In probability theory and statistics, the split normal distribution also known as the two-piece normal distribution results from joining at the mode the corresponding halves of two normal distributions with the same mode but different variances. It is claimed by Johnson et al. that this distribution was introduced by Gibbons and Mylroie and by John. But these are two of several independent rediscoveries of the Zweiseitige Gauss'sche Gesetz introduced in the posthumously published Kollektivmasslehre (1897) of Gustav Theodor Fechner (1801-1887), see Wallis (2014). Another rediscovery has appeared more recently in a finance journal.
Definition The split normal distribution arises from merging two opposite halves of two probability density functions (PDFs) of normal distributions in their common mode. The PDF of the split normal distribution is given by
f ( x ; μ , σ 1 , σ 2 ) = { A exp ( − ( x − μ ) 2 2 σ 1 2 ) if x < μ A exp ( − ( x − μ ) 2 2 σ 2 2 ) otherwise {\displaystyle f(x;\mu ,\sigma _{1},\sigma _{2})={\begin{cases}A\exp \left(-{\dfrac {(x-\mu )^{2}}{2\sigma _{1}^{2}}}\right)&{\text{if }}x<\mu \\[1ex]A\exp \left(-{\dfrac {(x-\mu )^{2}}{2\sigma _{2}^{2}}}\right)&{\text{otherwise}}\end{cases}}}
where
A = 2 / π ( σ 1 + σ 2 ) − 1 . {\displaystyle \quad A={\sqrt {2/\pi }}(\sigma _{1}+\sigma _{2})^{-1}.}
Discussion The split normal distribution results from merging two halves of normal distributions. In a general case the 'parent' normal distributions can have different variances which implies that the joined PDF would not be continuous. To ensure that the resulting PDF integrates to 1, the normalizing constant A is used. In a special case when σ 1 2 = σ 2 2 = σ ∗ 2 {\displaystyle \sigma _{1}^{2}=\sigma _{2}^{2}=\sigma _{*}^{2}} the split normal distribution reduces to normal distribution with variance σ ∗ 2 {\displaystyle \sigma _{*}^{2}} . When σ2≠σ1 the constant A is different from the constant of normal distribution. However, when σ 1 2 = σ 2 2 = σ ∗ 2 {\displaystyle \sigma _{1}^{2}=\sigma _{2}^{2}=\sigma _{*}^{2}} the constants are equal. The sign of its third central moment is determined by the difference (σ2-σ1). If this difference is positive, the distribution is skewed to the right and if negative, then it is skewed to the left. Other properties of the split normal density were discussed by Johnson et al. and Julio.
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