In statistics and probability theory, the nonparametric skew is a statistic occasionally used with random variables that take real values.Arnold BC, Groeneveld RA (1995) Measuring skewness with respect to the mode. The American Statistician 49 (1) 34–38 DOI:10.1080/00031305.1995.10476109Rubio F.J.; Steel M.F.J. (2012) "On the Marshall–Olkin transformation as a skewing mechanism". Computational Statistics & Data Analysis Preprint It is a measure of the skewness of a random variable's distribution—that is, the distribution's tendency to "lean" to one side or the other of the mean. Its calculation does not require any knowledge of the form of the underlying distribution—hence the name nonparametric. It has some desirable properties: it is zero for any symmetric distribution; it is unaffected by a scale shift; and it reveals either left- or right-skewness equally well. In some statistical samples it has been shown to be less powerfulTabor J (2010) Investigating the Investigative Task: Testing for skewness - An investigation of different test statistics and their power to detect skewness. J Stat Ed 18: 1–13 than the usual measures of skewness in detecting departures of the population from normality.
Properties
Definition
The nonparametric skew is defined as
where the mean (μ), median (ν) and standard deviation (σ) of the population have their usual meanings.
Properties
The nonparametric skew is one third of the Pearson 2 skewness coefficient and lies between −1 and +1 for any distribution.Hotelling H, Solomons LM (1932) The limits of a measure of skewness. Annals Math Stat 3, 141–114Garver (1932) Concerning the limits of a mesuare of skewness. Ann Math Stats 3(4) 141–142 This range is implied by the fact that the mean lies within one standard deviation of any median.O’Cinneide CA (1990) The mean is within one standard deviation of any median. Amer Statist 44, 292–293
Under an affine transformation of the variable (X), the value of S does not change except for a possible change in sign. In symbols
where a ≠ 0 and b are constants and S( X ) is the nonparametric skew of the variable X.
Sharper bounds
The bounds of this statistic ( ±1 ) were sharpened by MajindarMajindar KN (1962) "Improved bounds on a measure of skewness". Annals of Mathematical Statistics, 33, 1192–1194 who showed that its absolute value is bounded by
with
and
where X is a random variable with finite variance, E() is the expectation operator and Pr() is the probability of the event occurring.
When p = q = 0.5 the absolute value of this statistic is bounded by 1. With p = 0.1 and p = 0.01, the statistic's absolute value is bounded by 0.6 and 0.199 respectively.
Extensions
It is also known thatMallows CCC, Richter D (1969) "Inequalities of Chebyschev type involving conditional expectations". Annals of Mathematical Statistics, 40:1922–1932
where ν0 is any median and E(.) is the expectation operator.
It has been shown that
where xq is the qth quantile. Quantiles lie between 0 and 1: the median (the 0.5 quantile) has q = 0.5. This inequality has also been used to define a measure of skewness.Dziubinska R, Szynal D (1996) On functional measures of skewness. Applicationes Mathematicae 23(4) 395–403
This latter inequality has been sharpened further.Dharmadhikari SS (1991) Bounds on quantiles: a comment on O'Cinneide. The Am Statist 45: 257-58
Another extension for a distribution with a finite mean has been published:Gilat D, Hill TP(1993) Quantile-locating functions and the distance between the mean and quantiles. Statistica Neerlandica 47 (4) 279–283 DOI: 10.1111/j.1467-9574.1993.tb01424.x
The bounds in this last pair of inequalities are attained when and for fixed numbers a < b.
Finite samples
For a finite sample with sample size n ≥ 2 with xr is the rth order statistic, m the sample mean and s the sample standard deviation corrected for degrees of freedom,David HA (1991) Mean minus median: A comment on O'Cinneide. The Am Statist 45: 257
Replacing r with n / 2 gives the result appropriate for the sample median:Joarder AH, Laradji A (2004) Some inequalities in descriptive statistics. Technical Report Series TR 321
where a is the sample median.
Statistical tests
Hotelling and Solomons considered the distribution of the test statistic
where n is the sample size, m is the sample mean, a is the sample median and s is the sample's standard deviation.
Statistical tests of D have assumed that the null hypothesis being tested is that the distribution is symmetric .
Gastwirth estimated the asymptotic variance of n−1/2D.Gastwirth JL (1971) "On the sign test for symmetry". Journal of the American Statistical Association 66:821–823 If the distribution is unimodal and symmetric about 0, the asymptotic variance lies between 1/4 and 1. Assuming a conservative estimate (putting the variance equal to 1) can lead to a true level of significance well below the nominal level.
Assuming that the underlying distribution is symmetric Cabilio and Masaro have shown that the distribution of S is asymptotically normal.Cabilio P, Masaro J (1996) "A simple test of symmetry about an unknown median". Canadian Journal of Statistics-Revue Canadienne De Statistique, 24:349–361 The asymptotic variance depends on the underlying distribution: for the normal distribution, the asymptotic variance of S is 0.5708...
