MAD {rQCC}

Median absolute deviation (MAD)

Description

Calculates the unbiased median absolute deviation (MAD) estimator and the unbiased squared MAD under the normal distribution which are adjusted by the Fisher-consistency and finite-sample unbiasing factors.

Usage

mad.unbiased (x, center = median(x), constant=1.4826, na.rm = FALSE)

mad2.unbiased(x, center = median(x), constant=1.4826, na.rm = FALSE)

Arguments

x

a numeric vector of observations.

center

pptionally, the center: defaults to the median.

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.

Details

The unbiased MAD (mad.unbiased) is defined as the mad{stats} divided by c_5(n), where c_5(n) is the finite-sample unbiasing factor. Note that c_5(n) notation is used in Park et. al (2022), and c_5(n) is calculated using the function c4.factor{rQCC} with estimator="mad" option. The default value (constant=1.4826) ensures the Fisher-consistency under the normal distribution. Note that the original MAD was proposed by Hampel (1974).

The unbiased squared MAD (mad2.unbiased) is defined as the squared mad{stats} divided by w_5(n) where w_5(n) is the finite-sample unbiasing factor. Note that w_5(n) notation is used in Park et. al (2022), and w_5(n) is calculated using the function w4.factor{rQCC} with estimator="mad2" option. The default value (constant=1.4826) ensures the Fisher-consistency under the normal distribution. Note that the square of the conventional MAD is Fisher-consistent for the variance (\sigma^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

Hampel, F. R. (1974). The influence curve and its role in robust estimation. Journal of the American Statistical Association, 69, 383–393.

See Also

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

w4.factor{rQCC} for finite-sample unbiasing factor for the variance under the normal distribution.

shamos{rQCC} for robust Fisher-consistent estimator of the standard deviation under the normal distribution.

shamos.unbiased{rQCC} for robust finite-sample unbiased estimator of the standard deviation under the normal distribution.

mad{stats} for calculating the sample MAD.

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

Examples

x = c(0:10, 50)

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

# Unbiased MAD.
mad.unbiased(x)

# Fisher-consistent squared MAD, but not unbiased.
mad(x)^2

# Unbiased squared MAD.
mad2.unbiased(x)