unbiased with a finite-sample unbiasing factor.

Usage

shamos(x, constant=1.048358, na.rm = FALSE,  IncludeEqual=FALSE)

shamos.unbiased(x, constant=1.048358, na.rm = FALSE,  IncludeEqual=FALSE)

shamos2.unbiased(x, constant=1.048358, na.rm = FALSE, IncludeEqual=FALSE)

Arguments

x	a numeric vector of observations.
constant	Correction factor for the Fisher-consistency under the normal distribution
na.rm	a logical value indicating whether NA values should be stripped before the computation proceeds.
IncludeEqual	FALSE (default) calculates median of |Xi-Xj| with i < j, while TRUE calculates median of |Xi-Xj| with i ≤ j.

Details

The Shamos estimator is defined as

Shamos = constant * median of |Xi-Xj| over i<j

where i, j=1,2,...,n. The default value (constant=1.048358) ensures the Fisher-consistency under the normal distribution. Note that constant=1/(√(2)*Φ^(-1)(3/4)) = 1.048358 (approximately).

The unbiased Shamos is defined as

Shamos = constant * median of |Xi-Xj| over i<j divided by c6(n)

for i,j=1,2,...,n, where c6(n) is the finite-sample unbiasing factor. Note that c6(n) notation is used in Park et. al (2022), and c6(n) is calculated using the function c4.factor{rQCC} with estimator="shamos" option.

The unbiased squared Shamos is defined as the squared shamos{rQCC} divided by w6(n) where w6(n) is the finite-sample unbiasing factor. Note that w6(n) notation is used in Park et. al (2022), and w6(n) is calculated using the function w4.factor{rQCC} with estimator="shamos2" option. Note that the square of the conventional Shamos estimator is Fisher-consistent for the variance (σ^2) under the normal distribution, but it is not unbiased with a sample of finite size.

Value

They return a numeric value.

Author(s)

Chanseok Park and Min Wang

References

Park, C., H. Kim, and M. Wang (2022). Investigation of finite-sample properties of robust location and scale estimators. Communications in Statistics - Simulation and Computation, 51, 2619-2645.
doi: 10.1080/03610918.2019.1699114

Shamos, M. I. (1976). Geometry and statistics: Problems at the interface. In Traub, J. F., editor, Algorithms and Complexity: New Directions and Recent Results, pages 251–280. Academic Press, New York.

Lèvy-Leduc, C., Boistard, H., Moulines, E., Taqqu, M. S., and Reisen, V. A. (2011). Large sample behaviour of some well-known robust estimators under long-range dependence. Statistics, 45, 59–71.

See Also

mad.unbiased{rQCC} for calculating the unbiased sample MAD.

mad{stats} for calculating the Fisher-consistent sample MAD.

c4.factor{rQCC} for finite-sample unbiasing factor for the standard deviation (σ) under the normal distribution.

w4.factor{rQCC} for finite-sample unbiasing factor for the squared Shamos estimator of the variance (σ^2) under the normal distribution.

finite.breakdown{rQCC} for calculating the finite-sample breakdown point.

Examples

x = c(0:10, 50)

# Fisher-consistent Shamos, but not unbiased with a finite sample. 
shamos(x)

# Unbiased Shamos. 
shamos.unbiased(x)

# Fisher-consistent squared Shamos, but not unbiased with a finite sample. 
shamos(x)^2 

# Unbiased squared Shamos. 
shamos2.unbiased(x)

rQCC documentation built on Dec. 28, 2022, 1:49 a.m.