2.5 Other Measures
2.5.1 Skewness
Skewness is a measure that assesses the asymmetry of a frequency distribution. There are several definitions in the literature, of which three alternatives are presented. \[\begin{equation} g_1 = \dfrac{m_3}{m_2^{3/2}} = \dfrac{\frac{1}{n} \sum_{i=1}^{n} (x_{i} - \bar{x}_n)^3}{\left[ \frac{1}{n} \sum_{i=1}^{n} (x_{i} - \bar{x}_n)^2 \right]^{3/2}} \tag{2.41} \end{equation}\]
\[\begin{equation} b_1 = g_{1} \left( \dfrac{n-1}{n} \right)^{3/2} = \dfrac{m_3}{s^3} = \dfrac{\frac{1}{n} \sum_{i=1}^{n} (x_{i} - \bar{x}_n)^3 }{\left[ \frac{1}{n-1} \sum_{i=1}^{n} (x_{i} - \bar{x}_n)^2 \right]^{3/2}} \tag{2.42} \end{equation}\]
\[\begin{eqnarray} G_1 = g_{1} \sqrt{\dfrac{n(n-1)}{n-2}} = b_{1} \dfrac{n^2}{(n-1)(n-2)} \tag{2.43} \end{eqnarray}\]
set.seed(1); x <- rnorm(100) # Generating 100 values N(0,1) with fixed seed
e1071::skewness(x, type = 1) # Classical definition of skewness, Equation (2.25)## [1] -0.0722319## [1] -0.07333656## [1] -0.071151132.5.2 Kurtosis
Kurtosis is a measure of the flatness of a frequency distribution. As with skewness, three alternatives are presented among the various definitions of kurtosis. \[\begin{eqnarray} g_2 = \dfrac{m_4}{m_2^{2}} - 3 = \dfrac{\frac{1}{n} \sum_{i=1}^{n} (x_{i} - \bar{x}_n)^4}{\left[ \frac{1}{n} \sum_{i=1}^{n} (x_{i} - \bar{x}_n)^2 \right]^{2}} - 3 \tag{2.44} \end{eqnarray}\]
\[\begin{eqnarray} b_2 = (g_2 + 3) \left( 1 - \dfrac{1}{n} \right)^{2} - 3 = \dfrac{m_4}{s^4} - 3 = \dfrac{\frac{1}{n} \sum_{i=1}^{n} (x_{i} - \bar{x}_n)^4 }{\left[ \frac{1}{n-1} \sum_{i=1}^{n} (x_{i} - \bar{x}_n)^2 \right]^{2}} - 3 \tag{2.45} \end{eqnarray}\]
\[\begin{eqnarray} G_2 = \dfrac{ \left[ (n+1) g_2 + 6 \right] (n-1)}{(n-2)(n-3)} \tag{2.46} \end{eqnarray}\]
set.seed(1); x <- rnorm(100) # Generating 100 N(0,1) values with fixed seed
e1071::kurtosis(x, type = 1) # Classical definition of kurtosis, Equation (2.28)## [1] 0.007653206## [1] 0.07053697## [1] -0.05219909