Split normal distribution

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 (1897¹) of Gustav Theodor Fechner (1801-1887), see Wallis (2014). Another rediscovery has appeared more recently in a finance journal. {{Probability distribution | name = Split-normal | type = density | notation = \mathcal{SN}(\mu,\,\sigma_1,\sigma_2) | parameters = \mu \in \Re — mode (location, real) \sigma_1 > 0 — left-hand-side standard deviation (scale, real) \sigma_2 > 0 — right-hand-side standard deviation (scale, real) | support = x \in \Re | | pdf = A \exp \left(- \frac {(x-\mu)^2}{2 \sigma_1^2}\right) \quad \text{if } x A \exp \left(- \frac {(x-\mu)^2}{2 \sigma_2^2}\right) \quad \text{otherwise,} \text{where} \quad A= \sqrt{2/\pi} (\sigma_1+\sigma_2)^{-1} | mode = \mu | mean = \mu+\sqrt{2 / \pi}(\sigma_2-\sigma_1) | variance = (1-2 / \pi)(\sigma_2-\sigma_1)^2 + \sigma_1 \sigma_2 | skewness = \gamma_3 = \sqrt{\frac{2}{\pi}}(\sigma_2-\sigma_1)\left[\left(\frac{4}{\pi}-1\right)(…

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. {{Probability distribution | name = Split-normal | type = density | notation = \mathcal{SN}(\mu,\,\sigma_1,\sigma_2) | parameters = \mu \in \Re — mode (location, real) \sigma_1 > 0 — left-hand-side standard deviation (scale, real) \sigma_2 > 0 — right-hand-side standard deviation (scale, real) | support = x \in \Re | | pdf = A \exp \left(- \frac {(x-\mu)^2}{2 \sigma_1^2}\right) \quad \text{if } x A \exp \left(- \frac {(x-\mu)^2}{2 \sigma_2^2}\right) \quad \text{otherwise,} \text{where} \quad A= \sqrt{2/\pi} (\sigma_1+\sigma_2)^{-1} | mode = \mu | mean = \mu+\sqrt{2 / \pi}(\sigma_2-\sigma_1) | variance = (1-2 / \pi)(\sigma_2-\sigma_1)^2 + \sigma_1 \sigma_2 | skewness = \gamma_3 = \sqrt{\frac{2}{\pi}}(\sigma_2-\sigma_1)\left[\left(\frac{4}{\pi}-1\right)(\sigma_2-\sigma_1)^2 + \sigma_1 \sigma_2\right] }} f(x;\mu,\sigma_1,\sigma_2) = \begin{cases} A \exp \left(- \dfrac {(x-\mu)^2}{2 \sigma_1^2}\right) & \text{if } x where \quad A= \sqrt{2/\pi} (\sigma_1+\sigma_2)^{-1}. In a special case when \sigma_1^2=\sigma_2^2=\sigma_{*}^2 the split normal distribution reduces to normal distribution with variance \sigma_{*}^2 . When σ 2 ≠σ 1 the constant A is different from the constant of normal distribution. However, when \sigma_1^2=\sigma_2^2=\sigma_{*}^2 the constants are equal. The sign of its third central moment is determined by the difference (σ 2 -σ 1 ). 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. \sigma^2 &= \sigma_1^2(1+\gamma)= \sigma_2^2(1-\gamma) \\ \gamma &= \frac{\sigma_2^2-\sigma_1^2}{\sigma_2^2+\sigma_1^2} \\ \xi &=\sqrt{2 / \pi}(\sigma_2-\sigma_1) \\ \gamma &= \operatorname{sgn}(\xi) \sqrt{1-\left( \frac{\sqrt{1+2\beta}-1}{\beta} \right)^2}, \quad \text{where} \quad \beta = \frac{\pi\xi^2}{2\sigma^2}. \end{align} L(\mu) = -\left[\sum_{x_i: x_i \mu} (x_i-\mu)^2 \right]^{1/3} and has to be maximized numerically with respect to a single parameter μ only. Given the maximum likelihood estimator \hat{\mu} the other parameters take values: \hat{\sigma}_1^2 = \frac{-L(\mu)}{N} \left[\sum_{x_i: x_i \hat{\sigma}_2^2 = \frac{-L(\mu)}{N} \left[\sum_{x_i: x_i>\mu} (x_i-\mu)^2 \right]^{2/3}, where N is the number of observations. Villani and Larsson propose to use either maximum likelihood method or bayesian estimation and provide some analytical results for either univariate and multivariate case.

Text derived from https://en.wikipedia.org/w/index.php?oldid=1374403628 under CC BY-SA 4.0; modifications have been made.