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In probability theory
Probability theory

Probability theory is the branch of mathematics concerned with analysis of Statistical randomness phenomena. The central objects of probability theory are random variables, stochastic processes, and event s: mathematical abstractions of determinism events or measured quantities that may either be single occurrences or evolve over time in an a...
 and statistics
Statistics

Statistics is a Mathematics pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It also provides tools for prediction and forecasting based on data....
, the skew normal distribution is a continuous probability distribution
Continuous probability distribution

In probability theory, a probability distribution is called continuous if its cumulative distribution function is continuous function. This is equivalent to saying that for random variables X with the distribution in question, Pr[X = a] = 0 for all real numbers a, i.e.: the probability that X attains the value a is zer...
 that generalises the normal distribution
Normal distribution

The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields....
 to allow for non-zero skewness
Skewness

In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real number-valued random variable....
.

the cumulative distribution function
Cumulative distribution function

In probability theory and statistics, the cumulative distribution function or just distribution function, completely describes the probability distribution of a real-valued random variable X....
 given by

Then the probability density function of the skew-normal distribution with parameter α is given by

To add location
Location parameter

In statistics, a location family is a class of probability distributions parametrized by a scalar- or vector-valued parameter ?, which determines the "location" or shift of the distribution....
 and scale
Scale parameter

In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions....
 parameters to this, one makes the usual transform . One can verify that the normal distribution is recovered when , and that the absolute value of the skewness increases as the absolute value of increases.






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In probability theory
Probability theory

Probability theory is the branch of mathematics concerned with analysis of Statistical randomness phenomena. The central objects of probability theory are random variables, stochastic processes, and event s: mathematical abstractions of determinism events or measured quantities that may either be single occurrences or evolve over time in an a...
 and statistics
Statistics

Statistics is a Mathematics pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It also provides tools for prediction and forecasting based on data....
, the skew normal distribution is a continuous probability distribution
Continuous probability distribution

In probability theory, a probability distribution is called continuous if its cumulative distribution function is continuous function. This is equivalent to saying that for random variables X with the distribution in question, Pr[X = a] = 0 for all real numbers a, i.e.: the probability that X attains the value a is zer...
 that generalises the normal distribution
Normal distribution

The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields....
 to allow for non-zero skewness
Skewness

In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real number-valued random variable....
.

Definition


Let denote the standard normal probability density function
Probability density function

In mathematics, a probability density function is a function that represents a probability distribution in terms of integrals.Formally, a probability distribution has density ƒ, if ƒ is a non-negative Lebesgue integration function such that the probability of the interval [a, b] is given by...
 with the cumulative distribution function
Cumulative distribution function

In probability theory and statistics, the cumulative distribution function or just distribution function, completely describes the probability distribution of a real-valued random variable X....
 given by

Then the probability density function of the skew-normal distribution with parameter α is given by

To add location
Location parameter

In statistics, a location family is a class of probability distributions parametrized by a scalar- or vector-valued parameter ?, which determines the "location" or shift of the distribution....
 and scale
Scale parameter

In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions....
 parameters to this, one makes the usual transform . One can verify that the normal distribution is recovered when , and that the absolute value of the skewness increases as the absolute value of increases. The distribution is right skewed if and is left skewed if .

Estimation


Maximum likelihood
Maximum likelihood

Maximum likelihood estimation is a popular statistics method used for fitting a mathematical model to data. The modeling of real world data using estimation by maximum likelihood offers a way of tuning the free parameters of the model to provide a good fit....
 estimates for , , and can be computed numerically, but no closed-form expression for the estimates is available unless . If a closed-form expression is needed, the method of moments
Method of moments (statistics)

In statistics, the method of moments is a method of estimation of population parameters such as mean, variance, median, etc. , by equating sample moment with unobservable population moments and then solving those equations for the quantities to be estimated....
 can be applied to estimate from the sample skew, by inverting the skewness equation. This yields the estimate

where , and is the sample skew. The sign of is the same as the sign of . Consequently, .

See also

  • Normal distribution
    Normal distribution

    The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields....
  • Generalized normal distribution
    Generalized normal distribution

    The generalized normal distribution or generalized Gaussian distribution can refer to either of two families of parametric statistics continuous probability distributions on the real number line....
  • Lognormal distribution
  • Skewness
    Skewness

    In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real number-valued random variable....


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