rQCC (version 0.19.8.2)

unbiased.squared.shamos: Unbiased squared Shamos estimate

Description

Calculates the unbiased squared Shamos estimate under the normal distribution which is adjusted by the Fisher-consistency and finite-sample correction factors.

Usage

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 standard normal distribution

na.rm

a logical value indicating whether NA values should be stripped before the computation proceeds.

IncludeEqual

FALSE (default) calculates median of \(|X_i-X_j|\) with \(i < j\), while TRUE calculates median of \(|X_i-X_j|\) with \(i \le j\).

Value

shamos2.unbiased returns a numeric value.

Details

The unbiased squared Shamos estimator is defined as the squared rQCC::shamos divided by \(w_6(n)\) where the finite-sample correction factor \(w_6(n)\) is calculated by rQCC::w6.for.shamos2. The default value (constant=1.048358) ensures the Fisher-consistency under the standard normal. 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.

References

Park, C., H. Kim, and M. Wang (2019). Finite-sample properties of robust location and scale estimators. arXiv:1908.00462.

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.

L<U+00E8>vy-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

rQCC::w6.for.shamos2 for finite-sample correction factor for the squared Shamos estimator for the variance (\(\sigma^2\)) under the normal distribution.

rQCC::shamos.unbiased for robust finite-sample unbiased Shamos estimator for the standard deviation (\(\sigma\)) of a normal distribution.

rQCC::finite.breakdown for calculating the finite-sample breakdown point.

Examples

# NOT RUN {
x = c(0:10, 50)

# Unbiased squared Shamos
shamos2.unbiased(x)

# Fisher-consistent squared Shamos
shamos(x)^2
# }