medcouple {mrfDepth}

A robust measure of skewness for univariate data

Description

Computes the medcouple, a robust measure of skewness for univariate data. For multivariate data the medcouple is computed on each column of the data matrix.

Usage

medcouple(x, do.reflect = NULL)

Arguments

x

An n by p data matrix.

do.reflect

Logical indicating whether the medcouple should also be computed on the reflected sample -x, with final result (mc(x)-mc(-x))/2.
Defaults TRUE when n<=100 and to FALSE otherwise.

Details

The medcouple is a robust measure of skewness yielding values between -1 and 1. For left- and right-skewed data the medcouple is negative and positive respectively.

The medcouple is defined as the median of the kernel function h(x_i,x_j) = \frac{(x_j - med(x)) - (med(x)-x_i)}{x_j-x_i} evaluated over all couples (x_i,x_j) where x_i is smaller than the median of x and x_j larger than the median of x. When there are multiple observations tied to the median, the kernel is defined separately as the denominator is not defined for these observations. Let m_1 < ... < m_k denote the indices of the observations which are tied to the median. Then h(x_{m_i},x_{m_j}) is defined to equal -1 if i + j - 1 < k, 0 when i + j - 1 = k and +1 if i + j - 1 > k. To compute the medcouple an algorithm with time complexity O(n log(n)) is applied. For details, see https://en.wikipedia.org/wiki/Medcouple.

For numerical accuracy it is advised, for small data sets, to compute the medcouple on both x and -x. The final value of the medcouple may then be obtained as a linear combination of both calculations. This procedure is warranted by the properties of the medcouple. Indeed the medcouple of the distribution X equals minus the medcouple of the reflected distribution -X. Moreover the medcouple is location and scale invariant.

Note that missing values are not allowed.

Value

mc

A p-vector containing the medcouple of each column of the data matrix x.

Author(s)

P. Segaert with original code from M. Maechler and G. Brys.

References

Brys G., Hubert M., Struyf A. (2004). A robust measure of skewness. Journal of Computational and Graphical Statistics, 13, 996–1017.

Examples

# Calculate the medcouple of a bivariate data set.
# Note that the medcouple of each variable is returned.
data(bloodfat)
medcouple(bloodfat)

# For smaller data sets it is advisable to compute
# the medcouple on both the sample and the reflected sample.
small.data <- bloodfat[1:25,]
medcouple(small.data, do.reflect = FALSE)
-medcouple(-small.data, do.reflect = FALSE)
# Small difference are due to numerical instabilities. 
# Use the option do.reflect to increase expected accuracy. 
medcouple(small.data, do.reflect = TRUE)