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.
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)
They return a numeric value.
a numeric vector of observations.
Correction factor for the Fisher-consistency under the normal distribution
a logical value indicating whether NA values should be stripped before the computation proceeds.
FALSE (default) calculates median of
\(|X_i-X_j|\) with \(i < j\),
while TRUE calculates median of \(|X_i-X_j|\) with \(i \le j\).
Chanseok Park and Min Wang
The Shamos estimator is defined as
$$\textrm{Shamos}=\code{constant}\times\mathop{\mathrm{median}}_{i < j} \big(|X_i-X_j|\big)$$
where \(i,j=1,2,\ldots,n\).
The default value (constant=1.048358) ensures the Fisher-consistency under the normal distribution.
Note that
\(\code{constant}=1/\{\sqrt{2}\,\Phi^{-1}(3/4)\}\approx 1.048358\).
The unbiased Shamos is defined as
$$\textrm{Shamos}=\code{constant}\times\mathop{\mathrm{median}}_{i < j} \big(|X_i-X_j|\big)/{c_6(n)}$$
for \(i,j=1,2,\ldots,n\), where
\(c_6(n)\) is the finite-sample unbiasing factor.
Note that \(c_6(n)\) notation is used in Park et. al (2022), and
\(c_6(n)\) 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 \(w_6(n)\) where
\(w_6(n)\) is the finite-sample unbiasing factor.
Note that \(w_6(n)\) notation is used in Park et. al (2022), and
\(w_6(n)\) 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 (\(\sigma^2\)) under the normal distribution, but
it is not unbiased with a sample of finite size.
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.
tools:::Rd_expr_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.
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 (\(\sigma\)) under the normal distribution.
w4.factor{rQCC} for finite-sample unbiasing factor for the squared Shamos estimator
of the variance (\(\sigma^2\)) under the normal distribution.
finite.breakdown{rQCC} for calculating the finite-sample breakdown point.
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)
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