Nonparametric skew

The nonparametric skew is defined as

${\displaystyle S={\frac {\mu -\nu }{\sigma }}}$

where the mean (µ), median (ν) and standard deviation (σ) of the population have their usual meanings.

The nonparametric skew is one third of the Pearson 2 skewness coefficient and lies between −1 and +1 for any distribution.[5][6] This range is implied by the fact that the mean lies within one standard deviation of any median.[7]

Under an affine transformation of the variable (X), the value of S does not change except for a possible change in sign. In symbols

${\displaystyle S(aX+b)=\operatorname {sign} (a)\,S(X)}$

where a ≠ 0 and b are constants and S( X ) is the nonparametric skew of the variable X.

For a finite sample with sample size n ≥ 2 with x_r is the r^th order statistic, m the sample mean and s the sample standard deviation corrected for degrees of freedom,[13]

${\displaystyle {\frac {|m-x_{r}|}{s}}\leq {\text{max}}\left[{\sqrt {\frac {(n-1)(r-1)}{n(n-r+1)}}},{\sqrt {\frac {(n-1)(n-r)}{nr}}}\right]}$

Replacing r with n / 2 gives the result appropriate for the sample median:[14]

${\displaystyle {\frac {|m-a|}{s}}\leq {\sqrt {\frac {n^{2}-n}{n^{2}}}}={\sqrt {\frac {n-1}{n}}}}$

where a is the sample median.

For symmetric probability distributions the value of the nonparametric skew is 0.

It is positive for right skewed distributions and negative for left skewed distributions. Absolute values ≥ 0.2 indicate marked skewness.

It may be difficult to determine S for some distributions. This is usually because a closed form for the median is not known: examples of such distributions include the gamma distribution, inverse-chi-squared distribution, the inverse-gamma distribution and the scaled inverse chi-squared distribution.

The following values for S are known:

• Beta distribution: 1 < α < β where α and β are the parameters of the distribution, then to a good approximation[21]

${\displaystyle S={\frac {1}{3}}{\frac {(\alpha -2\beta )(\alpha +\beta +1)^{1/2}}{(\alpha +\beta -2/3)(\alpha \beta )^{1/2}}}}$

If 1 < β < α then the positions of α and β are reversed in the formula. S is always < 0.

• Binomial distribution: varies. If the mean is an integer then S = 0. If the mean is not an integer S may have either sign or be zero.[22] It is bounded by ±min{ max{ p, 1 − p }, log_e2 } / σ where σ is the standard deviation of the binomial distribution.[23]
• Burr distribution:
• Birnbaum–Saunders distribution:

${\displaystyle S={\frac {2}{\beta ^{2}(4+5\alpha ^{2})}}}$

where α is the shape parameter and β is the location parameter.

• Cantor distribution:

${\displaystyle {\frac {-4}{3}}\leq S\leq {\frac {4}{3}}}$

• Chi square distribution: Although S ≥ 0 its value depends on the numbers of degrees of freedom (k).

${\displaystyle S\approx {\frac {1-(1-{\frac {2}{k}})^{3}}{2}}}$

• Dagum distribution:
• Exponential distribution:

${\displaystyle S=1-\log _{e}(2)\approx 0.31}$

• Exponential distribution with two parameters:[24]

${\displaystyle S=1-\log _{e}(2)\approx 0.31}$

• Exponential-logarithmic distribution

${\displaystyle S=-{\frac {polylog(2,1-p)+\ln(1+{\sqrt {p}})\ln p}{\sqrt {-[2polylog(3,1-p)+polylog^{2}(2,1-p)]}}}}$

Here S is always > 0.

• Exponentially modified Gaussian distribution:

${\displaystyle 0\leq S\leq 1-\log _{e}(2)}$

• F distribution with n and n degrees of freedom ( n > 4 ):[25]

${\displaystyle S=n^{-3/2}{\sqrt {\frac {n-4}{n-2}}}+O(n^{-5/2})}$

• Fréchet distribution: The variance of this distribution is defined only for α > 2.