Assuming that the underlying distribution is symmetric, by considering the distribution of values above and below the median Zheng and Gastwirth have argued thatZheng T, Gastwirth J (2010) "On bootstrap tests of symmetry about an unknown median". Journal of Data Science, 8(3): 413–427
where n is the sample size, is distributed as a t distribution.
Related statistics
Antonietta Mira studied the distribution of the difference between the mean and the median.Mira A (1999) "Distribution-free test for symmetry based on Bonferroni’s measure", Journal of Applied Statistics, 26:959–972
where m is the sample mean and a is the median. If the underlying distribution is symmetrical γ1 itself is asymptotically normal. This statistic had been earlier suggested by Bonferroni.Bonferroni CE (1930) Elementi di statistica generale. Seeber, Firenze
Assuming a symmetric underlying distribution, a modification of S was studied by Miao, Gel and Gastwirth who modified the standard deviation to create their statistic.Miao W, Gel YR, Gastwirth JL (2006) "A new test of symmetry about an unknown median". In: Hsiung A, Zhang C-H, Ying Z, eds. Random Walk, Sequential Analysis and Related Topics — A Festschrift in honor of Yuan-Shih Chow. World Scientific; Singapore
where Xi are the sample values, || is the absolute value and the sum is taken over all n sample values.
The test statistic was
The scaled statistic T is asymptotically normal with a mean of zero for a symmetric distribution. Its asymptotic variance depends on the underlying distribution: the limiting values are, for the normal distribution = 0.5708... and, for the t distribution with three degrees of freedom, = 0.9689...
Values for individual distributions
Symmetric distributions
For symmetric probability distributions the value of the nonparametric skew is 0.
Asymmetric distributions
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 approximationKerman J (2011) "A closed-form approximation for the median of the beta distribution".
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.Kaas R, Buhrman JM (1980) Mean, median and mode in binomial distributions. Statistica Neerlandica 34 (1) 13–18 It is bounded by ±min{ max{ p, 1 − p }, loge2 } / σ where σ is the standard deviation of the binomial distribution.Hamza K (1995) "The smallest uniform upper bound on the distance between the mean and the median of the binomial and Poisson distributions". Statistics and Probability Letters, 23 (1) 21–25 Burr distribution: Birnbaum–Saunders distribution:
where α is the shape parameter and β is the location parameter.
Cantor distribution: despite the distribution being symmetric about its mean of , the median can be any value in as this central interval has zero probability
Chi square distribution: Although S ≥ 0 its value depends on the numbers of degrees of freedom (k).
Dagum distribution: Exponential distribution:
Exponential distribution with two parameters:
Exponential-logarithmic distribution
Here S is always > 0.
Exponentially modified Gaussian distribution:
F distribution with n and n degrees of freedom ( n > 4 ):Terrell GR (1986) "Pearson's rule for sample medians". Technical Report 86-2
Fréchet distribution: The variance of this distribution is defined only for α > 2.
Gamma distribution: The median can only be determined approximately for this distribution.Banneheka BMSG, Ekanayake GEMUPD (2009) A new point estimator for the median of Gamma distribution. Viyodaya J Science 14:95–103 If the shape parameter α is ≥ 1 then
where β > 0 is the rate parameter. Here S is always > 0.
Generalized normal distribution version 2
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.
Gumbel distribution:
where γ is Euler's constant.Ferguson T. "Asymptotic Joint Distribution of Sample Mean and a Sample Quantile", Unpublished
Half-normal distribution:
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 / .
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 )
Log-Weibull distribution:
Lomax distribution: S is defined only for α > 2
Maxwell–Boltzmann distribution:
Nakagami distribution
Pareto distribution: for α > 2 where α is the shape parameter of the distribution,
and S is always > 0.
Poisson distribution:
where λ is the parameter of the distribution.Choi KP (1994) "On the medians of Gamma distributions and an equation of Ramanujan". Proc Amer Math Soc 121 (1) 245–251
Rayleigh distribution:
Weibull distribution:
where k is the shape parameter of the distribution. Here S is always > 0.
History
In 1895 Pearson first suggested measuring skewness by standardizing the difference between the mean and the mode,Pearson K (1895) Contributions to the Mathematical Theory of Evolution–II. Skew variation in homogenous material. Phil Trans Roy Soc A. 186: 343–414 giving
where μ, θ and σ is the mean, mode and standard deviation of the distribution respectively. Estimates of the population mode from the sample data may be difficult but the difference between the mean and the mode for many distributions is approximately three times the difference between the mean and the medianStuart A, Ord JK (1994) Kendall’s advanced theory of statistics. Vol 1. Distribution theory. 6th Edition. Edward Arnold, London which suggested to Pearson a second skewness coefficient:
where ν is the median of the distribution. Bowley dropped the factor 3 from this formula in 1901 leading to the nonparametric skew statistic.