Skew normal distribution

Short description: Probability distribution

Skew Normal

Probability density function

Cumulative distribution function

Parameters|  ξ location (real)
ω scale (positive, real)
α shape (real)
Support|  x∈(−∞;+∞)
PDF|  2ω2πe−(x−ξ)22ω2∫−∞α(x−ξω)12πe−t22 dt
CDF|  Φ(x−ξω)−2T(x−ξω,α)
T(h,a) is Owen's T function
Mean|  ξ+ωδ2π where δ=α1+α2
Mode|  ξ+ωmo(α)
Variance|  ω2(1−2δ2π)
Skewness|  γ1=4−π2(δ2/π)3(1−2δ2/π)3/2
Kurtosis|  2(π−3)(δ2/π)4(1−2δ2/π)2
MGF|  MX(t)=2exp⁡(ξt+ω2t22)Φ(ωδt)
CF|  eitξ−t2ω2/2(1+iErfi(δωt2))

In probability theory and statistics, the skew normal distribution is a continuous probability distribution that generalises the normal distribution to allow for non-zero skewness.

## Contents

* 1 Definition
* 2 Estimation
* 3 Related distributions
* 4 See also
* 5 References
* 6 External links

## Definition

Let ϕ(x) denote the standard normal probability density function

ϕ(x)=12πe−x22

with the cumulative distribution function given by

Φ(x)=∫−∞xϕ(t) dt=12[1+erf⁡(x2)],

where "erf" is the error function. Then the probability density function (pdf) of the skew-normal distribution with parameter α is given by

f(x)=2ϕ(x)Φ(αx).

This distribution was first introduced by O'Hagan and Leonard (1976).[1] Alternative forms to this distribution, with the corresponding quantile function, have been given by Ashour and Abdel-Hamid[2] and by Mudholkar and Hutson.[3]

A stochastic process that underpins the distribution was described by Andel, Netuka and Zvara (1984).[4] Both the distribution and its stochastic process underpinnings were consequences of the symmetry argument developed in Chan and Tong (1986),[5] which applies to multivariate cases beyond normality, e.g. skew multivariate t distribution and others. The distribution is a particular case of a general class of distributions with probability density functions of the form f(x)=2ϕ(x)Φ(x) where ϕ(⋅) is any PDF symmetric about zero and Φ(⋅) is any CDF whose PDF is symmetric about zero.[6]

To add location and scale parameters to this, one makes the usual transform x->x−ξω. One can verify that the normal distribution is recovered when α=0, and that the absolute value of the skewness increases as the absolute value of α increases. The distribution is right skewed if α>0 and is left skewed if α<0. The probability density function with location ξ, scale ω, and parameter α becomes

f(x)=2ωϕ(x−ξω)Φ(α(x−ξω)).

The skewness (γ1) of the distribution is limited to slightly less than the interval (−1,1) (see Estimation).

As has been shown,[7] the mode (maximum) mo of the distribution is unique. For general α there is no analytic expression for mo, but a quite accurate (numerical) approximation is:

δ=α1+α2mo(α)≈2πδ−(1−π4)(2πδ)31−2πδ2−sgn(α)2e(−2π|α|)

## Estimation

Maximum likelihood estimates for ξ, ω, and α can be computed numerically, but no closed-form expression for the estimates is available unless α=0. In contrast, the method of moments has a closed-form expression since the skewness equation can be inverted with

|δ|=π2|γ1|23|γ1|23+((4−π)/2)23

where δ=α1+α2 and the sign of δ is the same as the sign of γ1. Consequently, α=δ1−δ2, ω=σ1−2δ2/π, and ξ=μ−ωδ2π where μ and σ are the mean and standard deviation. As long as the sample skewness γ^1 is not too large, these formulas provide method of moments estimates α^, ω^, and ξ^ based on a sample's μ^, σ^, and γ^1.

The maximum (theoretical) skewness is obtained by setting δ=1 in the skewness equation, giving γ1≈0.9952717. However it is possible that the sample skewness is larger, and then α cannot be determined from these equations. When using the method of moments in an automatic fashion, for example to give starting values for maximum likelihood iteration, one should therefore let (for example) |γ^1|=min⁡(0.99,|(1/n)∑((xi−μ^)/σ^)3|).

Concern has been expressed about the inference of skew normal distributions using the direct parameterization.[8]

## Related distributions

The exponentially modified normal distribution is another 3-parameter distribution that is a generalization of the normal distribution to skewed cases. The skew normal still has a normal-like tail in the direction of the skew, with a shorter tail in the other direction; that is, its density is asymptotically proportional to e−kx2 for some positive k. Thus, in terms of the seven states of randomness, it shows "proper mild randomness". In contrast, the exponentially modified normal has an exponential tail in the direction of the skew; its density is asymptotically proportional to e−k|x|. In the same terms, it shows "borderline mild randomness".