${\displaystyle S={\frac {\Gamma \left(1-{\frac {1}{\alpha }}\right)-{\frac {1}{{\sqrt {\alpha }}\log _{e}(2)}}}{\sqrt {\Gamma \left(1-{\frac {2}{\alpha }}\right)-\left(\Gamma \left(1-{\frac {1}{\alpha }}\right)\right)^{2}}}}}$

• Gamma distribution: The median can only be determined approximately for this distribution.[26] If the shape parameter α is ≥ 1 then

${\displaystyle S\approx {\frac {\beta }{3\alpha +0.2}}}$

where β > 0 is the rate parameter. Here S is always > 0.

• Generalized normal distribution version 2

${\displaystyle S=-{\frac {\exp({\frac {-k^{2}}{2}})-1}{\sqrt {\exp({\frac {k^{2}}{2}})-1}}}}$

S is always < 0.

• Generalized Pareto distribution: S is defined only when the shape parameter ( k ) is < 1/2. S is < 0 for this distribution.

${\displaystyle S=\left({\frac {2^{k}-1}{k}}-2^{k}\right)(1-2k)^{0.5}}$

• Gumbel distribution:

${\displaystyle {\frac {{\sqrt {6}}[\gamma +\log _{e}(\log _{e}(2))]}{\pi }}\approx 0.1643}$

where γ is Euler's constant.[27]

• Half-normal distribution:[24]

${\displaystyle S\approx {\frac {{\sqrt {2}}-0.6745{\sqrt {\pi }}}{\sqrt {\pi -2}}}\approx 0.36279}$

• Kumaraswamy distribution
• Log-logistic distribution (Fisk distribution): Let β be the shape parameter. The variance and mean of this distribution are only defined when β > 2. To simplify the notation let b = β / π.

${\displaystyle S={\frac {b-\sin(b)}{\sqrt {b\tan(b)-b^{2}}}}}$

The standard deviation does not exist for values of b > 4.932 (approximately). For values for which the standard deviation is defined, S is > 0.

• Log-normal distribution: With mean ( μ ) and variance ( σ² )

${\displaystyle S={\frac {1}{(e^{\frac {\sigma ^{2}}{2}}+1)(e^{\mu +\sigma ^{2}})}}}$

• Log-Weibull distribution:[24]

${\displaystyle S\approx {\frac {[\log _{e}(\log _{e}(2))-0.5772]{\sqrt {6}}}{\pi }}\approx -0.1643}$

• Lomax distribution: S is defined only for α > 2

${\displaystyle S={\frac {(\alpha -1)(\alpha -2)(1-(\alpha -1)(2^{1/\alpha }-1))}{\alpha ^{1/2}}}}$

• Maxwell–Boltzmann distribution:[24]

${\displaystyle S\approx {\frac {{\sqrt {2}}-1.5382\Gamma ({\frac {3}{2}})}{\sqrt {2(\Gamma ({\frac {5}{2}})-\Gamma ({\frac {3}{2}}))}}}\approx 0.0854}$

• Nakagami distribution

${\displaystyle S=-1}$

• Pareto distribution: for α > 2 where α is the shape parameter of the distribution,

${\displaystyle S=(\alpha -2^{1/\alpha }[\alpha -1])({\frac {\alpha -2}{\alpha }})^{1/2},}$

and S is always > 0.

• Poisson distribution:

${\displaystyle {\frac {-\log _{e}(2)}{\lambda ^{\frac {1}{2}}}}\leq S\leq {\frac {1}{3\lambda ^{\frac {1}{2}}}}}$

where λ is the parameter of the distribution.[28]

• Rayleigh distribution:

${\displaystyle S={\sqrt {\frac {2}{4-\pi }}}[({\frac {\pi }{2}})^{0.5}-\log _{e}(4)]\approx 0.1251}$

• Weibull distribution:

${\displaystyle S={\frac {\Gamma (1+1/k)-\log _{e}(2)^{1/k}}{(\Gamma (1+2/k)-\Gamma (1+1/k))^{1/2}}},}$

where k is the shape parameter of the distribution. Here S is always > 0.

The rule that for some distributions the difference between the mean and the mode is three times that between the mean and the median is due to Pearson who discovered it while investigating his Type 3 distributions. It is often applied to slightly asymmetric distributions that resemble a normal distribution but it is not always true.

In 1895 Pearson noted that for what is now known as the gamma distribution that the relation[29]

${\displaystyle \nu -\theta =2(\mu -\nu )}$

where θ, ν and µ are the mode, median and mean of the distribution respectively was approximately true for distributions with a large shape parameter.

