Nonparametric skew
In statistics and probability theory, the nonparametric skew is a statistic occasionally used with random variables that take real values.^{[1]}^{[2]} 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. Although its use has been mentioned in older textbooks^{[3]}^{[4]} it appears to have gone out of fashion. In statistical samples it has been shown to be less powerful^{[5]} than the usual measures of skewness in detecting departures of the population from normality.^{[6]}
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.^{[7]}^{[8]} This range is implied by the fact that the mean lies within one standard deviation of any median.^{[9]}
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 Majindar^{[10]} 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 that^{[11]}
where ν_{0} is any median and E(.) is the expectation operator.
It has been shown that
where x_{q} is the q^{th} quantile.^{[9]} 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.^{[12]}
This latter inequality has been sharpened further.^{[13]}
Another extension for a distribution with a finite mean has been published:^{[14]}
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 x_{r} is the r^{th} order statistic, m the sample mean and s the sample standard deviation corrected for degrees of freedom,^{[15]}
Replacing r with n / 2 gives the result appropriate for the sample median:^{[16]}
where a is the sample median.
Statistical tests
Hotelling and Solomons considered the distribution of the test statistic^{[7]}
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^{−1/2}D.^{[17]} 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.^{[18]} The asymptotic variance depends on the underlying distribution: for the normal distribution, the asymptotic variance of ( S√n ) 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 that^{[19]}
where n is the sample size, is distributed as a t distribution.
Related statistics
Mira studied the distribution of the difference between the mean and the median.^{[20]}
where m is the sample mean and a is the median. If the underlying distribution is symmetrical γ_{1} itself is asymptotically normal. This statistic had been earlier suggested by Bonferroni.^{[21]}
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.^{[22]}
where X_{i} 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√n ) 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 var( T√n ) = 0.5708 and, for the t distribution with three degrees of freedom, var( T√n ) = 0.9689.^{[22]}
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 approximation^{[23]}
- 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.^{[24]} It is bounded by ±min{ max{ p, 1 − p }, log_{e}2 } / σ where σ is the standard deviation of the binomial distribution.^{[25]}
- Burr distribution:
- Birnbaum–Saunders distribution:
- where α is the shape parameter and β is the location parameter.
- Chi square distribution: Although S ≥ 0 its value depends on the numbers of degrees of freedom (k).
- Exponential distribution with two parameters:^{[26]}
- Here S is always > 0.
- F distribution with n and n degrees of freedom ( n > 4 ):^{[27]}
- 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.^{[28]} 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.
- where γ is Euler's constant.^{[29]}
- Half-normal distribution:^{[26]}
- 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.
- 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 ( σ^{2} )
- Log-Weibull distribution:^{[26]}
- Lomax distribution: S is defined only for α > 2
- Maxwell–Boltzmann distribution:^{[26]}
- Pareto distribution: for α > 2 where α is the shape parameter of the distribution,
- and S is always > 0.
- where λ is the parameter of the distribution.^{[30]}
- 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,^{[31]} 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 median^{[32]} which suggested to Pearson a second skewness coefficient:
where ν is the median of the distribution. Bowley dropped the factor 3 is from this formula in 1901 leading to the nonparametric skew statistic.
The relationship between the median, the mean and the mode was first noted by Pearson when he was investigating his type III distributions.
Relationships between the mean, median and mode
For an arbitrary distribution the mode, median and mean may appear in any order.^{[33]}^{[34]}^{[35]}
Analyses have been made of some of the relationships between the mean, median, mode and standard deviation.^{[36]} and these relationships place some restrictions of the sign and magnitude of the nonparametric skew.
A simple example illustrating these relationships is the binomial distribution with n = 10 and p = 0.09.^{[37]} This distribution when plotted has a long right tail. The mean (0.9) is to the left of the median (1) but the skew (0.906) as defined by the third standardized moment is positive. In contrast the nonparametric skew is -0.110.
Pearson's rule
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^{[31]}
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.^{[38]} 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,
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.^{[39]} Haldane noted that in 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.^{[40]}
Doodson's relationship was studied by Kendall and Stuart in the log-normal distribution for which they found an exact relationship close to it.^{[41]}
Hall also showed that for a distribution with regularly varying tails and exponent α that^{[40]}
Unimodal distributions
Gauss showed in 1823 that for a unimodal distribution^{[42]}
and
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.^{[43]} 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
and for these negatively skewed unimodal distributions
This class includes the important F, beta and gamma distributions.
This rule does not hold for the unimodal Weibull distribution.^{[44]}
For a unimodal distribution the following bounds are known and are sharp:^{[45]}
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.
van Zwet condition
The following inequality,
where θ, ν and µ is the mode, median and mean of the distribution respectively, holds if
where F is the cumulative distribution function of the distribution.^{[46]} These conditions have since been generalised^{[35]} and extended to discrete distributions.^{[47]} Any distribution for which this holds has either a zero or a positive nonparametric skew.
Notes
Ordering of skewness
In 1964 van Zwet proposed a series of axioms for ordering measures of skewness.^{[48]} The nonparametric skew does not satisfy these axioms.
Benford's law
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.^{[49]}
Relation to Bowley's coefficient
This statistic can be derived from Bowley's coefficient of skewness^{[50]}
where Q_{i} is the ith quartile of the distribution.
Hinkley generalised this^{[51]}
where lies between 0 and 0.5. Bowley's coefficient is a special case with equal to 0.25.
Groeneveld and Meeden^{[52]} removed the dependence on by integrating over it.
The denominator is a measure of dispersion. Replacing the denominator with the standard deviation we obtain the nonparametric skew.
