Description Usage Arguments Details Value References See Also Examples
Compute Khattree-Bahuguna's Multivariate Skewness.
1 | kbMvtSkew(x)
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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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