Thus, the skew normal is useful for modeling skewed distributions which nevertheless have no more outliers than the normal, while the exponentially modified normal is useful for cases with an increased incidence of outliers in (just) one direction.

## See also

* Generalized normal distribution
* Log-normal distribution

## References

1. ↑ O'Hagan, A.; Leonard, Tom (1976). "Bayes estimation subject to uncertainty about parameter constraints". Biometrika 63 (1): 201–203. doi:10.1093/biomet/63.1.201. ISSN 0006-3444.
2. ↑ Ashour, Samir K.; Abdel-hameed, Mahmood A. (October 2010). "Approximate skew normal distribution". Journal of Advanced Research 1 (4): 341–350. doi:10.1016/j.jare.2010.06.004. ISSN 2090-1232.
3. ↑ Mudholkar, Govind S.; Hutson, Alan D. (February 2000). "The epsilon–skew–normal distribution for analyzing near-normal data". Journal of Statistical Planning and Inference 83 (2): 291–309. doi:10.1016/s0378-3758(99)00096-8. ISSN 0378-3758.
4. ↑ Andel, J., Netuka, I. and Zvara, K. (1984) On threshold autoregressive processes. Kybernetika, 20, 89-106
5. ↑ Chan, K. S.; Tong, H. (March 1986). "A note on certain integral equations associated with non-linear time series analysis". Probability Theory and Related Fields 73 (1): 153–158. doi:10.1007/bf01845999. ISSN 0178-8051.
6. ↑ Azzalini, A. (1985). "A class of distributions which includes the normal ones". Scandinavian Journal of Statistics 12: 171–178.
7. ↑ Azzalini, Adelchi; Capitanio, Antonella (2014). The skew-normal and related families. pp. 32–33. ISBN 978-1-107-02927-9.
8. ↑ Pewsey, Arthur (2000). "Problems of inference for Azzalini's skewnormal distribution". Journal of Applied Statistics 27 (7): 859–870. doi:10.1080/02664760050120542. Bibcode: 2000JApSt..27..859P.

## External links

* The multi-variate skew-normal distribution with an application to body mass, height and Body Mass Index
* A very brief introduction to the skew-normal distribution
* The Skew-Normal Probability Distribution (and related distributions, such as the skew-t)
* OWENS: Owen's T Function
* Closed-skew Distributions - Simulation, Inversion and Parameter Estimation

* v
* t
* e

Statistics

* Outline
* Index

Descriptive statistics

| Continuous data|

| Center|

* Mean
* arithmetic
* geometric
* harmonic
* Median
* Mode

|

Dispersion|

* Variance
* Standard deviation
* Coefficient of variation
* Percentile
* Range
* Interquartile range

Shape|

* Central limit theorem
* Moments
* Skewness
* Kurtosis
* L-moments

Count data|

* Index of dispersion

Summary tables|

* Grouped data
* Frequency distribution
* Contingency table

Dependence|

* Pearson product-moment correlation
* Rank correlation
* Spearman's rho
* Kendall's tau
* Partial correlation
* Scatter plot

Graphics|

* Bar chart
* Biplot
* Box plot
* Control chart
* Correlogram
* Fan chart
* Forest plot
* Histogram
* Pie chart
* Q–Q plot
* Run chart
* Scatter plot
* Stem-and-leaf display
* Radar chart
* Violin plot

Data collection

| Study design|

* Population
* Statistic
* Effect size
* Statistical power
* Optimal design
* Sample size determination
* Replication
* Missing data

|

Survey methodology|

* Sampling
* stratified
* cluster
* Standard error
* Opinion poll
* Questionnaire

Controlled experiments|

* Scientific control
* Randomized experiment
* Randomized controlled trial
* Random assignment
* Blocking
* Interaction
* Factorial experiment

