Skewness of Pareto Distribution
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Theorem
Let $X$ be a continuous random variable with the Pareto distribution with $a, b \in \R_{> 0}$.
Then the skewness $\gamma_1$ of $X$ is given by:
- $\gamma_1 = \begin {cases} \paren {\sqrt {\dfrac {\paren {a - 2} } a } } \paren { \dfrac {2 \paren {a + 1} } {a - 3} } & 3 < a \\ \text {does not exist} & 3 \ge a \end {cases}$
Proof
From Skewness in terms of Non-Central Moments, we have:
- $\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$
where:
- $\mu$ is the expectation of $X$.
- $\sigma$ is the standard deviation of $X$.
By Expectation of Pareto Distribution we have:
- $\mu = \dfrac {a b } {\paren {a - 1} }$
By Variance of Pareto Distribution we have:
- $\sigma = \dfrac {\sqrt a b } {\sqrt {\paren {a - 2} } \paren {a - 1} }$
From Raw Moment of Pareto Distribution, we have:
- $\expect {X^3} = \begin {cases} \dfrac {a b^3} {a - 3} & 3 < a \\ \text {does not exist} & 3 \ge a \end {cases}$
So:
\(\ds \gamma_1\) | \(=\) | \(\ds \frac {\paren {\dfrac {a b^3} {a - 3} } - 3 \paren {\dfrac {a b } {\paren {a - 1} } } \paren {\dfrac {a b^2 } {\paren {a - 2} \paren {a - 1}^2 } } - \paren {\dfrac {a b } {\paren {a - 1} } }^3} {\paren {\dfrac {\sqrt a b } {\sqrt {\paren {a - 2} } \paren {a - 1} } }^3 }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {\dfrac {a b^3} {a - 3} } \paren {\dfrac {\paren {a - 1} } {\paren {a - 1} } }^3 - 3 \paren {\dfrac {a^2 b^3 } {\paren {a - 2} \paren {a - 1}^3 } } - \paren {\dfrac {a b } {\paren {a - 1} } }^3} {\paren {\dfrac {\sqrt a b } {\sqrt {\paren {a - 2} } \paren {a - 1} } }^3 }\) | multiplying by $1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {\dfrac 1 {a - 3} } \paren { a - 1}^3 - 3 \paren {\dfrac a {\paren {a - 2} } } - a^2} {\paren {\dfrac {\sqrt a } {\paren {a - 2} \sqrt {\paren {a - 2} } } } }\) | canceling $\dfrac {a b^3} {\paren {a - 1}^3}$ from numerator and denominator | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\sqrt {\dfrac {\paren {a - 2} } a } } \paren {\paren {\dfrac {a - 2} {a - 3} } \paren { a - 1}^3 - 3 a - a^2 \paren {a - 2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\sqrt {\dfrac {\paren {a - 2} } a } } \paren {\paren {\dfrac {a - 2} {a - 3} } \paren { a^3 - 3 a^2 + 3 a - 1} - 3 a - a^3 + 2 a^2 }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\sqrt {\dfrac {\paren {a - 2} } a } } \paren {\paren {\dfrac {a - 2} {a - 3} } \paren { a^3 - 3 a^2 + 3 a - 1} - \paren {3 a + a^3 - 2 a^2} \paren {\dfrac {a - 3} {a - 3} } }\) | multiplying by $1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\sqrt {\dfrac {\paren {a - 2} } a } } \paren {\dfrac {a^4 - 3 a^3 + 3 a^2 - a - 2 a^3 + 6 a^2 - 6a + 2 - 3 a^2 - a^4 + 2 a^3 + 9a + 3 a^3 - 6 a^2} {a - 3} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\sqrt {\dfrac {\paren {a - 2} } a } } \paren {\dfrac {\paren {1 - 1} a^4 + \paren {-3 - 2 + 2 + 3} a^3 + \paren {3 + 6 - 3 - 6} a^2 + \paren {-1 - 6 + 9} a + 2} {a - 3} }\) | grouping terms | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\sqrt {\dfrac {\paren {a - 2} } a } } \paren { \dfrac {2 \paren {a + 1} } {a - 3} }\) |
$\blacksquare$