Description Usage Arguments Details Value References See Also Examples

Compute Khattree-Bahuguna's Multivariate Skewness.

1 | ```
kbMvtSkew(x)
``` |

`x` |
a matrix of original observations. |

Let *\mathbf{X}=(X_1,…,X_p)'* be the multivariate random vector and *(X_{i_1}, X_{i_2}, …, X_{i_p})'* be one of the *p!* permutations of *(X_1,…,X_p)'*. We predict *X_{i_j}* conditionally on subvector *(X_{i_1}, …,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 ≥ 2*, we have

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

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

*\vdots*

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

We calculate the sample skewness *\hat{δ}_{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{Δ}_{i} = ∑_{j=1}^{p} \hat{δ}_{i_j}, i = 1, 2, …, P* for all *P = p!* permutations of *(X_1,…,X_p)'*. The sample Khattree-Bahuguna's multivariate skewness is defined as

*\hat{Δ} = \frac{1}{P} ∑_{i=1}^{P} \hat{Δ}_{i}.*

Clearly, *0 ≤ \hat{Δ} ≤ \frac{p}{2}*.

`kbMvtSkew`

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

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.

`kbSkew`

for Khattree-Bahuguna's univariate skewness.

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