Compute Khattree-Bahuguna's Multivariate Skewness.
kbMvtSkew(x)a matrix of original observations.
kbMvtSkew computes the Khattree-Bahuguna's multivairate skewness for a \(p\)-dimensional data.
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, \(0 \le \hat{\Delta} \le \frac{p}{2}\).
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.
# NOT RUN {
# Compute Khattree-Bahuguna's multivairate skewness
data(OlymWomen)
kbMvtSkew(OlymWomen[, c("m800","m1500","m3000","marathon")])
# }
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