kbMvtSkew {KbMvtSkew}

Khattree-Bahuguna's Multivariate Skewness

Description

Compute Khattree-Bahuguna's Multivariate Skewness.

Usage

kbMvtSkew(x)

Arguments

x

a matrix of original observations.

Details

Let $\mathbf{X}=(X_1,\ldots,X_p)'$ be the multivariate random vector and $(X_{i_1}, X_{i_2}, \ldots, X_{i_p})'$ be one of the $p!$ permutations of $(X_1,\ldots,X_p)'$. We predict $X_{i_j}$ conditionally on subvector $(X_{i_1}, \ldots,X_{i_{j-1}})$ and compute the corresponding residual $V_{i_j}$ through a linear regression model for $j = 2, \cdots, p$. For $j=1$, we define $V_{i_1} = X_{i_1} - \bar{X}_{i_1}$, where $\bar{X}_{i_1}$ is the mean of $X_{i_1}$. For $j \ge 2$, we have

$\hat{X}_{i_2} = \hat{\beta}_0 + \hat{\beta}_1 X_{i_1}, \quad V_{i_2} = X_{i_2} - \hat{X}_{i_2}$

$\hat{X}_{i_3} = \hat{\beta}_0 + \hat{\beta}_1 X_{i_1} + \hat{\beta}_2 X_{i_2}, \quad V_{i_3} = X_{i_3} - \hat{X}_{i_3}$

$\vdots$

$\hat{X}_{i_p} = \hat{\beta}_0 + \hat{\beta}_1 X_{i_1} + \hat{\beta}_2 X_{i_2} + \cdots + \hat{\beta}_{p-1} X_{i_{p-1}}, \quad V_{i_p} = X_{i_p} - \hat{X}_{i_p}.$

We calculate the sample skewness $\hat{\delta}_{i_j}$ of $V_{i_j}$ by the sample Khattree-Bahuguna's univariate skewness formula (see details of kbSkew that follows) respectively for $j=1,\cdots,p$ and define $\hat{\Delta}_{i} = \sum_{j=1}^{p} \hat{\delta}_{i_j}, i = 1, 2, \ldots, P$ for all $P = p!$ permutations of $(X_1,\ldots,X_p)'$. The sample Khattree-Bahuguna's multivariate skewness is defined as

$\hat{\Delta} = \frac{1}{P} \sum_{i=1}^{P} \hat{\Delta}_{i}.$

Clearly, $0 \le \hat{\Delta} \le \frac{p}{2}$.

Value

kbMvtSkew computes the Khattree-Bahuguna's multivairate skewness for a $p$-dimensional data.

References

Khattree, R. and Bahuguna, M. (2019). An alternative data analytic approach to measure the univariate and multivariate skewness. International Journal of Data Science and Analytics, Vol. 7, No. 1, 1-16.

See Also

kbSkew for Khattree-Bahuguna's univariate skewness.

Examples

# Compute Khattree-Bahuguna's multivairate skewness

data(OlymWomen)
kbMvtSkew(OlymWomen[, c("m800","m1500","m3000","marathon")])