Adaptive Designs|

* Adaptive clinical trial
* Up-and-Down Designs
* Stochastic approximation

Observational Studies|

* Cross-sectional study
* Cohort study
* Natural experiment
* Quasi-experiment

Statistical inference

| Statistical theory|

* Population
* Statistic
* Probability distribution
* Sampling distribution
* Order statistic
* Empirical distribution
* Density estimation
* Statistical model
* Model specification
* Lp space
* Parameter
* location
* scale
* shape
* Parametric family
* Likelihood (monotone)
* Location–scale family
* Exponential family
* Completeness
* Sufficiency
* Statistical functional
* Bootstrap
* U
* V
* Optimal decision
* loss function
* Efficiency
* Statistical distance
* divergence
* Asymptotics
* Robustness

|

Frequentist inference|

| Point estimation|

* Estimating equations
* Maximum likelihood
* Method of moments
* M-estimator
* Minimum distance
* Unbiased estimators
* Mean-unbiased minimum-variance
* Rao–Blackwellization
* Lehmann–Scheffé theorem
* Median unbiased
* Plug-in

|

Interval estimation|

* Confidence interval
* Pivot
* Likelihood interval
* Prediction interval
* Tolerance interval
* Resampling
* Bootstrap
* Jackknife

Testing hypotheses|

* 1- & 2-tails
* Power
* Uniformly most powerful test
* Permutation test
* Randomization test
* Multiple comparisons

Parametric tests|

* Likelihood-ratio
* Score/Lagrange multiplier
* Wald

Specific tests|

* Z-test (normal)
* Student's t-test
* F-test

Goodness of fit|

* Chi-squared
* G-test
* Kolmogorov–Smirnov
* Anderson–Darling
* Lilliefors
* Jarque–Bera
* Normality (Shapiro–Wilk)
* Likelihood-ratio test
* Model selection
* Cross validation
* AIC
* BIC

Rank statistics|

* Sign
* Sample median
* Signed rank (Wilcoxon)
* Hodges–Lehmann estimator
* Rank sum (Mann–Whitney)
* Nonparametric anova
* 1-way (Kruskal–Wallis)
* 2-way (Friedman)
* Ordered alternative (Jonckheere–Terpstra)

Bayesian inference|

* Bayesian probability
* prior
* posterior
* Credible interval
* Bayes factor
* Bayesian estimator
* Maximum posterior estimator

* Correlation
* Regression analysis

| Correlation|

* Pearson product-moment
* Partial correlation
* Confounding variable
* Coefficient of determination

|

Regression analysis|

* Errors and residuals
* Regression validation
* Mixed effects models
* Simultaneous equations models
* Multivariate adaptive regression splines (MARS)

Linear regression|

* Simple linear regression
* Ordinary least squares
* General linear model
* Bayesian regression

Non-standard predictors|

* Nonlinear regression
* Nonparametric
* Semiparametric
* Isotonic
* Robust
* Heteroscedasticity
* Homoscedasticity

Generalized linear model|

* Exponential families
* Logistic (Bernoulli) / Binomial / Poisson regressions

Partition of variance|

* Analysis of variance (ANOVA, anova)
* Analysis of covariance
* Multivariate ANOVA
* Degrees of freedom

Categorical / Multivariate / Time-series / Survival analysis

| Categorical|

* Cohen's kappa
* Contingency table
* Graphical model
* Log-linear model
* McNemar's test

|

Multivariate|

* Regression
* Manova
* Principal components
* Canonical correlation
* Discriminant analysis
* Cluster analysis
* Classification
* Structural equation model
* Factor analysis
* Multivariate distributions
* Elliptical distributions
* Normal

Time-series|

| General|

* Decomposition
* Trend
* Stationarity
* Seasonal adjustment
* Exponential smoothing
* Cointegration
* Structural break
* Granger causality

|

Specific tests|

* Dickey–Fuller
* Johansen
* Q-statistic (Ljung–Box)
* Durbin–Watson
* Breusch–Godfrey

Time domain|

* Autocorrelation (ACF)
* partial (PACF)
* Cross-correlation (XCF)
* ARMA model
* ARIMA model (Box–Jenkins)
* Autoregressive conditional heteroskedasticity (ARCH)
* Vector autoregression (VAR)

Frequency domain|

* Spectral density estimation
* Fourier analysis
* Wavelet
* Whittle likelihood

Survival|

| Survival function|

* Kaplan–Meier estimator (product limit)
* Proportional hazards models
* Accelerated failure time (AFT) model
* First hitting time

|

Hazard function|

* Nelson–Aalen estimator

Test|

* Log-rank test

Applications

| Biostatistics|

* Bioinformatics
* Clinical trials / studies
* Epidemiology
* Medical statistics

|

Engineering statistics|

* Chemometrics
* Methods engineering
* Probabilistic design
* Process / quality control
* Reliability
* System identification

Social statistics|

* Actuarial science
* Census
* Crime statistics
* Demography
* Econometrics
* Jurimetrics
* National accounts
* Official statistics
* Population statistics
* Psychometrics

Spatial statistics|

* Cartography
* Environmental statistics
* Geographic information system
* Geostatistics
* Kriging

* Category
* '
* Commons
* WikiProject

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