References
- ↑ Arnold BC, Groeneveld RA (1995) Measuring skewness with respect to the mode. The American Statistician 49 (1) 34–38 DOI:10.1080/00031305.1995.10476109
- ↑ Rubio F.J.; Steel M.F.J. (2012) "On the Marshall–Olkin transformation as a skewing mechanism". Computational Statistics & Data Analysis Preprint
- ↑ Yule G.U.; Kendall M.G. (1950) An Introduction to the Theory of Statistics. 3rd edition. Harper Publishing Company pp 162–163
- ↑ Hildebrand DK (1986) Statistical thinking for behavioral scientists. Boston: Duxbury
- ↑ Tabor 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
- ↑ Doane, David P.; Seward, Lori E. (2011). "Measuring Skewness: A Forgotten Statistic?" (PDF). Journal of Statistics Education. 19 (2).
- 1 2 Hotelling H, Solomons LM (1932) The limits of a measure of skewness. Annals Math Stat 3, 141–114
- ↑ Garver (1932) Concerning the limits of a mesuare of skewness. Ann Math Stats 3(4) 141–142
- 1 2 O’Cinneide CA (1990) The mean is within one standard deviation of any median. Amer Statist 44, 292–293
- ↑ Majindar KN (1962) "Improved bounds on a measure of skewness". Annals of Mathematical Statistics, 33, 1192–1194 doi:10.1214/aoms/1177704482
- ↑ Mallows CCC, Richter D (1969) "Inequalities of Chebyschev type involving conditional expectations". Annals of Mathematical Statistics, 40:1922–1932
- ↑ Dziubinska R, Szynal D (1996) On functional measures of skewness. Applicationes Mathematicae 23(4) 395–403
- ↑ Dharmadhikari SS (1991) Bounds on on quantiles: a comment on O'Cinneide. The Am Statist 45: 257-58
- ↑ 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
- ↑ David HA (1991) Mean minus median: A comment on O'Cinneide. The Am Statist 45: 257
- ↑ Joarder AH, Laradji A (2004) Some inequalities in descriptive statistics. Technical Report Series TR 321
- ↑ Gastwirth JL (1971) "On the sign test for symmetry". Journal of the American Statistical Association 66:821–823
- ↑ Cabilio P, Masaro J (1996) "A simple test of symmetry about an unknown median". Canandian Journal of Statistics-Revue Canadienne De Statistique, 24:349–361
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- ↑ Bonferroni CE (1930) Elementi di statistica generale. Seeber, Firenze
- 1 2 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
- ↑ Kerman J (2011) "A closed-form approximation for the median of the beta distribution". arXiv:1111.0433v1
- ↑ Kaas R, Buhrman JM (1980) Mean, median and mode in binomial distributions. Statistica Neerlandica 34 (1) 13–18
- ↑ 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
- 1 2 3 4 http://web.ipac.caltech.edu/staff/fmasci/home/statistics_refs/UsefulDistributions.pdf
- ↑ Terrell GR (1986) "Pearson's rule for sample medians". Technical Report 86-2
- ↑ Banneheka BMSG, Ekanayake GEMUPD (2009) A new point estimator for the median of Gamma distribution. Viyodaya J Science 14:95–103
- ↑ Ferguson T. "Asymptotic Joint Distribution of Sample Mean and a Sample Quantile", Unpublished
- ↑ Choi KP (1994) "On the medians of Gamma distributions and an equation of Ramanujan". Proc Amer Math Soc 121 (1) 245–251
- 1 2 Pearson K (1895) Contributions to the Mathematical Theory of Evolution–II. Skew variation in homogenous material. Phil Trans Roy Soc A. 186: 343–414
- ↑ Stuart A, Ord JK (1994) Kendall’s advanced theory of statistics. Vol 1. Distribution theory. 6th Edition. Edward Arnold, London
- ↑ Relationship between the mean, median, mode, and standard deviation in a unimodal distribution
- ↑ von Hippel, Paul T. (2005) "Mean, Median, and Skew: Correcting a Textbook Rule", Journal of Statistics Education, 13(2)
- 1 2 Dharmadhikari SW, Joag-dev K (1983) Mean, Median, Mode III. Statistica Neerlandica, 33: 165–168
- ↑ Bottomly, H.(2002,2006) "Relationship between the mean, median, mode, and standard deviation in a unimodal distribution" Personal webpage
- ↑ Lesser LM (2005)."Letter to the editor" , [comment on von Hippel (2005)]. Journal of Statistics Education 13(2).
- ↑ Doodson AT (1917) "Relation of the mode, median and mean in frequency functions". Biometrika, 11 (4) 425–429 doi:10.1093/biomet/11.4.425
- ↑ Haldane JBS (1942) "The mode and median of a nearly normal distribution with given cumulants". Biometrika, 32: 294–299
- 1 2 Hall P (1980) "On the limiting behaviour of the mode and median of a sum of independent random variables". Annals of Probability 8: 419–430
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- ↑ Gauss C.F. Theoria Combinationis Observationum Erroribus Minimis Obnoxiae. Pars Prior. Pars Posterior. Supplementum. Theory of the Combination of Observations Least Subject to Errors. Part One. Part Two. Supplement. 1995. Translated by G.W. Stewart. Classics in Applied Mathematics Series, Society for Industrial and Applied Mathematics, Philadelphia
- ↑ MacGillivray HL (1981) The mean, median, mode inequality and skewness for a class of densities. Aust J Stat 23(2) 247–250
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- ↑ Johnson NL, Rogers CA (1951) "The moment problem for unimodal distributions". Annals of Mathematical Statistics, 22 (3) 433–439
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- ↑ van Zwet, W.R. (1964) "Convex transformations of random variables". Mathematics Centre Tract, 7, Mathematisch Centrum, Amsterdam
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