probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

and statistics, the normal-exponential-gamma distribution (sometimes called the NEG distribution) is a three-parameter family of continuous

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...

s. It has a

location parameter In geography, location or place are used to denote a region (point, line, or area) on Earth's surface or elsewhere. The term ''location'' generally implies a higher degree of certainty than ''place'', the latter often indicating an entity with an ...

\mu

scale parameter In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution. Definition If a family o ...

\theta

and a

shape parameter In probability theory and statistics, a shape parameter (also known as form parameter) is a kind of numerical parameter of a parametric family of probability distributionsEveritt B.S. (2002) Cambridge Dictionary of Statistics. 2nd Edition. CUP. t ...

k

Probability density function

The

probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...

(pdf) of the normal-exponential-gamma distribution is proportional to :

f(x;\mu, k,\theta) \propto \expD_\left(\frac\right)

, where ''D'' is a

parabolic cylinder function In mathematics, the parabolic cylinder functions are special functions defined as solutions to the differential equation This equation is found when the technique of separation of variables is used on Laplace's equation when expressed in parabo ...

.http://www.newton.ac.uk/programmes/SCB/seminars/121416154.html As for the

Laplace distribution In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two expo ...

, the pdf of the NEG distribution can be expressed as a mixture of

normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu i ...

s, :

f(x;\mu, k,\theta)=\int_0^\infty\int_0^\infty\ \mathrm(x,  \mu, \sigma^2)\mathrm(\sigma^2, \psi)\mathrm(\psi, k, 1/\theta^2) \, d\sigma^2 \, d\psi,

where, in this notation, the distribution-names should be interpreted as meaning the density functions of those distributions. Within this scale mixture, the scale's ''mixing distribution'' (an exponential with a gamma-distributed rate) actually is a

Lomax distribution The Lomax distribution, conditionally also called the Pareto Type II distribution, is a heavy-tail probability distribution used in business, economics, actuarial science, queueing theory and Internet traffic modeling. It is named after K. ...

Applications

The distribution has heavy tails and a sharp peak at

\mu

and, because of this, it has applications in variable selection.

References

Continuous distributions Compound probability distributions {{Statistics-stub

Probability density function

Applications

See also

References