Doodson in 1917 proved that the median lies between the mode and the mean for moderately skewed distributions with finite fourth moments.[36] This relationship holds for all the Pearson distributions and all of these distributions have a positive nonparametric skew.

Doodson also noted that for this family of distributions to a good approximation,

${\displaystyle \theta =3\nu -2\mu ,}$

where θ, ν and µ are the mode, median and mean of the distribution respectively. Doodson's approximation was further investigated and confirmed by Haldane.[37] Haldane noted that samples with identical and independent variates with a third cumulant had sample means that obeyed Pearson's relationship for large sample sizes. Haldane required a number of conditions for this relationship to hold including the existence of an Edgeworth expansion and the uniqueness of both the median and the mode. Under these conditions he found that mode and the median converged to 1/2 and 1/6 of the third moment respectively. This result was confirmed by Hall under weaker conditions using characteristic functions.[38]

Doodson's relationship was studied by Kendall and Stuart in the log-normal distribution for which they found an exact relationship close to it.[39]

Hall also showed that for a distribution with regularly varying tails and exponent α that[38]

${\displaystyle \mu -\theta =\alpha (\mu -\nu )}$

Gauss showed in 1823 that for a unimodal distribution[40]

${\displaystyle \sigma \leq \omega \leq 2\sigma }$

and

${\displaystyle |\nu -\mu |\leq {\sqrt {\frac {3}{4}}}\omega ,}$

where ω is the root mean square deviation from the mode.

For a large class of unimodal distributions that are positively skewed the mode, median and mean fall in that order.[41] Conversely for a large class of unimodal distributions that are negatively skewed the mean is less than the median which in turn is less than the mode. In symbols for these positively skewed unimodal distributions

${\displaystyle \theta \leq \nu \leq \mu }$

and for these negatively skewed unimodal distributions

${\displaystyle \mu \leq \nu \leq \theta }$

This class includes the important F, beta and gamma distributions.

This rule does not hold for the unimodal Weibull distribution.[42]

For a unimodal distribution the following bounds are known and are sharp:[43]

${\displaystyle {\frac {|\theta -\mu |}{\sigma }}\leq {\sqrt {3}},}$

${\displaystyle {\frac {|\nu -\mu |}{\sigma }}\leq {\sqrt {0.6}},}$

${\displaystyle {\frac {|\theta -\nu |}{\sigma }}\leq {\sqrt {3}},}$

where μ,ν and θ are the mean, median and mode respectively.

The middle bound limits the nonparametric skew of a unimodal distribution to approximately ±0.775.

The following inequality,

${\displaystyle \theta \leq \nu \leq \mu ,}$

where θ, ν and µ is the mode, median and mean of the distribution respectively, holds if

${\displaystyle F(\nu -x)+F(\nu +x)\geq 1{\text{ for all }}x,}$

where F is the cumulative distribution function of the distribution.[44] These conditions have since been generalised[33] and extended to discrete distributions.[45] Any distribution for which this holds has either a zero or a positive nonparametric skew.

In 1964 van Zwet proposed a series of axioms for ordering measures of skewness.[46] The nonparametric skew does not satisfy these axioms.

Benford's law is an empirical law concerning the distribution of digits in a list of numbers. It has been suggested that random variates from distributions with a positive nonparametric skew will obey this law.[47]

This statistic is very similar to Bowley's coefficient of skewness[48]

${\displaystyle SK_{2}={\frac {Q_{3}+Q_{1}-2Q_{2}}{Q_{3}-Q_{1}}}}$

where Q_i is the ith quartile of the distribution.

Hinkley generalised this[49]

${\displaystyle SK={\frac {F^{-1}(1-\alpha )+F^{-1}(\alpha )-2Q_{2}}{Q_{3}-Q_{1}}}}$

where ${\displaystyle \alpha }$ lies between 0 and 0.5. Bowley's coefficient is a special case with ${\displaystyle \alpha }$ equal to 0.25.

Groeneveld and Meeden[50] removed the dependence on ???(is this \alpha? needs fixing)??? by integrating over it.

${\displaystyle SK_{3}={\frac {\mu -Q_{2}}{E|y-Q_{2}|}}}$

The denominator is a measure of dispersion. Replacing the denominator with the standard deviation we obtain the nonparametric skew.